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

Percentage Accurate: 90.0% → 96.8%

Time: 10.5s

Alternatives: 16

Speedup: 0.5×

• 47.3% 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: 90.0% 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: 96.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(2.0 * N[(N[(N[(x * y), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(2.0 * N[(N[(c * b + a), $MachinePrecision] * N[((-i) * c), $MachinePrecision] + N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[((-z) * N[(N[(N[(N[(b * c + a), $MachinePrecision] * i), $MachinePrecision] / z), $MachinePrecision] * c + (-t)), $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 96.0%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

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

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

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

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

      9. lower-fma.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 99.1

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-fma.f64 99.1

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

      23. lift-*.f64 N/A

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

      24. *-commutative N/A

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

      25. lower-*.f64 99.1

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

   4. Applied rewrites 99.1%

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

   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. Add Preprocessing

   3. Taylor expanded in x around 0

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

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

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

      2. mul-1-neg N/A

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

      3. associate-*r* N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

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

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

      10. lower-fma.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-*.f64 14.3

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

   5. Applied rewrites 14.3%

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

   6. Taylor expanded in z around -inf

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

   8. Applied rewrites 64.3%

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

4. Final simplification 97.2%

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

Alternative 2: 57.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-2 \cdot \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right)\right) \cdot c\\ t_2 := \left(a + b \cdot c\right) \cdot c\\ t_3 := 2 \cdot \left(y \cdot x\right)\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{-24}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq -2 \cdot 10^{-257}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 10^{-160}:\\ \;\;\;\;t \cdot \left(z + z\right)\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+88}:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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))
        (t_3 (* 2.0 (* y x))))
   (if (<= t_2 -5e-24)
     t_1
     (if (<= t_2 -2e-257)
       t_3
       (if (<= t_2 1e-160) (* t (+ z z)) (if (<= t_2 5e+88) t_3 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;
	double t_3 = 2.0 * (y * x);
	double tmp;
	if (t_2 <= -5e-24) {
		tmp = t_1;
	} else if (t_2 <= -2e-257) {
		tmp = t_3;
	} else if (t_2 <= 1e-160) {
		tmp = t * (z + z);
	} else if (t_2 <= 5e+88) {
		tmp = t_3;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(-2.0 * Float64(fma(c, b, a) * i)) * c)
	t_2 = Float64(Float64(a + Float64(b * c)) * c)
	t_3 = Float64(2.0 * Float64(y * x))
	tmp = 0.0
	if (t_2 <= -5e-24)
		tmp = t_1;
	elseif (t_2 <= -2e-257)
		tmp = t_3;
	elseif (t_2 <= 1e-160)
		tmp = Float64(t * Float64(z + z));
	elseif (t_2 <= 5e+88)
		tmp = t_3;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(-2.0 * N[(N[(c * b + a), $MachinePrecision] * i), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]}, Block[{t$95$2 = N[(N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]}, Block[{t$95$3 = N[(2.0 * N[(y * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e-24], t$95$1, If[LessEqual[t$95$2, -2e-257], t$95$3, If[LessEqual[t$95$2, 1e-160], N[(t * N[(z + z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e+88], t$95$3, t$95$1]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (+.f64 a (*.f64 b c)) c) < -4.9999999999999998e-24 or 4.99999999999999997e88 < (*.f64 (+.f64 a (*.f64 b c)) c)

   1. Initial program 84.0%

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

   3. Taylor expanded in i around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*r* N/A

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

      5. distribute-lft-out N/A

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

      6. lower-*.f64 N/A

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

      7. distribute-lft-out N/A

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

      8. associate-*r* N/A

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

      9. distribute-rgt-in N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 71.3

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

   5. Applied rewrites 71.3%

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

   if -4.9999999999999998e-24 < (*.f64 (+.f64 a (*.f64 b c)) c) < -2e-257 or 9.9999999999999999e-161 < (*.f64 (+.f64 a (*.f64 b c)) c) < 4.99999999999999997e88

   1. Initial program 98.7%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 56.2

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

   5. Applied rewrites 56.2%

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

   if -2e-257 < (*.f64 (+.f64 a (*.f64 b c)) c) < 9.9999999999999999e-161

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 66.5

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

   5. Applied rewrites 66.5%

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

   7. Applied rewrites 66.5%

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

Alternative 3: 63.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-2 \cdot \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right)\right) \cdot c\\ t_2 := \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+256}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{-269}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(-i, a \cdot c, t \cdot z\right)\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+22}:\\ \;\;\;\;2 \cdot \left(y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 -5e+256)
     t_1
     (if (<= t_2 1e-269)
       (* 2.0 (fma (- i) (* a c) (* t z)))
       (if (<= t_2 2e+22) (* 2.0 (* 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 <= -5e+256) {
		tmp = t_1;
	} else if (t_2 <= 1e-269) {
		tmp = 2.0 * fma(-i, (a * c), (t * z));
	} else if (t_2 <= 2e+22) {
		tmp = 2.0 * (y * x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(-2.0 * Float64(fma(c, b, a) * i)) * c)
	t_2 = Float64(Float64(Float64(a + Float64(b * c)) * c) * i)
	tmp = 0.0
	if (t_2 <= -5e+256)
		tmp = t_1;
	elseif (t_2 <= 1e-269)
		tmp = Float64(2.0 * fma(Float64(-i), Float64(a * c), Float64(t * z)));
	elseif (t_2 <= 2e+22)
		tmp = Float64(2.0 * 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[(-2.0 * N[(N[(c * b + a), $MachinePrecision] * i), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * i), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+256], t$95$1, If[LessEqual[t$95$2, 1e-269], N[(2.0 * N[((-i) * N[(a * c), $MachinePrecision] + N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+22], N[(2.0 * N[(y * x), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -5.00000000000000015e256 or 2e22 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 78.9%

