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

Percentage Accurate: 90.2% → 96.0%

Time: 13.9s

Alternatives: 16

Speedup: 0.8×

• 47.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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: 90.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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: 96.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z, t, \mathsf{fma}\left(y, x, \left(-c\right) \cdot \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right)\right)\right) \cdot 2\\ \mathbf{if}\;c \leq -5.5 \cdot 10^{-73}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;c \leq 6.5 \cdot 10^{-163}:\\ \;\;\;\;\mathsf{fma}\left(\left(-a\right) \cdot c, i, \mathsf{fma}\left(y, x, t \cdot z\right)\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (fma z t (fma y x (* (- c) (* (fma c b a) i)))) 2.0)))
   (if (<= c -5.5e-73)
     t_1
     (if (<= c 6.5e-163) (* (fma (* (- a) c) i (fma y x (* t z))) 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(z, t, fma(y, x, (-c * (fma(c, b, a) * i)))) * 2.0;
	double tmp;
	if (c <= -5.5e-73) {
		tmp = t_1;
	} else if (c <= 6.5e-163) {
		tmp = fma((-a * c), i, fma(y, x, (t * z))) * 2.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(fma(z, t, fma(y, x, Float64(Float64(-c) * Float64(fma(c, b, a) * i)))) * 2.0)
	tmp = 0.0
	if (c <= -5.5e-73)
		tmp = t_1;
	elseif (c <= 6.5e-163)
		tmp = Float64(fma(Float64(Float64(-a) * c), i, fma(y, x, Float64(t * z))) * 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[(z * t + N[(y * x + N[((-c) * N[(N[(c * b + a), $MachinePrecision] * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]}, If[LessEqual[c, -5.5e-73], t$95$1, If[LessEqual[c, 6.5e-163], N[(N[(N[((-a) * c), $MachinePrecision] * i + N[(y * x + N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if c < -5.50000000000000006e-73 or 6.4999999999999999e-163 < c

   1. Initial program 90.5%

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

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

      4. associate--l+ N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. sub-neg N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

