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

Percentage Accurate: 89.9% → 96.1%

Time: 11.9s

Alternatives: 15

Speedup: 1.1×

• 46.4% 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

real(8) function code(x, y, z, t, a, b, c, i)
    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: 89.9% 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

real(8) function code(x, y, z, t, a, b, c, i)
    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: 96.1% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 93.2%

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

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

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

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

      8. lower-fma.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

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

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

      15. lower-*.f64 N/A

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

      16. lower-neg.f64 97.6

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-fma.f64 97.6

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

      22. lift-*.f64 N/A

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

      23. *-commutative N/A

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

      24. lower-*.f64 97.6

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

   4. Applied rewrites 97.6%

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

   if +inf.0 < (-.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 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 97.7%

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

Alternative 2: 41.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(i \cdot c\right) \cdot a\right) \cdot -2\\ t_2 := i \cdot \left(\left(c \cdot b + a\right) \cdot c\right)\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+102}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-90}:\\ \;\;\;\;\left(y \cdot x\right) \cdot 2\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+87}:\\ \;\;\;\;2 \cdot \left(t \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* (* i c) a) -2.0)) (t_2 (* i (* (+ (* c b) a) c))))
   (if (<= t_2 -1e+102)
     t_1
     (if (<= t_2 5e-90)
       (* (* y x) 2.0)
       (if (<= t_2 2e+87) (* 2.0 (* t z)) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = ((i * c) * a) * -2.0;
	double t_2 = i * (((c * b) + a) * c);
	double tmp;
	if (t_2 <= -1e+102) {
		tmp = t_1;
	} else if (t_2 <= 5e-90) {
		tmp = (y * x) * 2.0;
	} else if (t_2 <= 2e+87) {
		tmp = 2.0 * (t * z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i)
    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 = ((i * c) * a) * (-2.0d0)
    t_2 = i * (((c * b) + a) * c)
    if (t_2 <= (-1d+102)) then
        tmp = t_1
    else if (t_2 <= 5d-90) then
        tmp = (y * x) * 2.0d0
    else if (t_2 <= 2d+87) then
        tmp = 2.0d0 * (t * z)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = ((i * c) * a) * -2.0;
	double t_2 = i * (((c * b) + a) * c);
	double tmp;
	if (t_2 <= -1e+102) {
		tmp = t_1;
	} else if (t_2 <= 5e-90) {
		tmp = (y * x) * 2.0;
	} else if (t_2 <= 2e+87) {
		tmp = 2.0 * (t * z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = ((i * c) * a) * -2.0
	t_2 = i * (((c * b) + a) * c)
	tmp = 0
	if t_2 <= -1e+102:
		tmp = t_1
	elif t_2 <= 5e-90:
		tmp = (y * x) * 2.0
	elif t_2 <= 2e+87:
		tmp = 2.0 * (t * z)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(i * c) * a) * -2.0)
	t_2 = Float64(i * Float64(Float64(Float64(c * b) + a) * c))
	tmp = 0.0
	if (t_2 <= -1e+102)
		tmp = t_1;
	elseif (t_2 <= 5e-90)
		tmp = Float64(Float64(y * x) * 2.0);
	elseif (t_2 <= 2e+87)
		tmp = Float64(2.0 * Float64(t * z));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	t_1 = ((i * c) * a) * -2.0;
	t_2 = i * (((c * b) + a) * c);
	tmp = 0.0;
	if (t_2 <= -1e+102)
		tmp = t_1;
	elseif (t_2 <= 5e-90)
		tmp = (y * x) * 2.0;
	elseif (t_2 <= 2e+87)
		tmp = 2.0 * (t * z);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(i * c), $MachinePrecision] * a), $MachinePrecision] * -2.0), $MachinePrecision]}, Block[{t$95$2 = N[(i * N[(N[(N[(c * b), $MachinePrecision] + a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+102], t$95$1, If[LessEqual[t$95$2, 5e-90], N[(N[(y * x), $MachinePrecision] * 2.0), $MachinePrecision], If[LessEqual[t$95$2, 2e+87], N[(2.0 * N[(t * z), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -9.99999999999999977e101 or 1.9999999999999999e87 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 85.4%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 44.2

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

   5. Applied rewrites 44.2%

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

   if -9.99999999999999977e101 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 5.00000000000000019e-90

   1. Initial program 98.8%

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

   3. Taylor expanded in x around 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 61.8

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

   5. Applied rewrites 61.8%

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

   if 5.00000000000000019e-90 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 1.9999999999999999e87

   1. Initial program 99.7%

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

   3. Taylor expanded in 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 54.3

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

   5. Applied rewrites 54.3%

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

4. Final simplification 51.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq -1 \cdot 10^{+102}:\\ \;\;\;\;\left(\left(i \cdot c\right) \cdot a\right) \cdot -2\\ \mathbf{elif}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq 5 \cdot 10^{-90}:\\ \;\;\;\;\left(y \cdot x\right) \cdot 2\\ \mathbf{elif}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq 2 \cdot 10^{+87}:\\ \;\;\;\;2 \cdot \left(t \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(i \cdot c\right) \cdot a\right) \cdot -2\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := i \cdot \left(\left(c \cdot b + a\right) \cdot c\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+102}:\\ \;\;\;\;\mathsf{fma}\left(-i, \mathsf{fma}\left(c, b, a\right) \cdot c, y \cdot x\right) \cdot 2\\ \mathbf{elif}\;t\_1 \leq 10^{+201}:\\ \;\;\;\;\mathsf{fma}\left(z, t, \mathsf{fma}\left(-\left(i \cdot c\right) \cdot b, c, y \cdot x\right)\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-2 \cdot c\right) \cdot i\right) \cdot \mathsf{fma}\left(c, b, a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* i (* (+ (* c b) a) c))))
   (if (<= t_1 -1e+102)
     (* (fma (- i) (* (fma c b a) c) (* y x)) 2.0)
     (if (<= t_1 1e+201)
       (* (fma z t (fma (- (* (* i c) b)) c (* y x))) 2.0)
       (* (* (* -2.0 c) i) (fma c b a))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = i * (((c * b) + a) * c);
	double tmp;
	if (t_1 <= -1e+102) {
		tmp = fma(-i, (fma(c, b, a) * c), (y * x)) * 2.0;
	} else if (t_1 <= 1e+201) {
		tmp = fma(z, t, fma(-((i * c) * b), c, (y * x))) * 2.0;
	} else {
		tmp = ((-2.0 * c) * i) * fma(c, b, a);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(i * Float64(Float64(Float64(c * b) + a) * c))
	tmp = 0.0
	if (t_1 <= -1e+102)
		tmp = Float64(fma(Float64(-i), Float64(fma(c, b, a) * c), Float64(y * x)) * 2.0);
	elseif (t_1 <= 1e+201)
		tmp = Float64(fma(z, t, fma(Float64(-Float64(Float64(i * c) * b)), c, Float64(y * x))) * 2.0);
	else
		tmp = Float64(Float64(Float64(-2.0 * c) * i) * fma(c, b, a));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(i * N[(N[(N[(c * b), $MachinePrecision] + a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+102], N[(N[((-i) * N[(N[(c * b + a), $MachinePrecision] * c), $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], If[LessEqual[t$95$1, 1e+201], N[(N[(z * t + N[((-N[(N[(i * c), $MachinePrecision] * b), $MachinePrecision]) * c + N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(N[(-2.0 * c), $MachinePrecision] * i), $MachinePrecision] * N[(c * b + a), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -9.99999999999999977e101