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

   3. Taylor expanded in i around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*r* N/A

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

      5. distribute-lft-out N/A

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

      6. lower-*.f64 N/A

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

      7. distribute-lft-out N/A

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

      8. associate-*r* N/A

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

      9. distribute-rgt-in N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 81.6

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

   5. Applied rewrites 81.6%

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

   if -5.00000000000000015e256 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 9.9999999999999996e-270

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

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

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

      2. mul-1-neg N/A

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

      3. associate-*r* N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

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

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

      10. lower-fma.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-*.f64 64.8

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

   5. Applied rewrites 64.8%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 56.0%

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

   if 9.9999999999999996e-270 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 2e22

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 63.1

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

   5. Applied rewrites 63.1%

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

Alternative 4: 51.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+222}:\\ \;\;\;\;\left(c \cdot \left(c \cdot \left(i \cdot b\right)\right)\right) \cdot -2\\ \mathbf{elif}\;t\_1 \leq 10^{-269}:\\ \;\;\;\;t \cdot \left(z + z\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+90}:\\ \;\;\;\;2 \cdot \left(y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(c \cdot b\right) \cdot i\right) \cdot c\right) \cdot -2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* (+ a (* b c)) c) i)))
   (if (<= t_1 -5e+222)
     (* (* c (* c (* i b))) -2.0)
     (if (<= t_1 1e-269)
       (* t (+ z z))
       (if (<= t_1 5e+90) (* 2.0 (* y x)) (* (* (* (* c b) i) c) -2.0))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = ((a + (b * c)) * c) * i;
	double tmp;
	if (t_1 <= -5e+222) {
		tmp = (c * (c * (i * b))) * -2.0;
	} else if (t_1 <= 1e-269) {
		tmp = t * (z + z);
	} else if (t_1 <= 5e+90) {
		tmp = 2.0 * (y * x);
	} else {
		tmp = (((c * b) * i) * c) * -2.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(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 = ((a + (b * c)) * c) * i
    if (t_1 <= (-5d+222)) then
        tmp = (c * (c * (i * b))) * (-2.0d0)
    else if (t_1 <= 1d-269) then
        tmp = t * (z + z)
    else if (t_1 <= 5d+90) then
        tmp = 2.0d0 * (y * x)
    else
        tmp = (((c * b) * i) * c) * (-2.0d0)
    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 = ((a + (b * c)) * c) * i;
	double tmp;
	if (t_1 <= -5e+222) {
		tmp = (c * (c * (i * b))) * -2.0;
	} else if (t_1 <= 1e-269) {
		tmp = t * (z + z);
	} else if (t_1 <= 5e+90) {
		tmp = 2.0 * (y * x);
	} else {
		tmp = (((c * b) * i) * c) * -2.0;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = ((a + (b * c)) * c) * i
	tmp = 0
	if t_1 <= -5e+222:
		tmp = (c * (c * (i * b))) * -2.0
	elif t_1 <= 1e-269:
		tmp = t * (z + z)
	elif t_1 <= 5e+90:
		tmp = 2.0 * (y * x)
	else:
		tmp = (((c * b) * i) * c) * -2.0
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(a + Float64(b * c)) * c) * i)
	tmp = 0.0
	if (t_1 <= -5e+222)
		tmp = Float64(Float64(c * Float64(c * Float64(i * b))) * -2.0);
	elseif (t_1 <= 1e-269)
		tmp = Float64(t * Float64(z + z));
	elseif (t_1 <= 5e+90)
		tmp = Float64(2.0 * Float64(y * x));
	else
		tmp = Float64(Float64(Float64(Float64(c * b) * i) * c) * -2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	t_1 = ((a + (b * c)) * c) * i;
	tmp = 0.0;
	if (t_1 <= -5e+222)
		tmp = (c * (c * (i * b))) * -2.0;
	elseif (t_1 <= 1e-269)
		tmp = t * (z + z);
	elseif (t_1 <= 5e+90)
		tmp = 2.0 * (y * x);
	else
		tmp = (((c * b) * i) * c) * -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * i), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+222], N[(N[(c * N[(c * N[(i * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], If[LessEqual[t$95$1, 1e-269], N[(t * N[(z + z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+90], N[(2.0 * N[(y * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(c * b), $MachinePrecision] * i), $MachinePrecision] * c), $MachinePrecision] * -2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -5.00000000000000023e222

   1. Initial program 73.7%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 72.1

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

   5. Applied rewrites 72.1%

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

   7. Applied rewrites 74.5%

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

   if -5.00000000000000023e222 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 9.9999999999999996e-270

