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

Percentage Accurate: 90.4% → 97.3%

Time: 4.5s

Alternatives: 13

Speedup: 0.8×

• 46.1% 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: 90.4% 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: 97.3% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 94.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 98.4

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

   3. Applied rewrites 98.4%

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

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

   1. Initial program 0.0%

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

   2. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 37.7

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

   4. Applied rewrites 37.7%

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

   5. Taylor expanded in x around -inf

      \[\leadsto 2 \cdot \left(-1 \cdot \color{blue}{\left(x \cdot \left(-1 \cdot y + \frac{c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)}{x}\right)\right)}\right) \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto 2 \cdot \left(\left(-1 \cdot x\right) \cdot \left(-1 \cdot y + \color{blue}{\frac{c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)}{x}}\right)\right) \]

      2. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(\left(-1 \cdot x\right) \cdot \left(-1 \cdot y + \color{blue}{\frac{c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)}{x}}\right)\right) \]

      3. mul-1-neg N/A

         \[\leadsto 2 \cdot \left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(-1 \cdot y + \frac{\color{blue}{c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)}}{x}\right)\right) \]

      4. lower-neg.f64 N/A

         \[\leadsto 2 \cdot \left(\left(-x\right) \cdot \left(-1 \cdot y + \frac{\color{blue}{c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)}}{x}\right)\right) \]

      5. +-commutative N/A

         \[\leadsto 2 \cdot \left(\left(-x\right) \cdot \left(\frac{c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)}{x} + -1 \cdot \color{blue}{y}\right)\right) \]

      6. associate-/l* N/A

         \[\leadsto 2 \cdot \left(\left(-x\right) \cdot \left(c \cdot \frac{i \cdot \left(a + b \cdot c\right)}{x} + -1 \cdot y\right)\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto 2 \cdot \left(\left(-x\right) \cdot \mathsf{fma}\left(c, \frac{i \cdot \left(a + b \cdot c\right)}{\color{blue}{x}}, -1 \cdot y\right)\right) \]

      8. *-commutative N/A

         \[\leadsto 2 \cdot \left(\left(-x\right) \cdot \mathsf{fma}\left(c, \frac{i \cdot \left(a + c \cdot b\right)}{x}, -1 \cdot y\right)\right) \]

      9. +-commutative N/A

         \[\leadsto 2 \cdot \left(\left(-x\right) \cdot \mathsf{fma}\left(c, \frac{i \cdot \left(c \cdot b + a\right)}{x}, -1 \cdot y\right)\right) \]

      10. *-commutative N/A

          \[\leadsto 2 \cdot \left(\left(-x\right) \cdot \mathsf{fma}\left(c, \frac{\left(c \cdot b + a\right) \cdot i}{x}, -1 \cdot y\right)\right) \]

      11. lower-/.f64 N/A

          \[\leadsto 2 \cdot \left(\left(-x\right) \cdot \mathsf{fma}\left(c, \frac{\left(c \cdot b + a\right) \cdot i}{x}, -1 \cdot y\right)\right) \]

      12. lower-*.f64 N/A

          \[\leadsto 2 \cdot \left(\left(-x\right) \cdot \mathsf{fma}\left(c, \frac{\left(c \cdot b + a\right) \cdot i}{x}, -1 \cdot y\right)\right) \]

      13. *-commutative N/A

          \[\leadsto 2 \cdot \left(\left(-x\right) \cdot \mathsf{fma}\left(c, \frac{\left(b \cdot c + a\right) \cdot i}{x}, -1 \cdot y\right)\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto 2 \cdot \left(\left(-x\right) \cdot \mathsf{fma}\left(c, \frac{\mathsf{fma}\left(b, c, a\right) \cdot i}{x}, -1 \cdot y\right)\right) \]

      15. mul-1-neg N/A

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

      16. lower-neg.f64 72.9

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

   7. Applied rewrites 72.9%

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

Alternative 2: 96.1% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 94.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 98.4