   1. Initial program 92.4%

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

   3. Taylor expanded in 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. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

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

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

      7. mul-1-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 89.3

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

   5. Applied rewrites 89.3%

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

   if -9.99999999999999977e101 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 1.00000000000000004e201

   1. Initial program 99.1%

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

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

      5. sub-neg N/A

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

      6. associate-+l+ N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

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

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

      14. lower-fma.f64 N/A

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

   4. Applied rewrites 97.6%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 88.8

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

   7. Applied rewrites 88.8%

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

   if 1.00000000000000004e201 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 75.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-+.f64 N/A

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

      6. flip3-+ N/A

         \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(c \cdot i\right) \cdot \color{blue}{\frac{{a}^{3} + {\left(b \cdot c\right)}^{3}}{a \cdot a + \left(\left(b \cdot c\right) \cdot \left(b \cdot c\right) - a \cdot \left(b \cdot c\right)\right)}}\right) \]

      7. clear-num N/A

         \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(c \cdot i\right) \cdot \color{blue}{\frac{1}{\frac{a \cdot a + \left(\left(b \cdot c\right) \cdot \left(b \cdot c\right) - a \cdot \left(b \cdot c\right)\right)}{{a}^{3} + {\left(b \cdot c\right)}^{3}}}}\right) \]

      8. un-div-inv N/A

         \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \color{blue}{\frac{c \cdot i}{\frac{a \cdot a + \left(\left(b \cdot c\right) \cdot \left(b \cdot c\right) - a \cdot \left(b \cdot c\right)\right)}{{a}^{3} + {\left(b \cdot c\right)}^{3}}}}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \color{blue}{\frac{c \cdot i}{\frac{a \cdot a + \left(\left(b \cdot c\right) \cdot \left(b \cdot c\right) - a \cdot \left(b \cdot c\right)\right)}{{a}^{3} + {\left(b \cdot c\right)}^{3}}}}\right) \]

      10. *-commutative N/A

          \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \frac{\color{blue}{i \cdot c}}{\frac{a \cdot a + \left(\left(b \cdot c\right) \cdot \left(b \cdot c\right) - a \cdot \left(b \cdot c\right)\right)}{{a}^{3} + {\left(b \cdot c\right)}^{3}}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \frac{\color{blue}{i \cdot c}}{\frac{a \cdot a + \left(\left(b \cdot c\right) \cdot \left(b \cdot c\right) - a \cdot \left(b \cdot c\right)\right)}{{a}^{3} + {\left(b \cdot c\right)}^{3}}}\right) \]

      12. clear-num N/A

          \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \frac{i \cdot c}{\color{blue}{\frac{1}{\frac{{a}^{3} + {\left(b \cdot c\right)}^{3}}{a \cdot a + \left(\left(b \cdot c\right) \cdot \left(b \cdot c\right) - a \cdot \left(b \cdot c\right)\right)}}}}\right) \]

      13. flip3-+ N/A

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

      14. lift-+.f64 N/A

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

      15. lower-/.f64 89.7

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-fma.f64 89.7

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

   4. Applied rewrites 89.7%

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

   5. Taylor expanded in i around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 78.4

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

   7. Applied rewrites 78.4%

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

   9. Applied rewrites 89.5%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq -1 \cdot 10^{+102}:\\ \;\;\;\;\mathsf{fma}\left(-i, \mathsf{fma}\left(c, b, a\right) \cdot c, y \cdot x\right) \cdot 2\\ \mathbf{elif}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq 10^{+201}:\\ \;\;\;\;\mathsf{fma}\left(z, t, \mathsf{fma}\left(-\left(i \cdot c\right) \cdot b, c, y \cdot x\right)\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-2 \cdot c\right) \cdot i\right) \cdot \mathsf{fma}\left(c, b, a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := i \cdot \left(\left(c \cdot b + a\right) \cdot c\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+42}:\\ \;\;\;\;\mathsf{fma}\left(-i, \mathsf{fma}\left(c, b, a\right) \cdot c, y \cdot x\right) \cdot 2\\ \mathbf{elif}\;t\_1 \leq 10^{+201}:\\ \;\;\;\;\mathsf{fma}\left(t, z, y \cdot x\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-2 \cdot c\right) \cdot i\right) \cdot \mathsf{fma}\left(c, b, a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* i (* (+ (* c b) a) c))))
   (if (<= t_1 -2e+42)
     (* (fma (- i) (* (fma c b a) c) (* y x)) 2.0)
     (if (<= t_1 1e+201)
       (* (fma t z (* y x)) 2.0)
       (* (* (* -2.0 c) i) (fma c b a))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = i * (((c * b) + a) * c);
	double tmp;
	if (t_1 <= -2e+42) {
		tmp = fma(-i, (fma(c, b, a) * c), (y * x)) * 2.0;
	} else if (t_1 <= 1e+201) {
		tmp = fma(t, z, (y * x)) * 2.0;
	} else {
		tmp = ((-2.0 * c) * i) * fma(c, b, a);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(i * Float64(Float64(Float64(c * b) + a) * c))
	tmp = 0.0
	if (t_1 <= -2e+42)
		tmp = Float64(fma(Float64(-i), Float64(fma(c, b, a) * c), Float64(y * x)) * 2.0);
	elseif (t_1 <= 1e+201)
		tmp = Float64(fma(t, z, Float64(y * x)) * 2.0);
	else
		tmp = Float64(Float64(Float64(-2.0 * c) * i) * fma(c, b, a));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(i * N[(N[(N[(c * b), $MachinePrecision] + a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+42], N[(N[((-i) * N[(N[(c * b + a), $MachinePrecision] * c), $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], If[LessEqual[t$95$1, 1e+201], N[(N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(N[(-2.0 * c), $MachinePrecision] * i), $MachinePrecision] * N[(c * b + a), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -2.00000000000000009e42