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 47.0

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

   5. Applied rewrites 47.0%

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

   7. Applied rewrites 47.0%

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

   if 9.9999999999999996e-270 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 5.0000000000000004e90

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 58.9

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

   5. Applied rewrites 58.9%

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

   if 5.0000000000000004e90 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 83.6%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 50.2

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

   5. Applied rewrites 50.2%

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

   7. Applied rewrites 57.3%

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

   9. Applied rewrites 59.0%

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

Alternative 5: 51.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(c \cdot \left(c \cdot \left(i \cdot b\right)\right)\right) \cdot -2\\ t_2 := \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+222}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{-269}:\\ \;\;\;\;t \cdot \left(z + z\right)\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+90}:\\ \;\;\;\;2 \cdot \left(y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 -5e+222)
     t_1
     (if (<= t_2 1e-269)
       (* t (+ z z))
       (if (<= t_2 5e+90) (* 2.0 (* 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 <= -5e+222) {
		tmp = t_1;
	} else if (t_2 <= 1e-269) {
		tmp = t * (z + z);
	} else if (t_2 <= 5e+90) {
		tmp = 2.0 * (y * x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b, c, i)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (c * (c * (i * b))) * (-2.0d0)
    t_2 = ((a + (b * c)) * c) * i
    if (t_2 <= (-5d+222)) then
        tmp = t_1
    else if (t_2 <= 1d-269) then
        tmp = t * (z + z)
    else if (t_2 <= 5d+90) then
        tmp = 2.0d0 * (y * x)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (c * (c * (i * b))) * -2.0;
	double t_2 = ((a + (b * c)) * c) * i;
	double tmp;
	if (t_2 <= -5e+222) {
		tmp = t_1;
	} else if (t_2 <= 1e-269) {
		tmp = t * (z + z);
	} else if (t_2 <= 5e+90) {
		tmp = 2.0 * (y * x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = (c * (c * (i * b))) * -2.0
	t_2 = ((a + (b * c)) * c) * i
	tmp = 0
	if t_2 <= -5e+222:
		tmp = t_1
	elif t_2 <= 1e-269:
		tmp = t * (z + z)
	elif t_2 <= 5e+90:
		tmp = 2.0 * (y * x)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(c * Float64(c * Float64(i * b))) * -2.0)
	t_2 = Float64(Float64(Float64(a + Float64(b * c)) * c) * i)
	tmp = 0.0
	if (t_2 <= -5e+222)
		tmp = t_1;
	elseif (t_2 <= 1e-269)
		tmp = Float64(t * Float64(z + z));
	elseif (t_2 <= 5e+90)
		tmp = Float64(2.0 * Float64(y * x));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	t_1 = (c * (c * (i * b))) * -2.0;
	t_2 = ((a + (b * c)) * c) * i;
	tmp = 0.0;
	if (t_2 <= -5e+222)
		tmp = t_1;
	elseif (t_2 <= 1e-269)
		tmp = t * (z + z);
	elseif (t_2 <= 5e+90)
		tmp = 2.0 * (y * x);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(c * N[(c * N[(i * b), $MachinePrecision]), $MachinePrecision]), $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, -5e+222], t$95$1, If[LessEqual[t$95$2, 1e-269], N[(t * N[(z + z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e+90], N[(2.0 * N[(y * x), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -5.00000000000000023e222 or 5.0000000000000004e90 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 78.5%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 61.5

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

   5. Applied rewrites 61.5%

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

   7. Applied rewrites 66.5%

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

   if -5.00000000000000023e222 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 9.9999999999999996e-270

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 47.0

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

   5. Applied rewrites 47.0%

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

   7. Applied rewrites 47.0%

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

   if 9.9999999999999996e-270 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 5.0000000000000004e90

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 58.9

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

   5. Applied rewrites 58.9%

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

Alternative 6: 94.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\\ \mathbf{if}\;t\_1 \leq -\infty \lor \neg \left(t\_1 \leq 2 \cdot 10^{+305}\right):\\ \;\;\;\;\left(-2 \cdot \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right)\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(y, x, t \cdot z - i \cdot \left(\mathsf{fma}\left(c, b, a\right) \cdot c\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* (+ a (* b c)) c) i)))
   (if (or (<= t_1 (- INFINITY)) (not (<= t_1 2e+305)))
     (* (* -2.0 (* (fma c b a) i)) c)
     (* 2.0 (fma y x (- (* t z) (* i (* (fma c b a) c))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = ((a + (b * c)) * c) * i;
	double tmp;
	if ((t_1 <= -((double) INFINITY)) || !(t_1 <= 2e+305)) {
		tmp = (-2.0 * (fma(c, b, a) * i)) * c;
	} else {
		tmp = 2.0 * fma(y, x, ((t * z) - (i * (fma(c, b, a) * c))));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(a + Float64(b * c)) * c) * i)
	tmp = 0.0
	if ((t_1 <= Float64(-Inf)) || !(t_1 <= 2e+305))
		tmp = Float64(Float64(-2.0 * Float64(fma(c, b, a) * i)) * c);
	else
		tmp = Float64(2.0 * fma(y, x, Float64(Float64(t * z) - Float64(i * Float64(fma(c, b, a) * c)))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * i), $MachinePrecision]}, If[Or[LessEqual[t$95$1, (-Infinity)], N[Not[LessEqual[t$95$1, 2e+305]], $MachinePrecision]], N[(N[(-2.0 * N[(N[(c * b + a), $MachinePrecision] * i), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision], N[(2.0 * N[(y * x + N[(N[(t * z), $MachinePrecision] - N[(i * N[(N[(c * b + a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -inf.0 or 1.9999999999999999e305 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 72.5%