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

   3. Applied rewrites 98.4%

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

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

   1. Initial program 0.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 11.9

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

   3. Applied rewrites 11.9%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. associate-*r* N/A

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

      15. associate--l+ N/A

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

      16. *-commutative N/A

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

      17. *-commutative N/A

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

      18. *-commutative N/A

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

      19. *-commutative N/A

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

      20. +-commutative N/A

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

      21. *-commutative N/A

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

   5. Applied rewrites 48.8%

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

Alternative 3: 88.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+299}:\\ \;\;\;\;2 \cdot \left(y \cdot x - \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right)\\ \mathbf{elif}\;t\_1 \leq 20000000000000:\\ \;\;\;\;2 \cdot \left(\mathsf{fma}\left(t, z, y \cdot x\right) - \left(i \cdot c\right) \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t \cdot z - \mathsf{fma}\left(c, b, a\right) \cdot \left(i \cdot c\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* (+ a (* b c)) c) i)))
   (if (<= t_1 -2e+299)
     (* 2.0 (- (* y x) (* (* (fma c b a) i) c)))
     (if (<= t_1 20000000000000.0)
       (* 2.0 (- (fma t z (* y x)) (* (* i c) a)))
       (* 2.0 (- (* t z) (* (fma c b a) (* i c))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = ((a + (b * c)) * c) * i;
	double tmp;
	if (t_1 <= -2e+299) {
		tmp = 2.0 * ((y * x) - ((fma(c, b, a) * i) * c));
	} else if (t_1 <= 20000000000000.0) {
		tmp = 2.0 * (fma(t, z, (y * x)) - ((i * c) * a));
	} else {
		tmp = 2.0 * ((t * z) - (fma(c, b, a) * (i * c)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(a + Float64(b * c)) * c) * i)
	tmp = 0.0
	if (t_1 <= -2e+299)
		tmp = Float64(2.0 * Float64(Float64(y * x) - Float64(Float64(fma(c, b, a) * i) * c)));
	elseif (t_1 <= 20000000000000.0)
		tmp = Float64(2.0 * Float64(fma(t, z, Float64(y * x)) - Float64(Float64(i * c) * a)));
	else
		tmp = Float64(2.0 * Float64(Float64(t * z) - Float64(fma(c, b, a) * Float64(i * c))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * i), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+299], N[(2.0 * N[(N[(y * x), $MachinePrecision] - N[(N[(N[(c * b + a), $MachinePrecision] * i), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 20000000000000.0], N[(2.0 * N[(N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision] - N[(N[(i * c), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(t * z), $MachinePrecision] - N[(N[(c * b + a), $MachinePrecision] * N[(i * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -2.0000000000000001e299

   1. Initial program 75.1%

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

   2. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 88.3

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

   4. Applied rewrites 88.3%

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

   if -2.0000000000000001e299 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 2e13

   1. Initial program 99.1%

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

   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. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 91.8

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

   4. Applied rewrites 91.8%

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

   if 2e13 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 85.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 91.6