   1. Initial program 93.3%

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

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

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

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

      7. mul-1-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 87.9

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

   5. Applied rewrites 87.9%

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

   if -2.00000000000000009e42 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 1.00000000000000004e201

   1. Initial program 99.1%

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

   3. Taylor expanded in c around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 85.0

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

   5. Applied rewrites 85.0%

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

   if 1.00000000000000004e201 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 75.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-+.f64 N/A

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

      6. flip3-+ N/A

         \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(c \cdot i\right) \cdot \color{blue}{\frac{{a}^{3} + {\left(b \cdot c\right)}^{3}}{a \cdot a + \left(\left(b \cdot c\right) \cdot \left(b \cdot c\right) - a \cdot \left(b \cdot c\right)\right)}}\right) \]

      7. clear-num N/A

         \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(c \cdot i\right) \cdot \color{blue}{\frac{1}{\frac{a \cdot a + \left(\left(b \cdot c\right) \cdot \left(b \cdot c\right) - a \cdot \left(b \cdot c\right)\right)}{{a}^{3} + {\left(b \cdot c\right)}^{3}}}}\right) \]

      8. un-div-inv N/A

         \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \color{blue}{\frac{c \cdot i}{\frac{a \cdot a + \left(\left(b \cdot c\right) \cdot \left(b \cdot c\right) - a \cdot \left(b \cdot c\right)\right)}{{a}^{3} + {\left(b \cdot c\right)}^{3}}}}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \color{blue}{\frac{c \cdot i}{\frac{a \cdot a + \left(\left(b \cdot c\right) \cdot \left(b \cdot c\right) - a \cdot \left(b \cdot c\right)\right)}{{a}^{3} + {\left(b \cdot c\right)}^{3}}}}\right) \]

      10. *-commutative N/A

          \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \frac{\color{blue}{i \cdot c}}{\frac{a \cdot a + \left(\left(b \cdot c\right) \cdot \left(b \cdot c\right) - a \cdot \left(b \cdot c\right)\right)}{{a}^{3} + {\left(b \cdot c\right)}^{3}}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \frac{\color{blue}{i \cdot c}}{\frac{a \cdot a + \left(\left(b \cdot c\right) \cdot \left(b \cdot c\right) - a \cdot \left(b \cdot c\right)\right)}{{a}^{3} + {\left(b \cdot c\right)}^{3}}}\right) \]

      12. clear-num N/A

          \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \frac{i \cdot c}{\color{blue}{\frac{1}{\frac{{a}^{3} + {\left(b \cdot c\right)}^{3}}{a \cdot a + \left(\left(b \cdot c\right) \cdot \left(b \cdot c\right) - a \cdot \left(b \cdot c\right)\right)}}}}\right) \]

      13. flip3-+ N/A

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

      14. lift-+.f64 N/A

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

      15. lower-/.f64 89.7

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-fma.f64 89.7

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

   4. Applied rewrites 89.7%

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

   5. Taylor expanded in i around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 78.4

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

   7. Applied rewrites 78.4%

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

   9. Applied rewrites 89.5%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq -2 \cdot 10^{+42}:\\ \;\;\;\;\mathsf{fma}\left(-i, \mathsf{fma}\left(c, b, a\right) \cdot c, y \cdot x\right) \cdot 2\\ \mathbf{elif}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq 10^{+201}:\\ \;\;\;\;\mathsf{fma}\left(t, z, y \cdot x\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-2 \cdot c\right) \cdot i\right) \cdot \mathsf{fma}\left(c, b, a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := i \cdot \left(\left(c \cdot b + a\right) \cdot c\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+102}:\\ \;\;\;\;\mathsf{fma}\left(-i, \mathsf{fma}\left(c, b, a\right) \cdot c, t \cdot z\right) \cdot 2\\ \mathbf{elif}\;t\_1 \leq 10^{+201}:\\ \;\;\;\;\mathsf{fma}\left(t, z, y \cdot x\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-2 \cdot c\right) \cdot i\right) \cdot \mathsf{fma}\left(c, b, a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* i (* (+ (* c b) a) c))))
   (if (<= t_1 -1e+102)
     (* (fma (- i) (* (fma c b a) c) (* t z)) 2.0)
     (if (<= t_1 1e+201)
       (* (fma t z (* y x)) 2.0)
       (* (* (* -2.0 c) i) (fma c b a))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = i * (((c * b) + a) * c);
	double tmp;
	if (t_1 <= -1e+102) {
		tmp = fma(-i, (fma(c, b, a) * c), (t * z)) * 2.0;
	} else if (t_1 <= 1e+201) {
		tmp = fma(t, z, (y * x)) * 2.0;
	} else {
		tmp = ((-2.0 * c) * i) * fma(c, b, a);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(i * Float64(Float64(Float64(c * b) + a) * c))
	tmp = 0.0
	if (t_1 <= -1e+102)
		tmp = Float64(fma(Float64(-i), Float64(fma(c, b, a) * c), Float64(t * z)) * 2.0);
	elseif (t_1 <= 1e+201)
		tmp = Float64(fma(t, z, Float64(y * x)) * 2.0);
	else
		tmp = Float64(Float64(Float64(-2.0 * c) * i) * fma(c, b, a));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(i * N[(N[(N[(c * b), $MachinePrecision] + a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+102], N[(N[((-i) * N[(N[(c * b + a), $MachinePrecision] * c), $MachinePrecision] + N[(t * z), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], If[LessEqual[t$95$1, 1e+201], N[(N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(N[(-2.0 * c), $MachinePrecision] * i), $MachinePrecision] * N[(c * b + a), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -9.99999999999999977e101

   1. Initial program 92.4%

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

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

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

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

      7. mul-1-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-*.f64 84.8

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

   5. Applied rewrites 84.8%

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

   if -9.99999999999999977e101 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 1.00000000000000004e201

   1. Initial program 99.1%

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

   3. Taylor expanded in c around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 83.8

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

   5. Applied rewrites 83.8%

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

   if 1.00000000000000004e201 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 75.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-+.f64 N/A

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

      6. flip3-+ N/A

         \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(c \cdot i\right) \cdot \color{blue}{\frac{{a}^{3} + {\left(b \cdot c\right)}^{3}}{a \cdot a + \left(\left(b \cdot c\right) \cdot \left(b \cdot c\right) - a \cdot \left(b \cdot c\right)\right)}}\right) \]