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

   3. Taylor expanded in i around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*r* N/A

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

      5. distribute-lft-out N/A

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

      6. lower-*.f64 N/A

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

      7. distribute-lft-out N/A

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

      8. associate-*r* N/A

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

      9. distribute-rgt-in N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 89.6

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

   5. Applied rewrites 89.6%

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

   if -inf.0 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 1.9999999999999999e305

   1. Initial program 99.9%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 99.9

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 99.9

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 99.9

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 99.9

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

   4. Applied rewrites 99.9%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(a + b \cdot c\right) \cdot c\right) \cdot i \leq -\infty \lor \neg \left(\left(\left(a + b \cdot c\right) \cdot c\right) \cdot i \leq 2 \cdot 10^{+305}\right):\\ \;\;\;\;\left(-2 \cdot \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right)\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(y, x, t \cdot z - i \cdot \left(\mathsf{fma}\left(c, b, a\right) \cdot c\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 96.0% accurate, 0.5× speedup

Math

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

FPCore

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

Julia

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

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(2.0 * N[(N[(N[(x * y), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(2.0 * N[(N[(c * b + a), $MachinePrecision] * N[((-i) * c), $MachinePrecision] + N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 * N[(N[(c * b + a), $MachinePrecision] * i), $MachinePrecision]), $MachinePrecision] * c), $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 96.0%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

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

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

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

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

      9. lower-fma.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 99.1

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-fma.f64 99.1

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

      23. lift-*.f64 N/A

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

      24. *-commutative N/A

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

      25. lower-*.f64 99.1

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

   4. Applied rewrites 99.1%

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

   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. Add Preprocessing

   3. Taylor expanded in i around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*r* N/A

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

      5. distribute-lft-out N/A

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

      6. lower-*.f64 N/A

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

      7. distribute-lft-out N/A

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

      8. associate-*r* N/A

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

      9. distribute-rgt-in N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 64.3

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

   5. Applied rewrites 64.3%

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

4. Final simplification 97.2%

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

Alternative 8: 86.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\\ \mathbf{if}\;t\_1 \leq -\infty \lor \neg \left(t\_1 \leq 5 \cdot 10^{+130}\right):\\ \;\;\;\;\left(-2 \cdot \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right)\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(-i, c \cdot a, \mathsf{fma}\left(t, z, y \cdot x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* (+ a (* b c)) c) i)))
   (if (or (<= t_1 (- INFINITY)) (not (<= t_1 5e+130)))
     (* (* -2.0 (* (fma c b a) i)) c)
     (* 2.0 (fma (- i) (* c a) (fma t z (* y x)))))))

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 <= -((double) INFINITY)) || !(t_1 <= 5e+130)) {
		tmp = (-2.0 * (fma(c, b, a) * i)) * c;
	} else {
		tmp = 2.0 * fma(-i, (c * a), fma(t, z, (y * x)));
	}
	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 <= Float64(-Inf)) || !(t_1 <= 5e+130))
		tmp = Float64(Float64(-2.0 * Float64(fma(c, b, a) * i)) * c);
	else
		tmp = Float64(2.0 * fma(Float64(-i), Float64(c * a), fma(t, z, Float64(y * x))));
	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[Or[LessEqual[t$95$1, (-Infinity)], N[Not[LessEqual[t$95$1, 5e+130]], $MachinePrecision]], N[(N[(-2.0 * N[(N[(c * b + a), $MachinePrecision] * i), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision], N[(2.0 * N[((-i) * N[(c * a), $MachinePrecision] + N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -inf.0 or 4.9999999999999996e130 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 74.9%

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

   3. Taylor expanded in i around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*r* N/A

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

      5. distribute-lft-out N/A

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

      6. lower-*.f64 N/A

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

      7. distribute-lft-out N/A

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

      8. associate-*r* N/A

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

      9. distribute-rgt-in N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 87.9

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

   5. Applied rewrites 87.9%

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

   if -inf.0 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 4.9999999999999996e130

   1. Initial program 99.8%

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

   3. Taylor expanded in b around 0

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

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

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

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

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

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

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

      13. lower-fma.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-fma.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 90.1

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

   5. Applied rewrites 90.1%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(a + b \cdot c\right) \cdot c\right) \cdot i \leq -\infty \lor \neg \left(\left(\left(a + b \cdot c\right) \cdot c\right) \cdot i \leq 5 \cdot 10^{+130}\right):\\ \;\;\;\;\left(-2 \cdot \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right)\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(-i, c \cdot a, \mathsf{fma}\left(t, z, y \cdot x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 87.3% accurate, 0.5× speedup