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

   3. Applied rewrites 91.6%

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

   4. Taylor expanded in x around 0

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

      1. lift-*.f64 83.1

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

   6. Applied rewrites 83.1%

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

Alternative 4: 88.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 \cdot \left(y \cdot x - \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right)\\ t_2 := \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+299}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+260}:\\ \;\;\;\;2 \cdot \left(\mathsf{fma}\left(t, z, y \cdot x\right) - \left(i \cdot c\right) \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* 2.0 (- (* y x) (* (* (fma c b a) i) c))))
        (t_2 (* (* (+ a (* b c)) c) i)))
   (if (<= t_2 -2e+299)
     t_1
     (if (<= t_2 2e+260) (* 2.0 (- (fma t z (* y x)) (* (* i c) a))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = 2.0 * ((y * x) - ((fma(c, b, a) * i) * c));
	double t_2 = ((a + (b * c)) * c) * i;
	double tmp;
	if (t_2 <= -2e+299) {
		tmp = t_1;
	} else if (t_2 <= 2e+260) {
		tmp = 2.0 * (fma(t, z, (y * x)) - ((i * c) * a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(2.0 * Float64(Float64(y * x) - Float64(Float64(fma(c, b, a) * i) * c)))
	t_2 = Float64(Float64(Float64(a + Float64(b * c)) * c) * i)
	tmp = 0.0
	if (t_2 <= -2e+299)
		tmp = t_1;
	elseif (t_2 <= 2e+260)
		tmp = Float64(2.0 * Float64(fma(t, z, Float64(y * x)) - Float64(Float64(i * c) * a)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(2.0 * N[(N[(y * x), $MachinePrecision] - N[(N[(N[(c * b + a), $MachinePrecision] * i), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * i), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+299], t$95$1, If[LessEqual[t$95$2, 2e+260], N[(2.0 * N[(N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision] - N[(N[(i * c), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -2.0000000000000001e299 or 2.00000000000000013e260 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 76.5%

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

   2. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 86.5

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

   4. Applied rewrites 86.5%

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

   if -2.0000000000000001e299 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 2.00000000000000013e260

   1. Initial program 99.1%

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

   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. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 89.5

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

   4. Applied rewrites 89.5%

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

Alternative 5: 82.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+86}:\\ \;\;\;\;2 \cdot \left(y \cdot x - \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+138}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(t, z, y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(b, c, a\right) \cdot c\right) \cdot i\right) \cdot -2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* (+ a (* b c)) c) i)))
   (if (<= t_1 -1e+86)
     (* 2.0 (- (* y x) (* (* (fma c b a) i) c)))
     (if (<= t_1 5e+138)
       (* 2.0 (fma t z (* y x)))
       (* (* (* (fma b c a) c) i) -2.0)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = ((a + (b * c)) * c) * i;
	double tmp;
	if (t_1 <= -1e+86) {
		tmp = 2.0 * ((y * x) - ((fma(c, b, a) * i) * c));
	} else if (t_1 <= 5e+138) {
		tmp = 2.0 * fma(t, z, (y * x));
	} else {
		tmp = ((fma(b, c, a) * c) * i) * -2.0;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(a + Float64(b * c)) * c) * i)
	tmp = 0.0
	if (t_1 <= -1e+86)
		tmp = Float64(2.0 * Float64(Float64(y * x) - Float64(Float64(fma(c, b, a) * i) * c)));
	elseif (t_1 <= 5e+138)
		tmp = Float64(2.0 * fma(t, z, Float64(y * x)));
	else
		tmp = Float64(Float64(Float64(fma(b, c, a) * c) * i) * -2.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * i), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+86], N[(2.0 * N[(N[(y * x), $MachinePrecision] - N[(N[(N[(c * b + a), $MachinePrecision] * i), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+138], N[(2.0 * N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(b * c + a), $MachinePrecision] * c), $MachinePrecision] * i), $MachinePrecision] * -2.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -1e86

   1. Initial program 82.2%

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

   2. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 81.2

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

   4. Applied rewrites 81.2%

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

   if -1e86 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 5.00000000000000016e138

   1. Initial program 99.0%

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

   2. Taylor expanded in c around 0

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

      1. lower-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 85.9

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

   4. Applied rewrites 85.9%

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

   if 5.00000000000000016e138 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 81.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 89.7

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

   3. Applied rewrites 89.7%

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

   4. Taylor expanded in i around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 78.3

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

   6. Applied rewrites 78.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. +-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   8. Applied rewrites 78.3%