      7. clear-num N/A

         \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(c \cdot i\right) \cdot \color{blue}{\frac{1}{\frac{a \cdot a + \left(\left(b \cdot c\right) \cdot \left(b \cdot c\right) - a \cdot \left(b \cdot c\right)\right)}{{a}^{3} + {\left(b \cdot c\right)}^{3}}}}\right) \]

      8. un-div-inv N/A

         \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \color{blue}{\frac{c \cdot i}{\frac{a \cdot a + \left(\left(b \cdot c\right) \cdot \left(b \cdot c\right) - a \cdot \left(b \cdot c\right)\right)}{{a}^{3} + {\left(b \cdot c\right)}^{3}}}}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \color{blue}{\frac{c \cdot i}{\frac{a \cdot a + \left(\left(b \cdot c\right) \cdot \left(b \cdot c\right) - a \cdot \left(b \cdot c\right)\right)}{{a}^{3} + {\left(b \cdot c\right)}^{3}}}}\right) \]

      10. *-commutative N/A

          \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \frac{\color{blue}{i \cdot c}}{\frac{a \cdot a + \left(\left(b \cdot c\right) \cdot \left(b \cdot c\right) - a \cdot \left(b \cdot c\right)\right)}{{a}^{3} + {\left(b \cdot c\right)}^{3}}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \frac{\color{blue}{i \cdot c}}{\frac{a \cdot a + \left(\left(b \cdot c\right) \cdot \left(b \cdot c\right) - a \cdot \left(b \cdot c\right)\right)}{{a}^{3} + {\left(b \cdot c\right)}^{3}}}\right) \]

      12. clear-num N/A

          \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \frac{i \cdot c}{\color{blue}{\frac{1}{\frac{{a}^{3} + {\left(b \cdot c\right)}^{3}}{a \cdot a + \left(\left(b \cdot c\right) \cdot \left(b \cdot c\right) - a \cdot \left(b \cdot c\right)\right)}}}}\right) \]

      13. flip3-+ N/A

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

      14. lift-+.f64 N/A

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

      15. lower-/.f64 89.7

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-fma.f64 89.7

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

   4. Applied rewrites 89.7%

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

   5. Taylor expanded in i around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 78.4

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

   7. Applied rewrites 78.4%

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

   9. Applied rewrites 89.5%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq -1 \cdot 10^{+102}:\\ \;\;\;\;\mathsf{fma}\left(-i, \mathsf{fma}\left(c, b, a\right) \cdot c, t \cdot z\right) \cdot 2\\ \mathbf{elif}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq 10^{+201}:\\ \;\;\;\;\mathsf{fma}\left(t, z, y \cdot x\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-2 \cdot c\right) \cdot i\right) \cdot \mathsf{fma}\left(c, b, a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := i \cdot \left(\left(c \cdot b + a\right) \cdot c\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+153}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(b, c, a\right) \cdot c\right) \cdot -2\right) \cdot i\\ \mathbf{elif}\;t\_1 \leq 10^{+201}:\\ \;\;\;\;\mathsf{fma}\left(t, z, y \cdot x\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-2 \cdot c\right) \cdot i\right) \cdot \mathsf{fma}\left(c, b, a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* i (* (+ (* c b) a) c))))
   (if (<= t_1 -1e+153)
     (* (* (* (fma b c a) c) -2.0) i)
     (if (<= t_1 1e+201)
       (* (fma t z (* y x)) 2.0)
       (* (* (* -2.0 c) i) (fma c b a))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = i * (((c * b) + a) * c);
	double tmp;
	if (t_1 <= -1e+153) {
		tmp = ((fma(b, c, a) * c) * -2.0) * i;
	} else if (t_1 <= 1e+201) {
		tmp = fma(t, z, (y * x)) * 2.0;
	} else {
		tmp = ((-2.0 * c) * i) * fma(c, b, a);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(i * Float64(Float64(Float64(c * b) + a) * c))
	tmp = 0.0
	if (t_1 <= -1e+153)
		tmp = Float64(Float64(Float64(fma(b, c, a) * c) * -2.0) * i);
	elseif (t_1 <= 1e+201)
		tmp = Float64(fma(t, z, Float64(y * x)) * 2.0);
	else
		tmp = Float64(Float64(Float64(-2.0 * c) * i) * fma(c, b, a));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(i * N[(N[(N[(c * b), $MachinePrecision] + a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+153], N[(N[(N[(N[(b * c + a), $MachinePrecision] * c), $MachinePrecision] * -2.0), $MachinePrecision] * i), $MachinePrecision], If[LessEqual[t$95$1, 1e+201], N[(N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(N[(-2.0 * c), $MachinePrecision] * i), $MachinePrecision] * N[(c * b + a), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -1e153

   1. Initial program 91.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-+.f64 N/A

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

      6. flip3-+ N/A

         \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(c \cdot i\right) \cdot \color{blue}{\frac{{a}^{3} + {\left(b \cdot c\right)}^{3}}{a \cdot a + \left(\left(b \cdot c\right) \cdot \left(b \cdot c\right) - a \cdot \left(b \cdot c\right)\right)}}\right) \]

      7. clear-num N/A

         \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(c \cdot i\right) \cdot \color{blue}{\frac{1}{\frac{a \cdot a + \left(\left(b \cdot c\right) \cdot \left(b \cdot c\right) - a \cdot \left(b \cdot c\right)\right)}{{a}^{3} + {\left(b \cdot c\right)}^{3}}}}\right) \]

      8. un-div-inv N/A

         \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \color{blue}{\frac{c \cdot i}{\frac{a \cdot a + \left(\left(b \cdot c\right) \cdot \left(b \cdot c\right) - a \cdot \left(b \cdot c\right)\right)}{{a}^{3} + {\left(b \cdot c\right)}^{3}}}}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \color{blue}{\frac{c \cdot i}{\frac{a \cdot a + \left(\left(b \cdot c\right) \cdot \left(b \cdot c\right) - a \cdot \left(b \cdot c\right)\right)}{{a}^{3} + {\left(b \cdot c\right)}^{3}}}}\right) \]

      10. *-commutative N/A

          \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \frac{\color{blue}{i \cdot c}}{\frac{a \cdot a + \left(\left(b \cdot c\right) \cdot \left(b \cdot c\right) - a \cdot \left(b \cdot c\right)\right)}{{a}^{3} + {\left(b \cdot c\right)}^{3}}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \frac{\color{blue}{i \cdot c}}{\frac{a \cdot a + \left(\left(b \cdot c\right) \cdot \left(b \cdot c\right) - a \cdot \left(b \cdot c\right)\right)}{{a}^{3} + {\left(b \cdot c\right)}^{3}}}\right) \]