Math

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

FPCore

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

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 <= -((double) INFINITY)) {
		tmp = (-2.0 * (fma(c, b, a) * i)) * c;
	} else if (t_1 <= 1e+106) {
		tmp = 2.0 * fma(-i, (c * a), fma(t, z, (y * x)));
	} else {
		tmp = 2.0 * fma(-i, (fma(c, b, a) * c), (y * x));
	}
	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 <= Float64(-Inf))
		tmp = Float64(Float64(-2.0 * Float64(fma(c, b, a) * i)) * c);
	elseif (t_1 <= 1e+106)
		tmp = Float64(2.0 * fma(Float64(-i), Float64(c * a), fma(t, z, Float64(y * x))));
	else
		tmp = Float64(2.0 * fma(Float64(-i), Float64(fma(c, b, a) * c), Float64(y * x)));
	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, (-Infinity)], N[(N[(-2.0 * N[(N[(c * b + a), $MachinePrecision] * i), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision], If[LessEqual[t$95$1, 1e+106], N[(2.0 * N[((-i) * N[(c * a), $MachinePrecision] + N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[((-i) * N[(N[(c * b + a), $MachinePrecision] * c), $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 69.4%

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

   3. Taylor expanded in i around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*r* N/A

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

      5. distribute-lft-out N/A

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

      6. lower-*.f64 N/A

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

      7. distribute-lft-out N/A

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

      8. associate-*r* N/A

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

      9. distribute-rgt-in N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 87.8

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

   5. Applied rewrites 87.8%

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

   if -inf.0 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 1.00000000000000009e106

   1. Initial program 99.9%

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

   3. Taylor expanded in b around 0

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

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

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

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

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

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

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

      13. lower-fma.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-fma.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 90.4

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

   5. Applied rewrites 90.4%

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

   if 1.00000000000000009e106 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 82.6%

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

   3. Taylor expanded in z around 0

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

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

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

      2. mul-1-neg N/A

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

      3. associate-*r* N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

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

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

      10. lower-fma.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 88.2

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

   5. Applied rewrites 88.2%

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

Alternative 10: 65.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(c \cdot c\right) \cdot b\\ t_2 := 2 \cdot \mathsf{fma}\left(-i, t\_1, t \cdot z\right)\\ t_3 := 2 \cdot \mathsf{fma}\left(-i, t\_1, y \cdot x\right)\\ \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{-71}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;x \cdot y \leq -5 \cdot 10^{-230}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \cdot y \leq 4 \cdot 10^{-307}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(-i, a \cdot c, t \cdot z\right)\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+30}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* c c) b))
        (t_2 (* 2.0 (fma (- i) t_1 (* t z))))
        (t_3 (* 2.0 (fma (- i) t_1 (* y x)))))
   (if (<= (* x y) -1e-71)
     t_3
     (if (<= (* x y) -5e-230)
       t_2
       (if (<= (* x y) 4e-307)
         (* 2.0 (fma (- i) (* a c) (* t z)))
         (if (<= (* x y) 5e+30) t_2 t_3))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (c * c) * b;
	double t_2 = 2.0 * fma(-i, t_1, (t * z));
	double t_3 = 2.0 * fma(-i, t_1, (y * x));
	double tmp;
	if ((x * y) <= -1e-71) {
		tmp = t_3;
	} else if ((x * y) <= -5e-230) {
		tmp = t_2;
	} else if ((x * y) <= 4e-307) {
		tmp = 2.0 * fma(-i, (a * c), (t * z));
	} else if ((x * y) <= 5e+30) {
		tmp = t_2;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(c * c) * b)
	t_2 = Float64(2.0 * fma(Float64(-i), t_1, Float64(t * z)))
	t_3 = Float64(2.0 * fma(Float64(-i), t_1, Float64(y * x)))
	tmp = 0.0
	if (Float64(x * y) <= -1e-71)
		tmp = t_3;
	elseif (Float64(x * y) <= -5e-230)
		tmp = t_2;
	elseif (Float64(x * y) <= 4e-307)
		tmp = Float64(2.0 * fma(Float64(-i), Float64(a * c), Float64(t * z)));
	elseif (Float64(x * y) <= 5e+30)
		tmp = t_2;
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(c * c), $MachinePrecision] * b), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 * N[((-i) * t$95$1 + N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(2.0 * N[((-i) * t$95$1 + N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * y), $MachinePrecision], -1e-71], t$95$3, If[LessEqual[N[(x * y), $MachinePrecision], -5e-230], t$95$2, If[LessEqual[N[(x * y), $MachinePrecision], 4e-307], N[(2.0 * N[((-i) * N[(a * c), $MachinePrecision] + N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 5e+30], t$95$2, t$95$3]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -9.9999999999999992e-72 or 4.9999999999999998e30 < (*.f64 x y)

   1. Initial program 87.1%

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

   3. Taylor expanded in z around 0

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

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

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

      2. mul-1-neg N/A

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

      3. associate-*r* N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

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

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

      10. lower-fma.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 82.0

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

   5. Applied rewrites 82.0%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 71.7%

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

   if -9.9999999999999992e-72 < (*.f64 x y) < -5.00000000000000035e-230 or 3.99999999999999964e-307 < (*.f64 x y) < 4.9999999999999998e30