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

Alternative 6: 81.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+299}:\\ \;\;\;\;-2 \cdot \left(\left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+138}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(t, z, y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(b, c, a\right) \cdot c\right) \cdot i\right) \cdot -2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* (+ a (* b c)) c) i)))
   (if (<= t_1 -2e+299)
     (* -2.0 (* (* (fma c b a) i) c))
     (if (<= t_1 5e+138)
       (* 2.0 (fma t z (* y x)))
       (* (* (* (fma b c a) c) i) -2.0)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = ((a + (b * c)) * c) * i;
	double tmp;
	if (t_1 <= -2e+299) {
		tmp = -2.0 * ((fma(c, b, a) * i) * c);
	} else if (t_1 <= 5e+138) {
		tmp = 2.0 * fma(t, z, (y * x));
	} else {
		tmp = ((fma(b, c, a) * c) * i) * -2.0;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(a + Float64(b * c)) * c) * i)
	tmp = 0.0
	if (t_1 <= -2e+299)
		tmp = Float64(-2.0 * Float64(Float64(fma(c, b, a) * i) * c));
	elseif (t_1 <= 5e+138)
		tmp = Float64(2.0 * fma(t, z, Float64(y * x)));
	else
		tmp = Float64(Float64(Float64(fma(b, c, a) * c) * i) * -2.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * i), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+299], N[(-2.0 * N[(N[(N[(c * b + a), $MachinePrecision] * i), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+138], N[(2.0 * N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(b * c + a), $MachinePrecision] * c), $MachinePrecision] * i), $MachinePrecision] * -2.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -2.0000000000000001e299

   1. Initial program 75.1%

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

   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. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 88.5

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

   4. Applied rewrites 88.5%

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

   if -2.0000000000000001e299 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 5.00000000000000016e138

   1. Initial program 99.1%

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

   2. Taylor expanded in c around 0

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

      1. lower-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 80.3

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

   4. Applied rewrites 80.3%

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

   if 5.00000000000000016e138 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 81.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 89.7

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

   3. Applied rewrites 89.7%

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

   4. Taylor expanded in i around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 78.3

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

   6. Applied rewrites 78.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. +-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   8. Applied rewrites 78.3%

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

Alternative 7: 81.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -2 \cdot \left(\left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right)\\ t_2 := \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+299}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+138}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(t, z, y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* -2.0 (* (* (fma c b a) i) c))) (t_2 (* (* (+ a (* b c)) c) i)))
   (if (<= t_2 -2e+299)
     t_1
     (if (<= t_2 5e+138) (* 2.0 (fma t z (* y x))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = -2.0 * ((fma(c, b, a) * i) * c);
	double t_2 = ((a + (b * c)) * c) * i;
	double tmp;
	if (t_2 <= -2e+299) {
		tmp = t_1;
	} else if (t_2 <= 5e+138) {
		tmp = 2.0 * fma(t, z, (y * x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(-2.0 * Float64(Float64(fma(c, b, a) * i) * c))
	t_2 = Float64(Float64(Float64(a + Float64(b * c)) * c) * i)
	tmp = 0.0
	if (t_2 <= -2e+299)
		tmp = t_1;
	elseif (t_2 <= 5e+138)
		tmp = Float64(2.0 * fma(t, z, Float64(y * x)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(-2.0 * N[(N[(N[(c * b + a), $MachinePrecision] * i), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * i), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+299], t$95$1, If[LessEqual[t$95$2, 5e+138], N[(2.0 * N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -2.0000000000000001e299 or 5.00000000000000016e138 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 79.0%

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

   2. Taylor expanded in i around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 82.6

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

   4. Applied rewrites 82.6%

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

   if -2.0000000000000001e299 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 5.00000000000000016e138