      12. clear-num N/A

          \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \frac{i \cdot c}{\color{blue}{\frac{1}{\frac{{a}^{3} + {\left(b \cdot c\right)}^{3}}{a \cdot a + \left(\left(b \cdot c\right) \cdot \left(b \cdot c\right) - a \cdot \left(b \cdot c\right)\right)}}}}\right) \]

      13. flip3-+ N/A

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

      14. lift-+.f64 N/A

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

      15. lower-/.f64 92.6

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-fma.f64 92.6

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

   4. Applied rewrites 92.6%

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

   5. Taylor expanded in i around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 87.5

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

   7. Applied rewrites 87.5%

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

   if -1e153 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 1.00000000000000004e201

   1. Initial program 99.2%

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

   3. Taylor expanded in c around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 81.4

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

   5. Applied rewrites 81.4%

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

   if 1.00000000000000004e201 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 75.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-+.f64 N/A

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

      6. flip3-+ N/A

         \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(c \cdot i\right) \cdot \color{blue}{\frac{{a}^{3} + {\left(b \cdot c\right)}^{3}}{a \cdot a + \left(\left(b \cdot c\right) \cdot \left(b \cdot c\right) - a \cdot \left(b \cdot c\right)\right)}}\right) \]

      7. clear-num N/A

         \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(c \cdot i\right) \cdot \color{blue}{\frac{1}{\frac{a \cdot a + \left(\left(b \cdot c\right) \cdot \left(b \cdot c\right) - a \cdot \left(b \cdot c\right)\right)}{{a}^{3} + {\left(b \cdot c\right)}^{3}}}}\right) \]

      8. un-div-inv N/A

         \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \color{blue}{\frac{c \cdot i}{\frac{a \cdot a + \left(\left(b \cdot c\right) \cdot \left(b \cdot c\right) - a \cdot \left(b \cdot c\right)\right)}{{a}^{3} + {\left(b \cdot c\right)}^{3}}}}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \color{blue}{\frac{c \cdot i}{\frac{a \cdot a + \left(\left(b \cdot c\right) \cdot \left(b \cdot c\right) - a \cdot \left(b \cdot c\right)\right)}{{a}^{3} + {\left(b \cdot c\right)}^{3}}}}\right) \]

      10. *-commutative N/A

          \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \frac{\color{blue}{i \cdot c}}{\frac{a \cdot a + \left(\left(b \cdot c\right) \cdot \left(b \cdot c\right) - a \cdot \left(b \cdot c\right)\right)}{{a}^{3} + {\left(b \cdot c\right)}^{3}}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \frac{\color{blue}{i \cdot c}}{\frac{a \cdot a + \left(\left(b \cdot c\right) \cdot \left(b \cdot c\right) - a \cdot \left(b \cdot c\right)\right)}{{a}^{3} + {\left(b \cdot c\right)}^{3}}}\right) \]

      12. clear-num N/A

          \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \frac{i \cdot c}{\color{blue}{\frac{1}{\frac{{a}^{3} + {\left(b \cdot c\right)}^{3}}{a \cdot a + \left(\left(b \cdot c\right) \cdot \left(b \cdot c\right) - a \cdot \left(b \cdot c\right)\right)}}}}\right) \]

      13. flip3-+ N/A

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

      14. lift-+.f64 N/A

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

      15. lower-/.f64 89.7

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-fma.f64 89.7

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

   4. Applied rewrites 89.7%

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

   5. Taylor expanded in i around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 78.4

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

   7. Applied rewrites 78.4%

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

   9. Applied rewrites 89.5%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq -1 \cdot 10^{+153}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(b, c, a\right) \cdot c\right) \cdot -2\right) \cdot i\\ \mathbf{elif}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq 10^{+201}:\\ \;\;\;\;\mathsf{fma}\left(t, z, y \cdot x\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-2 \cdot c\right) \cdot i\right) \cdot \mathsf{fma}\left(c, b, a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := i \cdot \left(\left(c \cdot b + a\right) \cdot c\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+153}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(b, c, a\right) \cdot c\right) \cdot -2\right) \cdot i\\ \mathbf{elif}\;t\_1 \leq 10^{+201}:\\ \;\;\;\;\mathsf{fma}\left(t, z, y \cdot x\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot -2\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* i (* (+ (* c b) a) c))))
   (if (<= t_1 -1e+153)
     (* (* (* (fma b c a) c) -2.0) i)
     (if (<= t_1 1e+201)
       (* (fma t z (* y x)) 2.0)
       (* (* (* (fma c b a) i) -2.0) c)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = i * (((c * b) + a) * c);
	double tmp;
	if (t_1 <= -1e+153) {
		tmp = ((fma(b, c, a) * c) * -2.0) * i;
	} else if (t_1 <= 1e+201) {
		tmp = fma(t, z, (y * x)) * 2.0;
	} else {
		tmp = ((fma(c, b, a) * i) * -2.0) * c;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(i * Float64(Float64(Float64(c * b) + a) * c))
	tmp = 0.0
	if (t_1 <= -1e+153)
		tmp = Float64(Float64(Float64(fma(b, c, a) * c) * -2.0) * i);
	elseif (t_1 <= 1e+201)
		tmp = Float64(fma(t, z, Float64(y * x)) * 2.0);
	else
		tmp = Float64(Float64(Float64(fma(c, b, a) * i) * -2.0) * c);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(i * N[(N[(N[(c * b), $MachinePrecision] + a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+153], N[(N[(N[(N[(b * c + a), $MachinePrecision] * c), $MachinePrecision] * -2.0), $MachinePrecision] * i), $MachinePrecision], If[LessEqual[t$95$1, 1e+201], N[(N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(N[(N[(c * b + a), $MachinePrecision] * i), $MachinePrecision] * -2.0), $MachinePrecision] * c), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -1e153

   1. Initial program 91.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-+.f64 N/A

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

      6. flip3-+ N/A

         \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(c \cdot i\right) \cdot \color{blue}{\frac{{a}^{3} + {\left(b \cdot c\right)}^{3}}{a \cdot a + \left(\left(b \cdot c\right) \cdot \left(b \cdot c\right) - a \cdot \left(b \cdot c\right)\right)}}\right) \]