   1. Initial program 93.6%

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

   3. Taylor expanded in x around 0

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

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

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

      2. mul-1-neg N/A

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

      3. associate-*r* N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

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

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

      10. lower-fma.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-*.f64 85.5

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

   5. Applied rewrites 85.5%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 72.6%

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

   if -5.00000000000000035e-230 < (*.f64 x y) < 3.99999999999999964e-307

   1. Initial program 97.6%

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

   3. Taylor expanded in x around 0

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

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

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

      2. mul-1-neg N/A

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

      3. associate-*r* N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

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

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

      10. lower-fma.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-*.f64 96.6

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

   5. Applied rewrites 96.6%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 85.7%

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

Alternative 11: 63.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-2 \cdot \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right)\right) \cdot c\\ \mathbf{if}\;c \leq -2.4 \cdot 10^{+92}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;c \leq -8.5 \cdot 10^{-223}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(-i, c \cdot a, y \cdot x\right)\\ \mathbf{elif}\;c \leq 5.6 \cdot 10^{-57}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(-i, a \cdot c, t \cdot z\right)\\ \mathbf{elif}\;c \leq 2.15 \cdot 10^{+70}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(-i, \left(c \cdot c\right) \cdot b, t \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* -2.0 (* (fma c b a) i)) c)))
   (if (<= c -2.4e+92)
     t_1
     (if (<= c -8.5e-223)
       (* 2.0 (fma (- i) (* c a) (* y x)))
       (if (<= c 5.6e-57)
         (* 2.0 (fma (- i) (* a c) (* t z)))
         (if (<= c 2.15e+70)
           (* 2.0 (fma (- i) (* (* c c) b) (* 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 * (fma(c, b, a) * i)) * c;
	double tmp;
	if (c <= -2.4e+92) {
		tmp = t_1;
	} else if (c <= -8.5e-223) {
		tmp = 2.0 * fma(-i, (c * a), (y * x));
	} else if (c <= 5.6e-57) {
		tmp = 2.0 * fma(-i, (a * c), (t * z));
	} else if (c <= 2.15e+70) {
		tmp = 2.0 * fma(-i, ((c * c) * b), (t * z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(-2.0 * Float64(fma(c, b, a) * i)) * c)
	tmp = 0.0
	if (c <= -2.4e+92)
		tmp = t_1;
	elseif (c <= -8.5e-223)
		tmp = Float64(2.0 * fma(Float64(-i), Float64(c * a), Float64(y * x)));
	elseif (c <= 5.6e-57)
		tmp = Float64(2.0 * fma(Float64(-i), Float64(a * c), Float64(t * z)));
	elseif (c <= 2.15e+70)
		tmp = Float64(2.0 * fma(Float64(-i), Float64(Float64(c * c) * b), Float64(t * z)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(-2.0 * N[(N[(c * b + a), $MachinePrecision] * i), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]}, If[LessEqual[c, -2.4e+92], t$95$1, If[LessEqual[c, -8.5e-223], N[(2.0 * N[((-i) * N[(c * a), $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 5.6e-57], N[(2.0 * N[((-i) * N[(a * c), $MachinePrecision] + N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 2.15e+70], N[(2.0 * N[((-i) * N[(N[(c * c), $MachinePrecision] * b), $MachinePrecision] + N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -2.40000000000000005e92 or 2.15e70 < c