   1. Initial program 99.1%

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

   2. Taylor expanded in c around 0

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

      1. lower-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 80.3

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

   4. Applied rewrites 80.3%

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

Alternative 8: 73.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\left(c \cdot c\right) \cdot i\right) \cdot b\right) \cdot -2\\ t_2 := \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+299}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+279}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(t, z, y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* (* (* c c) i) b) -2.0)) (t_2 (* (* (+ a (* b c)) c) i)))
   (if (<= t_2 -2e+299)
     t_1
     (if (<= t_2 5e+279) (* 2.0 (fma t z (* y x))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (((c * c) * i) * b) * -2.0;
	double t_2 = ((a + (b * c)) * c) * i;
	double tmp;
	if (t_2 <= -2e+299) {
		tmp = t_1;
	} else if (t_2 <= 5e+279) {
		tmp = 2.0 * fma(t, z, (y * x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(Float64(c * c) * i) * b) * -2.0)
	t_2 = Float64(Float64(Float64(a + Float64(b * c)) * c) * i)
	tmp = 0.0
	if (t_2 <= -2e+299)
		tmp = t_1;
	elseif (t_2 <= 5e+279)
		tmp = Float64(2.0 * fma(t, z, Float64(y * x)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(N[(c * c), $MachinePrecision] * i), $MachinePrecision] * b), $MachinePrecision] * -2.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * i), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+299], t$95$1, If[LessEqual[t$95$2, 5e+279], N[(2.0 * N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -2.0000000000000001e299 or 5.0000000000000002e279 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 76.1%

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

   2. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 68.1

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

   4. Applied rewrites 68.1%

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

   if -2.0000000000000001e299 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 5.0000000000000002e279

   1. Initial program 99.1%

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

   2. Taylor expanded in c around 0

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

      1. lower-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 76.6

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

   4. Applied rewrites 76.6%

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

Alternative 9: 57.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 \cdot \mathsf{fma}\left(t, z, y \cdot x\right)\\ t_2 := x \cdot y + z \cdot t\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+48}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{-22}:\\ \;\;\;\;\left(\left(i \cdot c\right) \cdot a\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* 2.0 (fma t z (* y x)))) (t_2 (+ (* x y) (* z t))))
   (if (<= t_2 -5e+48) t_1 (if (<= t_2 1e-22) (* (* (* i c) a) -2.0) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = 2.0 * fma(t, z, (y * x));
	double t_2 = (x * y) + (z * t);
	double tmp;
	if (t_2 <= -5e+48) {
		tmp = t_1;
	} else if (t_2 <= 1e-22) {
		tmp = ((i * c) * a) * -2.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(2.0 * fma(t, z, Float64(y * x)))
	t_2 = Float64(Float64(x * y) + Float64(z * t))
	tmp = 0.0
	if (t_2 <= -5e+48)
		tmp = t_1;
	elseif (t_2 <= 1e-22)
		tmp = Float64(Float64(Float64(i * c) * a) * -2.0);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(2.0 * N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * y), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+48], t$95$1, If[LessEqual[t$95$2, 1e-22], N[(N[(N[(i * c), $MachinePrecision] * a), $MachinePrecision] * -2.0), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 x y) (*.f64 z t)) < -4.99999999999999973e48 or 1e-22 < (+.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 89.2%

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

   2. Taylor expanded in c around 0

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

      1. lower-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 65.2

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

   4. Applied rewrites 65.2%

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

   if -4.99999999999999973e48 < (+.f64 (*.f64 x y) (*.f64 z t)) < 1e-22

   1. Initial program 93.7%

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

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 37.0

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

   4. Applied rewrites 37.0%

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

Alternative 10: 94.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 \cdot \mathsf{fma}\left(z, t, \mathsf{fma}\left(-c, \mathsf{fma}\left(b, c, a\right) \cdot i, y \cdot x\right)\right)\\ \mathbf{if}\;c \leq -8.6 \cdot 10^{-106}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;c \leq 10^{-183}:\\ \;\;\;\;2 \cdot \left(\mathsf{fma}\left(t, z, y \cdot x\right) - \left(i \cdot c\right) \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* 2.0 (fma z t (fma (- c) (* (fma b c a) i) (* y x))))))
   (if (<= c -8.6e-106)
     t_1
     (if (<= c 1e-183) (* 2.0 (- (fma t z (* y x)) (* (* i c) a))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = 2.0 * fma(z, t, fma(-c, (fma(b, c, a) * i), (y * x)));
	double tmp;
	if (c <= -8.6e-106) {
		tmp = t_1;
	} else if (c <= 1e-183) {
		tmp = 2.0 * (fma(t, z, (y * x)) - ((i * c) * a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(2.0 * fma(z, t, fma(Float64(-c), Float64(fma(b, c, a) * i), Float64(y * x))))
	tmp = 0.0
	if (c <= -8.6e-106)
		tmp = t_1;
	elseif (c <= 1e-183)
		tmp = Float64(2.0 * Float64(fma(t, z, Float64(y * x)) - Float64(Float64(i * c) * a)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(2.0 * N[(z * t + N[((-c) * N[(N[(b * c + a), $MachinePrecision] * i), $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -8.6e-106], t$95$1, If[LessEqual[c, 1e-183], N[(2.0 * N[(N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision] - N[(N[(i * c), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if c < -8.6000000000000004e-106 or 1.00000000000000001e-183 < c