      7. clear-num N/A

         \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(c \cdot i\right) \cdot \color{blue}{\frac{1}{\frac{a \cdot a + \left(\left(b \cdot c\right) \cdot \left(b \cdot c\right) - a \cdot \left(b \cdot c\right)\right)}{{a}^{3} + {\left(b \cdot c\right)}^{3}}}}\right) \]

      8. un-div-inv N/A

         \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \color{blue}{\frac{c \cdot i}{\frac{a \cdot a + \left(\left(b \cdot c\right) \cdot \left(b \cdot c\right) - a \cdot \left(b \cdot c\right)\right)}{{a}^{3} + {\left(b \cdot c\right)}^{3}}}}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \color{blue}{\frac{c \cdot i}{\frac{a \cdot a + \left(\left(b \cdot c\right) \cdot \left(b \cdot c\right) - a \cdot \left(b \cdot c\right)\right)}{{a}^{3} + {\left(b \cdot c\right)}^{3}}}}\right) \]

      10. *-commutative N/A

          \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \frac{\color{blue}{i \cdot c}}{\frac{a \cdot a + \left(\left(b \cdot c\right) \cdot \left(b \cdot c\right) - a \cdot \left(b \cdot c\right)\right)}{{a}^{3} + {\left(b \cdot c\right)}^{3}}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \frac{\color{blue}{i \cdot c}}{\frac{a \cdot a + \left(\left(b \cdot c\right) \cdot \left(b \cdot c\right) - a \cdot \left(b \cdot c\right)\right)}{{a}^{3} + {\left(b \cdot c\right)}^{3}}}\right) \]

      12. clear-num N/A

          \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \frac{i \cdot c}{\color{blue}{\frac{1}{\frac{{a}^{3} + {\left(b \cdot c\right)}^{3}}{a \cdot a + \left(\left(b \cdot c\right) \cdot \left(b \cdot c\right) - a \cdot \left(b \cdot c\right)\right)}}}}\right) \]

      13. flip3-+ N/A

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

      14. lift-+.f64 N/A

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

      15. lower-/.f64 92.6

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-fma.f64 92.6

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

   4. Applied rewrites 92.6%

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

   5. Taylor expanded in i around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 87.5

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

   7. Applied rewrites 87.5%

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

   if -1e153 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 1.00000000000000004e201

   1. Initial program 99.2%

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

   3. Taylor expanded in c around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 81.4

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

   5. Applied rewrites 81.4%

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

   if 1.00000000000000004e201 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 75.7%

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

   3. Taylor expanded in 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 84.5

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

   5. Applied rewrites 84.5%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq -1 \cdot 10^{+153}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(b, c, a\right) \cdot c\right) \cdot -2\right) \cdot i\\ \mathbf{elif}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq 10^{+201}:\\ \;\;\;\;\mathsf{fma}\left(t, z, y \cdot x\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot -2\right) \cdot c\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 81.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot -2\right) \cdot c\\ t_2 := i \cdot \left(\left(c \cdot b + a\right) \cdot c\right)\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+153}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{+201}:\\ \;\;\;\;\mathsf{fma}\left(t, z, y \cdot x\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* (* (fma c b a) i) -2.0) c)) (t_2 (* i (* (+ (* c b) a) c))))
   (if (<= t_2 -1e+153)
     t_1
     (if (<= t_2 1e+201) (* (fma t z (* y x)) 2.0) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = ((fma(c, b, a) * i) * -2.0) * c;
	double t_2 = i * (((c * b) + a) * c);
	double tmp;
	if (t_2 <= -1e+153) {
		tmp = t_1;
	} else if (t_2 <= 1e+201) {
		tmp = fma(t, z, (y * x)) * 2.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(fma(c, b, a) * i) * -2.0) * c)
	t_2 = Float64(i * Float64(Float64(Float64(c * b) + a) * c))
	tmp = 0.0
	if (t_2 <= -1e+153)
		tmp = t_1;
	elseif (t_2 <= 1e+201)
		tmp = Float64(fma(t, z, Float64(y * x)) * 2.0);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(N[(c * b + a), $MachinePrecision] * i), $MachinePrecision] * -2.0), $MachinePrecision] * c), $MachinePrecision]}, Block[{t$95$2 = N[(i * N[(N[(N[(c * b), $MachinePrecision] + a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+153], t$95$1, If[LessEqual[t$95$2, 1e+201], N[(N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -1e153 or 1.00000000000000004e201 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 82.5%

      \[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]
   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 84.3

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

   5. Applied rewrites 84.3%

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

   if -1e153 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 1.00000000000000004e201

   1. Initial program 99.2%

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

   3. Taylor expanded in c around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 81.4

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

   5. Applied rewrites 81.4%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq -1 \cdot 10^{+153}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot -2\right) \cdot c\\ \mathbf{elif}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq 10^{+201}:\\ \;\;\;\;\mathsf{fma}\left(t, z, y \cdot x\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot -2\right) \cdot c\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 72.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\left(c \cdot c\right) \cdot i\right) \cdot b\right) \cdot -2\\ t_2 := i \cdot \left(\left(c \cdot b + a\right) \cdot c\right)\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+154}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{+201}:\\ \;\;\;\;\mathsf{fma}\left(t, z, y \cdot x\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* (* (* c c) i) b) -2.0)) (t_2 (* i (* (+ (* c b) a) c))))
   (if (<= t_2 -1e+154)
     t_1
     (if (<= t_2 1e+201) (* (fma t z (* y x)) 2.0) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (((c * c) * i) * b) * -2.0;
	double t_2 = i * (((c * b) + a) * c);
	double tmp;
	if (t_2 <= -1e+154) {
		tmp = t_1;
	} else if (t_2 <= 1e+201) {
		tmp = fma(t, z, (y * x)) * 2.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(Float64(c * c) * i) * b) * -2.0)
	t_2 = Float64(i * Float64(Float64(Float64(c * b) + a) * c))
	tmp = 0.0
	if (t_2 <= -1e+154)
		tmp = t_1;
	elseif (t_2 <= 1e+201)
		tmp = Float64(fma(t, z, Float64(y * x)) * 2.0);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(N[(c * c), $MachinePrecision] * i), $MachinePrecision] * b), $MachinePrecision] * -2.0), $MachinePrecision]}, Block[{t$95$2 = N[(i * N[(N[(N[(c * b), $MachinePrecision] + a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+154], t$95$1, If[LessEqual[t$95$2, 1e+201], N[(N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -1.00000000000000004e154 or 1.00000000000000004e201 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 82.3%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