   1. Initial program 79.9%

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

   3. Taylor expanded in i around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*r* N/A

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

      5. distribute-lft-out N/A

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

      6. lower-*.f64 N/A

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

      7. distribute-lft-out N/A

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

      8. associate-*r* N/A

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

      9. distribute-rgt-in N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 83.8

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

   5. Applied rewrites 83.8%

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

   if -2.40000000000000005e92 < c < -8.5000000000000003e-223

   1. Initial program 94.3%

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

   3. Taylor expanded in z around 0

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

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

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

      2. mul-1-neg N/A

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

      3. associate-*r* N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

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

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

      10. lower-fma.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 76.0

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

   5. Applied rewrites 76.0%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 67.8%

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

   if -8.5000000000000003e-223 < c < 5.5999999999999999e-57

   1. Initial program 98.4%

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

   3. Taylor expanded in x around 0

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

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

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

      2. mul-1-neg N/A

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

      3. associate-*r* N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

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

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

      10. lower-fma.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-*.f64 66.2

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

   5. Applied rewrites 66.2%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 66.2%

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

   if 5.5999999999999999e-57 < c < 2.15e70

   1. Initial program 96.8%

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

   3. Taylor expanded in x around 0

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

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

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

      2. mul-1-neg N/A

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

      3. associate-*r* N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

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

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

      10. lower-fma.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-*.f64 76.4

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

   5. Applied rewrites 76.4%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 66.4%

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

Alternative 12: 63.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 \cdot \mathsf{fma}\left(-i, c \cdot a, y \cdot x\right)\\ t_2 := \left(-2 \cdot \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right)\right) \cdot c\\ \mathbf{if}\;c \leq -2.4 \cdot 10^{+92}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;c \leq -8.5 \cdot 10^{-223}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;c \leq 2 \cdot 10^{-219}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(-i, a \cdot c, t \cdot z\right)\\ \mathbf{elif}\;c \leq 0.000112:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* 2.0 (fma (- i) (* c a) (* y x))))
        (t_2 (* (* -2.0 (* (fma c b a) i)) c)))
   (if (<= c -2.4e+92)
     t_2
     (if (<= c -8.5e-223)
       t_1
       (if (<= c 2e-219)
         (* 2.0 (fma (- i) (* a c) (* t z)))
         (if (<= c 0.000112) t_1 t_2))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = 2.0 * fma(-i, (c * a), (y * x));
	double t_2 = (-2.0 * (fma(c, b, a) * i)) * c;
	double tmp;
	if (c <= -2.4e+92) {
		tmp = t_2;
	} else if (c <= -8.5e-223) {
		tmp = t_1;
	} else if (c <= 2e-219) {
		tmp = 2.0 * fma(-i, (a * c), (t * z));
	} else if (c <= 0.000112) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(2.0 * fma(Float64(-i), Float64(c * a), Float64(y * x)))
	t_2 = Float64(Float64(-2.0 * Float64(fma(c, b, a) * i)) * c)
	tmp = 0.0
	if (c <= -2.4e+92)
		tmp = t_2;
	elseif (c <= -8.5e-223)
		tmp = t_1;
	elseif (c <= 2e-219)
		tmp = Float64(2.0 * fma(Float64(-i), Float64(a * c), Float64(t * z)));
	elseif (c <= 0.000112)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(2.0 * N[((-i) * N[(c * a), $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(-2.0 * N[(N[(c * b + a), $MachinePrecision] * i), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]}, If[LessEqual[c, -2.4e+92], t$95$2, If[LessEqual[c, -8.5e-223], t$95$1, If[LessEqual[c, 2e-219], N[(2.0 * N[((-i) * N[(a * c), $MachinePrecision] + N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 0.000112], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if c < -2.40000000000000005e92 or 1.11999999999999998e-4 < c

   1. Initial program 82.0%

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

   3. Taylor expanded in i around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*r* N/A

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

      5. distribute-lft-out N/A

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

      6. lower-*.f64 N/A

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

      7. distribute-lft-out N/A

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

      8. associate-*r* N/A

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

      9. distribute-rgt-in N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 79.6

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

   5. Applied rewrites 79.6%

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

   if -2.40000000000000005e92 < c < -8.5000000000000003e-223 or 2.0000000000000001e-219 < c < 1.11999999999999998e-4

   1. Initial program 95.9%

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

   3. Taylor expanded in z around 0

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

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

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

      2. mul-1-neg N/A

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

      3. associate-*r* N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

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

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

      10. lower-fma.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 73.0

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

   5. Applied rewrites 73.0%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 65.1%

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

   if -8.5000000000000003e-223 < c < 2.0000000000000001e-219

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

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

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

      2. mul-1-neg N/A

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

      3. associate-*r* N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

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

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

      10. lower-fma.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-*.f64 77.4

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

   5. Applied rewrites 77.4%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 77.4%

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

Alternative 13: 63.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{+63} \lor \neg \left(x \leq 2.85 \cdot 10^{-118}\right):\\ \;\;\;\;2 \cdot \mathsf{fma}\left(-i, \left(c \cdot c\right) \cdot b, y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(-i\right) \cdot c, b \cdot c, z \cdot t\right) \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (or (<= x -1.5e+63) (not (<= x 2.85e-118)))
   (* 2.0 (fma (- i) (* (* c c) b) (* y x)))
   (* (fma (* (- i) c) (* b c) (* z t)) 2.0)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((x <= -1.5e+63) || !(x <= 2.85e-118)) {
		tmp = 2.0 * fma(-i, ((c * c) * b), (y * x));
	} else {
		tmp = fma((-i * c), (b * c), (z * t)) * 2.0;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if ((x <= -1.5e+63) || !(x <= 2.85e-118))
		tmp = Float64(2.0 * fma(Float64(-i), Float64(Float64(c * c) * b), Float64(y * x)));
	else
		tmp = Float64(fma(Float64(Float64(-i) * c), Float64(b * c), Float64(z * t)) * 2.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[Or[LessEqual[x, -1.5e+63], N[Not[LessEqual[x, 2.85e-118]], $MachinePrecision]], N[(2.0 * N[((-i) * N[(N[(c * c), $MachinePrecision] * b), $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[((-i) * c), $MachinePrecision] * N[(b * c), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.5e63 or 2.85000000000000006e-118 < x

   1. Initial program 87.9%

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

   3. Taylor expanded in z around 0

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

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

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

      2. mul-1-neg N/A

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

      3. associate-*r* N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

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

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

      10. lower-fma.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 79.2

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

   5. Applied rewrites 79.2%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 65.3%

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

   if -1.5e63 < x < 2.85000000000000006e-118

   1. Initial program 94.6%

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

   3. Taylor expanded in x around 0

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

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

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

      2. mul-1-neg N/A

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

      3. associate-*r* N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

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

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

      10. lower-fma.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-*.f64 81.2

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

   5. Applied rewrites 81.2%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 71.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 71.7