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

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

      16. lower-*.f64 N/A

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

   4. Applied rewrites 97.2%

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

   if -5.50000000000000006e-73 < c < 6.4999999999999999e-163

   1. Initial program 93.5%

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

   3. Taylor expanded in c around 0

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. neg-mul-1 N/A

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

      7. lower-neg.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 96.5

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

   5. Applied rewrites 96.5%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -5.5 \cdot 10^{-73}:\\ \;\;\;\;\mathsf{fma}\left(z, t, \mathsf{fma}\left(y, x, \left(-c\right) \cdot \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right)\right)\right) \cdot 2\\ \mathbf{elif}\;c \leq 6.5 \cdot 10^{-163}:\\ \;\;\;\;\mathsf{fma}\left(\left(-a\right) \cdot c, i, \mathsf{fma}\left(y, x, t \cdot z\right)\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t, \mathsf{fma}\left(y, x, \left(-c\right) \cdot \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right)\right)\right) \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 41.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(i \cdot c\right) \cdot a\right) \cdot -2\\ t_2 := \left(\left(b \cdot c + a\right) \cdot c\right) \cdot i\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+88}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{-163}:\\ \;\;\;\;\left(t \cdot z\right) \cdot 2\\ \mathbf{elif}\;t\_2 \leq 1000000000:\\ \;\;\;\;\left(x \cdot y\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* (* i c) a) -2.0)) (t_2 (* (* (+ (* b c) a) c) i)))
   (if (<= t_2 -2e+88)
     t_1
     (if (<= t_2 -5e-163)
       (* (* t z) 2.0)
       (if (<= t_2 1000000000.0) (* (* x y) 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 = (((b * c) + a) * c) * i;
	double tmp;
	if (t_2 <= -2e+88) {
		tmp = t_1;
	} else if (t_2 <= -5e-163) {
		tmp = (t * z) * 2.0;
	} else if (t_2 <= 1000000000.0) {
		tmp = (x * y) * 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) :: t_2
    real(8) :: tmp
    t_1 = ((i * c) * a) * (-2.0d0)
    t_2 = (((b * c) + a) * c) * i
    if (t_2 <= (-2d+88)) then
        tmp = t_1
    else if (t_2 <= (-5d-163)) then
        tmp = (t * z) * 2.0d0
    else if (t_2 <= 1000000000.0d0) then
        tmp = (x * y) * 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 = ((i * c) * a) * -2.0;
	double t_2 = (((b * c) + a) * c) * i;
	double tmp;
	if (t_2 <= -2e+88) {
		tmp = t_1;
	} else if (t_2 <= -5e-163) {
		tmp = (t * z) * 2.0;
	} else if (t_2 <= 1000000000.0) {
		tmp = (x * y) * 2.0;
	} 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 = (((b * c) + a) * c) * i
	tmp = 0
	if t_2 <= -2e+88:
		tmp = t_1
	elif t_2 <= -5e-163:
		tmp = (t * z) * 2.0
	elif t_2 <= 1000000000.0:
		tmp = (x * y) * 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(Float64(Float64(Float64(b * c) + a) * c) * i)
	tmp = 0.0
	if (t_2 <= -2e+88)
		tmp = t_1;
	elseif (t_2 <= -5e-163)
		tmp = Float64(Float64(t * z) * 2.0);
	elseif (t_2 <= 1000000000.0)
		tmp = Float64(Float64(x * y) * 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 = ((i * c) * a) * -2.0;
	t_2 = (((b * c) + a) * c) * i;
	tmp = 0.0;
	if (t_2 <= -2e+88)
		tmp = t_1;
	elseif (t_2 <= -5e-163)
		tmp = (t * z) * 2.0;
	elseif (t_2 <= 1000000000.0)
		tmp = (x * y) * 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[(N[(N[(i * c), $MachinePrecision] * a), $MachinePrecision] * -2.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(b * c), $MachinePrecision] + a), $MachinePrecision] * c), $MachinePrecision] * i), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+88], t$95$1, If[LessEqual[t$95$2, -5e-163], N[(N[(t * z), $MachinePrecision] * 2.0), $MachinePrecision], If[LessEqual[t$95$2, 1000000000.0], N[(N[(x * y), $MachinePrecision] * 2.0), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -1.99999999999999992e88 or 1e9 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 86.3%

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

   3. Taylor expanded in 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 47.3

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

   5. Applied rewrites 47.3%

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

   if -1.99999999999999992e88 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -4.99999999999999977e-163

   1. Initial program 96.8%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 51.4

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

   5. Applied rewrites 51.4%

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

   if -4.99999999999999977e-163 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 1e9