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

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

      8. lower-fma.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

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

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

      15. lower-*.f64 N/A

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

      16. lower-neg.f64 90.9

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-fma.f64 90.9

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

      22. lift-*.f64 N/A

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

      23. *-commutative N/A

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

      24. lower-*.f64 90.9

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

   4. Applied rewrites 90.9%

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

   5. Taylor expanded in b around inf

      \[\leadsto \color{blue}{-2 \cdot \left(b \cdot \left({c}^{2} \cdot i\right)\right)} \]
   6. 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 64.4

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

   7. Applied rewrites 64.4%

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

   if -1.00000000000000004e154 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 1.00000000000000004e201

   1. Initial program 99.2%

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

   3. Taylor expanded in c around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 80.9

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

   5. Applied rewrites 80.9%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq -1 \cdot 10^{+154}:\\ \;\;\;\;\left(\left(\left(c \cdot c\right) \cdot i\right) \cdot b\right) \cdot -2\\ \mathbf{elif}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq 10^{+201}:\\ \;\;\;\;\mathsf{fma}\left(t, z, y \cdot x\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(c \cdot c\right) \cdot i\right) \cdot b\right) \cdot -2\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 72.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := i \cdot \left(\left(c \cdot b + a\right) \cdot c\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+199}:\\ \;\;\;\;\left(\left(\left(i \cdot b\right) \cdot c\right) \cdot c\right) \cdot -2\\ \mathbf{elif}\;t\_1 \leq 10^{+201}:\\ \;\;\;\;\mathsf{fma}\left(t, z, y \cdot x\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(i \cdot c\right) \cdot \left(c \cdot b\right)\right) \cdot -2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* i (* (+ (* c b) a) c))))
   (if (<= t_1 -5e+199)
     (* (* (* (* i b) c) c) -2.0)
     (if (<= t_1 1e+201)
       (* (fma t z (* y x)) 2.0)
       (* (* (* i c) (* c b)) -2.0)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = i * (((c * b) + a) * c);
	double tmp;
	if (t_1 <= -5e+199) {
		tmp = (((i * b) * c) * c) * -2.0;
	} else if (t_1 <= 1e+201) {
		tmp = fma(t, z, (y * x)) * 2.0;
	} else {
		tmp = ((i * c) * (c * b)) * -2.0;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(i * Float64(Float64(Float64(c * b) + a) * c))
	tmp = 0.0
	if (t_1 <= -5e+199)
		tmp = Float64(Float64(Float64(Float64(i * b) * c) * c) * -2.0);
	elseif (t_1 <= 1e+201)
		tmp = Float64(fma(t, z, Float64(y * x)) * 2.0);
	else
		tmp = Float64(Float64(Float64(i * c) * Float64(c * b)) * -2.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(i * N[(N[(N[(c * b), $MachinePrecision] + a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+199], N[(N[(N[(N[(i * b), $MachinePrecision] * c), $MachinePrecision] * c), $MachinePrecision] * -2.0), $MachinePrecision], If[LessEqual[t$95$1, 1e+201], N[(N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(N[(i * c), $MachinePrecision] * N[(c * b), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -4.9999999999999998e199

   1. Initial program 90.0%

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

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

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

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

      8. lower-fma.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

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

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

      15. lower-*.f64 N/A

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

      16. lower-neg.f64 92.0

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-fma.f64 92.0

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

      22. lift-*.f64 N/A

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

      23. *-commutative N/A

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

      24. lower-*.f64 92.0

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

   4. Applied rewrites 92.0%

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

   5. Taylor expanded in b around inf

      \[\leadsto \color{blue}{-2 \cdot \left(b \cdot \left({c}^{2} \cdot i\right)\right)} \]
   6. 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 69.7

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

   7. Applied rewrites 69.7%

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

   9. Applied rewrites 67.7%

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

   if -4.9999999999999998e199 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 1.00000000000000004e201

   1. Initial program 99.2%

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

   3. Taylor expanded in c around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 79.3

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

   5. Applied rewrites 79.3%

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

   if 1.00000000000000004e201 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 75.7%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

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

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

      8. lower-fma.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

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

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

      15. lower-*.f64 N/A

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

      16. lower-neg.f64 89.6

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-fma.f64 89.6

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

      22. lift-*.f64 N/A

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

      23. *-commutative N/A

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

      24. lower-*.f64 89.6

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

   4. Applied rewrites 89.6%

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

   5. Taylor expanded in b around inf

      \[\leadsto \color{blue}{-2 \cdot \left(b \cdot \left({c}^{2} \cdot i\right)\right)} \]
   6. 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 62.8

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

   7. Applied rewrites 62.8%

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

   9. Applied rewrites 61.2%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq -5 \cdot 10^{+199}:\\ \;\;\;\;\left(\left(\left(i \cdot b\right) \cdot c\right) \cdot c\right) \cdot -2\\ \mathbf{elif}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq 10^{+201}:\\ \;\;\;\;\mathsf{fma}\left(t, z, y \cdot x\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(i \cdot c\right) \cdot \left(c \cdot b\right)\right) \cdot -2\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 73.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\left(i \cdot b\right) \cdot c\right) \cdot c\right) \cdot -2\\ t_2 := i \cdot \left(\left(c \cdot b + a\right) \cdot c\right)\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+199}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{+201}:\\ \;\;\;\;\mathsf{fma}\left(t, z, y \cdot x\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* (* (* i b) c) c) -2.0)) (t_2 (* i (* (+ (* c b) a) c))))
   (if (<= t_2 -5e+199)
     t_1
     (if (<= t_2 1e+201) (* (fma t z (* y x)) 2.0) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (((i * b) * c) * c) * -2.0;
	double t_2 = i * (((c * b) + a) * c);
	double tmp;
	if (t_2 <= -5e+199) {
		tmp = t_1;
	} else if (t_2 <= 1e+201) {
		tmp = fma(t, z, (y * x)) * 2.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(Float64(i * b) * c) * c) * -2.0)
	t_2 = Float64(i * Float64(Float64(Float64(c * b) + a) * c))
	tmp = 0.0
	if (t_2 <= -5e+199)
		tmp = t_1;
	elseif (t_2 <= 1e+201)
		tmp = Float64(fma(t, z, Float64(y * x)) * 2.0);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(N[(i * b), $MachinePrecision] * c), $MachinePrecision] * c), $MachinePrecision] * -2.0), $MachinePrecision]}, Block[{t$95$2 = N[(i * N[(N[(N[(c * b), $MachinePrecision] + a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+199], t$95$1, If[LessEqual[t$95$2, 1e+201], N[(N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -4.9999999999999998e199 or 1.00000000000000004e201 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 81.7%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