   1. Initial program 87.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 93.1

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

   3. Applied rewrites 93.1%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. associate-*r* N/A

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

      15. associate--l+ N/A

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

      16. *-commutative N/A

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

      17. *-commutative N/A

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

      18. *-commutative N/A

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

      19. *-commutative N/A

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

      20. +-commutative N/A

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

      21. *-commutative N/A

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

   5. Applied rewrites 93.7%

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

   if -8.6000000000000004e-106 < c < 1.00000000000000001e-183

   1. Initial program 98.5%

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

   2. Taylor expanded in b around 0

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

      1. lower--.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 96.2

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

   4. Applied rewrites 96.2%

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

Alternative 11: 93.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t, z, y \cdot x\right)\\ t_2 := 2 \cdot \mathsf{fma}\left(-c, \mathsf{fma}\left(c, b, a\right) \cdot i, t\_1\right)\\ \mathbf{if}\;c \leq -8.6 \cdot 10^{-106}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;c \leq 10^{-183}:\\ \;\;\;\;2 \cdot \left(t\_1 - \left(i \cdot c\right) \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma t z (* y x)))
        (t_2 (* 2.0 (fma (- c) (* (fma c b a) i) t_1))))
   (if (<= c -8.6e-106)
     t_2
     (if (<= c 1e-183) (* 2.0 (- t_1 (* (* i c) a))) t_2))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(t, z, (y * x));
	double t_2 = 2.0 * fma(-c, (fma(c, b, a) * i), t_1);
	double tmp;
	if (c <= -8.6e-106) {
		tmp = t_2;
	} else if (c <= 1e-183) {
		tmp = 2.0 * (t_1 - ((i * c) * a));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(t, z, Float64(y * x))
	t_2 = Float64(2.0 * fma(Float64(-c), Float64(fma(c, b, a) * i), t_1))
	tmp = 0.0
	if (c <= -8.6e-106)
		tmp = t_2;
	elseif (c <= 1e-183)
		tmp = Float64(2.0 * Float64(t_1 - Float64(Float64(i * c) * a)));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 * N[((-c) * N[(N[(c * b + a), $MachinePrecision] * i), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -8.6e-106], t$95$2, If[LessEqual[c, 1e-183], N[(2.0 * N[(t$95$1 - N[(N[(i * c), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if c < -8.6000000000000004e-106 or 1.00000000000000001e-183 < c

   1. Initial program 87.4%

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

   2. Taylor expanded in x around 0

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

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

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

      2. mul-1-neg N/A

         \[\leadsto 2 \cdot \left(\left(t \cdot z + x \cdot y\right) + \left(-1 \cdot c\right) \cdot \left(\color{blue}{i} \cdot \left(a + b \cdot c\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto 2 \cdot \left(\left(t \cdot z + x \cdot y\right) + -1 \cdot \color{blue}{\left(c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto 2 \cdot \left(-1 \cdot \left(c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right) + \color{blue}{\left(t \cdot z + x \cdot y\right)}\right) \]