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

   10. Applied rewrites 72.6%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{+63} \lor \neg \left(x \leq 2.85 \cdot 10^{-118}\right):\\ \;\;\;\;2 \cdot \mathsf{fma}\left(-i, \left(c \cdot c\right) \cdot b, y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(-i\right) \cdot c, b \cdot c, z \cdot t\right) \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 62.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.4 \cdot 10^{+62} \lor \neg \left(x \leq 2.85 \cdot 10^{-118}\right):\\ \;\;\;\;2 \cdot \mathsf{fma}\left(-i, \left(c \cdot c\right) \cdot b, y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(-i, \left(b \cdot c\right) \cdot c, t \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (or (<= x -6.4e+62) (not (<= x 2.85e-118)))
   (* 2.0 (fma (- i) (* (* c c) b) (* y x)))
   (* 2.0 (fma (- i) (* (* b c) c) (* t z)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((x <= -6.4e+62) || !(x <= 2.85e-118)) {
		tmp = 2.0 * fma(-i, ((c * c) * b), (y * x));
	} else {
		tmp = 2.0 * fma(-i, ((b * c) * c), (t * z));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if ((x <= -6.4e+62) || !(x <= 2.85e-118))
		tmp = Float64(2.0 * fma(Float64(-i), Float64(Float64(c * c) * b), Float64(y * x)));
	else
		tmp = Float64(2.0 * fma(Float64(-i), Float64(Float64(b * c) * c), Float64(t * z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[Or[LessEqual[x, -6.4e+62], N[Not[LessEqual[x, 2.85e-118]], $MachinePrecision]], N[(2.0 * N[((-i) * N[(N[(c * c), $MachinePrecision] * b), $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[((-i) * N[(N[(b * c), $MachinePrecision] * c), $MachinePrecision] + N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -6.39999999999999968e62 or 2.85000000000000006e-118 < x

   1. Initial program 88.0%

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

   3. Taylor expanded in z around 0

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

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

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

      2. mul-1-neg N/A

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

      3. associate-*r* N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

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

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

      10. lower-fma.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 79.3

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

   5. Applied rewrites 79.3%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 65.5%

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

   if -6.39999999999999968e62 < x < 2.85000000000000006e-118

   1. Initial program 94.6%

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

   3. Taylor expanded in x around 0

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

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

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

      2. mul-1-neg N/A

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

      3. associate-*r* N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

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

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

      10. lower-fma.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-*.f64 81.9

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

   5. Applied rewrites 81.9%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 72.3%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.4 \cdot 10^{+62} \lor \neg \left(x \leq 2.85 \cdot 10^{-118}\right):\\ \;\;\;\;2 \cdot \mathsf{fma}\left(-i, \left(c \cdot c\right) \cdot b, y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(-i, \left(b \cdot c\right) \cdot c, t \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 39.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{+63} \lor \neg \left(x \leq 2.85 \cdot 10^{-118}\right):\\ \;\;\;\;2 \cdot \left(y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(z + z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (or (<= x -1.1e+63) (not (<= x 2.85e-118)))
   (* 2.0 (* y x))
   (* t (+ z z))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((x <= -1.1e+63) || !(x <= 2.85e-118)) {
		tmp = 2.0 * (y * x);
	} else {
		tmp = t * (z + z);
	}
	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) :: tmp
    if ((x <= (-1.1d+63)) .or. (.not. (x <= 2.85d-118))) then
        tmp = 2.0d0 * (y * x)
    else
        tmp = t * (z + z)
    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 tmp;
	if ((x <= -1.1e+63) || !(x <= 2.85e-118)) {
		tmp = 2.0 * (y * x);
	} else {
		tmp = t * (z + z);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if (x <= -1.1e+63) or not (x <= 2.85e-118):
		tmp = 2.0 * (y * x)
	else:
		tmp = t * (z + z)
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if ((x <= -1.1e+63) || !(x <= 2.85e-118))
		tmp = Float64(2.0 * Float64(y * x));
	else
		tmp = Float64(t * Float64(z + z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if ((x <= -1.1e+63) || ~((x <= 2.85e-118)))
		tmp = 2.0 * (y * x);
	else
		tmp = t * (z + z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[Or[LessEqual[x, -1.1e+63], N[Not[LessEqual[x, 2.85e-118]], $MachinePrecision]], N[(2.0 * N[(y * x), $MachinePrecision]), $MachinePrecision], N[(t * N[(z + z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.0999999999999999e63 or 2.85000000000000006e-118 < x

   1. Initial program 88.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 44.3

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

   5. Applied rewrites 44.3%

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

   if -1.0999999999999999e63 < x < 2.85000000000000006e-118

   1. Initial program 94.6%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 40.9

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

   5. Applied rewrites 40.9%

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

   7. Applied rewrites 40.9%

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

4. Final simplification 42.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{+63} \lor \neg \left(x \leq 2.85 \cdot 10^{-118}\right):\\ \;\;\;\;2 \cdot \left(y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(z + z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 29.5% accurate, 4.4× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.8%

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

3. Taylor expanded in z around inf

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 28.3

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

5. Applied rewrites 28.3%

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

7. Applied rewrites 28.3%

   \[\leadsto t \cdot \color{blue}{\left(z + z\right)} \]
8. 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 2024360 
(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))))