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

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

      8. lower-fma.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

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

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

      15. lower-*.f64 N/A

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

      16. lower-neg.f64 90.6

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-fma.f64 90.6

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

      22. lift-*.f64 N/A

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

      23. *-commutative N/A

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

      24. lower-*.f64 90.6

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

   4. Applied rewrites 90.6%

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

   5. Taylor expanded in b around inf

      \[\leadsto \color{blue}{-2 \cdot \left(b \cdot \left({c}^{2} \cdot i\right)\right)} \]
   6. 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 65.7

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

   7. Applied rewrites 65.7%

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

   9. Applied rewrites 62.0%

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

   if -4.9999999999999998e199 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 1.00000000000000004e201

   1. Initial program 99.2%

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

   3. Taylor expanded in c around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 79.3

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

   5. Applied rewrites 79.3%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq -5 \cdot 10^{+199}:\\ \;\;\;\;\left(\left(\left(i \cdot b\right) \cdot c\right) \cdot c\right) \cdot -2\\ \mathbf{elif}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq 10^{+201}:\\ \;\;\;\;\mathsf{fma}\left(t, z, y \cdot x\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(i \cdot b\right) \cdot c\right) \cdot c\right) \cdot -2\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 63.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(i \cdot c\right) \cdot a\right) \cdot -2\\ t_2 := i \cdot \left(\left(c \cdot b + a\right) \cdot c\right)\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+153}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+212}:\\ \;\;\;\;\mathsf{fma}\left(t, z, y \cdot x\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* (* i c) a) -2.0)) (t_2 (* i (* (+ (* c b) a) c))))
   (if (<= t_2 -1e+153)
     t_1
     (if (<= t_2 5e+212) (* (fma t z (* y x)) 2.0) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = ((i * c) * a) * -2.0;
	double t_2 = i * (((c * b) + a) * c);
	double tmp;
	if (t_2 <= -1e+153) {
		tmp = t_1;
	} else if (t_2 <= 5e+212) {
		tmp = fma(t, z, (y * x)) * 2.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(i * c) * a) * -2.0)
	t_2 = Float64(i * Float64(Float64(Float64(c * b) + a) * c))
	tmp = 0.0
	if (t_2 <= -1e+153)
		tmp = t_1;
	elseif (t_2 <= 5e+212)
		tmp = Float64(fma(t, z, Float64(y * x)) * 2.0);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(i * c), $MachinePrecision] * a), $MachinePrecision] * -2.0), $MachinePrecision]}, Block[{t$95$2 = N[(i * N[(N[(N[(c * b), $MachinePrecision] + a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+153], t$95$1, If[LessEqual[t$95$2, 5e+212], N[(N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -1e153 or 4.99999999999999992e212 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 82.2%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 46.9

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

   5. Applied rewrites 46.9%

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

   if -1e153 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 4.99999999999999992e212

   1. Initial program 99.2%

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

   3. Taylor expanded in c around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 80.3

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

   5. Applied rewrites 80.3%

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

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq -1 \cdot 10^{+153}:\\ \;\;\;\;\left(\left(i \cdot c\right) \cdot a\right) \cdot -2\\ \mathbf{elif}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq 5 \cdot 10^{+212}:\\ \;\;\;\;\mathsf{fma}\left(t, z, y \cdot x\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(i \cdot c\right) \cdot a\right) \cdot -2\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 92.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 91.3%

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

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

   5. sub-neg N/A

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

   6. associate-+l+ N/A

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

   7. lift-*.f64 N/A

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

   8. lower-fma.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lift-*.f64 N/A

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

   12. associate-*r* N/A

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

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

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

   14. lower-fma.f64 N/A

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

4. Applied rewrites 93.6%

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

5. Final simplification 93.6%

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

Alternative 14: 44.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 \cdot \left(t \cdot z\right)\\ \mathbf{if}\;t \cdot z \leq -5 \cdot 10^{+88}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \cdot z \leq 2 \cdot 10^{+142}:\\ \;\;\;\;\left(y \cdot x\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* 2.0 (* t z))))
   (if (<= (* t z) -5e+88) t_1 (if (<= (* t z) 2e+142) (* (* y x) 2.0) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = 2.0 * (t * z);
	double tmp;
	if ((t * z) <= -5e+88) {
		tmp = t_1;
	} else if ((t * z) <= 2e+142) {
		tmp = (y * x) * 2.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 2.0d0 * (t * z)
    if ((t * z) <= (-5d+88)) then
        tmp = t_1
    else if ((t * z) <= 2d+142) then
        tmp = (y * x) * 2.0d0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = 2.0 * (t * z);
	double tmp;
	if ((t * z) <= -5e+88) {
		tmp = t_1;
	} else if ((t * z) <= 2e+142) {
		tmp = (y * x) * 2.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = 2.0 * (t * z)
	tmp = 0
	if (t * z) <= -5e+88:
		tmp = t_1
	elif (t * z) <= 2e+142:
		tmp = (y * x) * 2.0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(2.0 * Float64(t * z))
	tmp = 0.0
	if (Float64(t * z) <= -5e+88)
		tmp = t_1;
	elseif (Float64(t * z) <= 2e+142)
		tmp = Float64(Float64(y * x) * 2.0);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	t_1 = 2.0 * (t * z);
	tmp = 0.0;
	if ((t * z) <= -5e+88)
		tmp = t_1;
	elseif ((t * z) <= 2e+142)
		tmp = (y * x) * 2.0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(2.0 * N[(t * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t * z), $MachinePrecision], -5e+88], t$95$1, If[LessEqual[N[(t * z), $MachinePrecision], 2e+142], N[(N[(y * x), $MachinePrecision] * 2.0), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -4.99999999999999997e88 or 2.0000000000000001e142 < (*.f64 z t)

   1. Initial program 90.6%

      \[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]
   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 58.7

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

   5. Applied rewrites 58.7%

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

   if -4.99999999999999997e88 < (*.f64 z t) < 2.0000000000000001e142

   1. Initial program 91.7%

      \[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]
   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 39.7

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

   5. Applied rewrites 39.7%

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

4. Final simplification 45.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot z \leq -5 \cdot 10^{+88}:\\ \;\;\;\;2 \cdot \left(t \cdot z\right)\\ \mathbf{elif}\;t \cdot z \leq 2 \cdot 10^{+142}:\\ \;\;\;\;\left(y \cdot x\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 29.8% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.3%

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

3. Taylor expanded in z around 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 24.0

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

5. Applied rewrites 24.0%

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

Developer Target 1: 94.1% 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

real(8) function code(x, y, z, t, a, b, c, i)
    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 2024332 
(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))))