      5. associate-*r* N/A

         \[\leadsto 2 \cdot \left(\left(-1 \cdot c\right) \cdot \left(i \cdot \left(a + b \cdot c\right)\right) + \left(\color{blue}{t \cdot z} + x \cdot y\right)\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(-1 \cdot c, \color{blue}{i \cdot \left(a + b \cdot c\right)}, t \cdot z + x \cdot y\right) \]

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-fma.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 92.4

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

   4. Applied rewrites 92.4%

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

   if -8.6000000000000004e-106 < c < 1.00000000000000001e-183

   1. Initial program 98.5%

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

   2. Taylor expanded in b around 0

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

      1. lower--.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 96.2

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

   4. Applied rewrites 96.2%

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

Alternative 12: 44.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 \cdot \left(y \cdot x\right)\\ \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+48}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{-13}:\\ \;\;\;\;2 \cdot \left(t \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* 2.0 (* y x))))
   (if (<= (* x y) -5e+48) t_1 (if (<= (* x y) 2e-13) (* 2.0 (* t z)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = 2.0 * (y * x);
	double tmp;
	if ((x * y) <= -5e+48) {
		tmp = t_1;
	} else if ((x * y) <= 2e-13) {
		tmp = 2.0 * (t * z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b, c, i)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 2.0d0 * (y * x)
    if ((x * y) <= (-5d+48)) then
        tmp = t_1
    else if ((x * y) <= 2d-13) then
        tmp = 2.0d0 * (t * z)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = 2.0 * (y * x);
	double tmp;
	if ((x * y) <= -5e+48) {
		tmp = t_1;
	} else if ((x * y) <= 2e-13) {
		tmp = 2.0 * (t * z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = 2.0 * (y * x)
	tmp = 0
	if (x * y) <= -5e+48:
		tmp = t_1
	elif (x * y) <= 2e-13:
		tmp = 2.0 * (t * z)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(2.0 * Float64(y * x))
	tmp = 0.0
	if (Float64(x * y) <= -5e+48)
		tmp = t_1;
	elseif (Float64(x * y) <= 2e-13)
		tmp = Float64(2.0 * Float64(t * z));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	t_1 = 2.0 * (y * x);
	tmp = 0.0;
	if ((x * y) <= -5e+48)
		tmp = t_1;
	elseif ((x * y) <= 2e-13)
		tmp = 2.0 * (t * z);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(2.0 * N[(y * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * y), $MachinePrecision], -5e+48], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], 2e-13], N[(2.0 * N[(t * z), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -4.99999999999999973e48 or 2.0000000000000001e-13 < (*.f64 x y)

   1. Initial program 88.4%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 50.7

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

   4. Applied rewrites 50.7%

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

   if -4.99999999999999973e48 < (*.f64 x y) < 2.0000000000000001e-13

   1. Initial program 92.3%

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

   2. Taylor expanded in z around inf

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

      1. lower-*.f64 38.2

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

   4. Applied rewrites 38.2%

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

Alternative 13: 29.7% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left(t \cdot z\right) \end{array} \]

FPCore

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

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return 2.0 * (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 = 2.0d0 * (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 2.0 * (t * z);
}

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := N[(2.0 * N[(t * z), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 90.4%

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

2. Taylor expanded in z around inf

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

   1. lower-*.f64 29.7

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

4. Applied rewrites 29.7%

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

Developer Target 1: 94.4% accurate, 1.0× speedup

Math

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

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(a + Float64(b * c)) * Float64(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[(a + N[(b * c), $MachinePrecision]), $MachinePrecision] * N[(c * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2025096 
(FPCore (x y z t a b c i)
  :name "Diagrams.ThreeD.Shapes:frustum from diagrams-lib-1.3.0.3, A"
  :precision binary64

  :alt
  (! :herbie-platform default (* 2 (- (+ (* x y) (* z t)) (* (+ a (* b c)) (* c i)))))

  (* 2.0 (- (+ (* x y) (* z t)) (* (* (+ a (* b c)) c) i))))