   1. Initial program 98.7%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 61.8

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

   5. Applied rewrites 61.8%

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

4. Final simplification 52.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(b \cdot c + a\right) \cdot c\right) \cdot i \leq -2 \cdot 10^{+88}:\\ \;\;\;\;\left(\left(i \cdot c\right) \cdot a\right) \cdot -2\\ \mathbf{elif}\;\left(\left(b \cdot c + a\right) \cdot c\right) \cdot i \leq -5 \cdot 10^{-163}:\\ \;\;\;\;\left(t \cdot z\right) \cdot 2\\ \mathbf{elif}\;\left(\left(b \cdot c + a\right) \cdot c\right) \cdot i \leq 1000000000:\\ \;\;\;\;\left(x \cdot y\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(i \cdot c\right) \cdot a\right) \cdot -2\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 39.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(b \cdot c + a\right) \cdot c\right) \cdot i\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+88}:\\ \;\;\;\;\left(\left(a \cdot i\right) \cdot c\right) \cdot -2\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-163}:\\ \;\;\;\;\left(t \cdot z\right) \cdot 2\\ \mathbf{elif}\;t\_1 \leq 1000000000:\\ \;\;\;\;\left(x \cdot y\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a \cdot c\right) \cdot i\right) \cdot -2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* (+ (* b c) a) c) i)))
   (if (<= t_1 -2e+88)
     (* (* (* a i) c) -2.0)
     (if (<= t_1 -5e-163)
       (* (* t z) 2.0)
       (if (<= t_1 1000000000.0) (* (* x y) 2.0) (* (* (* a c) i) -2.0))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (((b * c) + a) * c) * i;
	double tmp;
	if (t_1 <= -2e+88) {
		tmp = ((a * i) * c) * -2.0;
	} else if (t_1 <= -5e-163) {
		tmp = (t * z) * 2.0;
	} else if (t_1 <= 1000000000.0) {
		tmp = (x * y) * 2.0;
	} else {
		tmp = ((a * c) * i) * -2.0;
	}
	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 = (((b * c) + a) * c) * i
    if (t_1 <= (-2d+88)) then
        tmp = ((a * i) * c) * (-2.0d0)
    else if (t_1 <= (-5d-163)) then
        tmp = (t * z) * 2.0d0
    else if (t_1 <= 1000000000.0d0) then
        tmp = (x * y) * 2.0d0
    else
        tmp = ((a * c) * i) * (-2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (((b * c) + a) * c) * i;
	double tmp;
	if (t_1 <= -2e+88) {
		tmp = ((a * i) * c) * -2.0;
	} else if (t_1 <= -5e-163) {
		tmp = (t * z) * 2.0;
	} else if (t_1 <= 1000000000.0) {
		tmp = (x * y) * 2.0;
	} else {
		tmp = ((a * c) * i) * -2.0;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = (((b * c) + a) * c) * i
	tmp = 0
	if t_1 <= -2e+88:
		tmp = ((a * i) * c) * -2.0
	elif t_1 <= -5e-163:
		tmp = (t * z) * 2.0
	elif t_1 <= 1000000000.0:
		tmp = (x * y) * 2.0
	else:
		tmp = ((a * c) * i) * -2.0
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(Float64(b * c) + a) * c) * i)
	tmp = 0.0
	if (t_1 <= -2e+88)
		tmp = Float64(Float64(Float64(a * i) * c) * -2.0);
	elseif (t_1 <= -5e-163)
		tmp = Float64(Float64(t * z) * 2.0);
	elseif (t_1 <= 1000000000.0)
		tmp = Float64(Float64(x * y) * 2.0);
	else
		tmp = Float64(Float64(Float64(a * c) * i) * -2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	t_1 = (((b * c) + a) * c) * i;
	tmp = 0.0;
	if (t_1 <= -2e+88)
		tmp = ((a * i) * c) * -2.0;
	elseif (t_1 <= -5e-163)
		tmp = (t * z) * 2.0;
	elseif (t_1 <= 1000000000.0)
		tmp = (x * y) * 2.0;
	else
		tmp = ((a * c) * i) * -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(N[(b * c), $MachinePrecision] + a), $MachinePrecision] * c), $MachinePrecision] * i), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+88], N[(N[(N[(a * i), $MachinePrecision] * c), $MachinePrecision] * -2.0), $MachinePrecision], If[LessEqual[t$95$1, -5e-163], N[(N[(t * z), $MachinePrecision] * 2.0), $MachinePrecision], If[LessEqual[t$95$1, 1000000000.0], N[(N[(x * y), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(N[(a * c), $MachinePrecision] * i), $MachinePrecision] * -2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -1.99999999999999992e88

   1. Initial program 87.9%

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

   3. Taylor expanded in 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 53.7

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

   5. Applied rewrites 53.7%

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

   7. Applied rewrites 49.5%

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

   if -1.99999999999999992e88 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -4.99999999999999977e-163

   1. Initial program 96.8%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 51.4

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

   5. Applied rewrites 51.4%

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

   if -4.99999999999999977e-163 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 1e9

   1. Initial program 98.7%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 61.8

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

   5. Applied rewrites 61.8%

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

   if 1e9 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 85.0%

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

   3. Taylor expanded in 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 42.0

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

   5. Applied rewrites 42.0%

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

   7. Applied rewrites 39.6%

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

4. Final simplification 50.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(b \cdot c + a\right) \cdot c\right) \cdot i \leq -2 \cdot 10^{+88}:\\ \;\;\;\;\left(\left(a \cdot i\right) \cdot c\right) \cdot -2\\ \mathbf{elif}\;\left(\left(b \cdot c + a\right) \cdot c\right) \cdot i \leq -5 \cdot 10^{-163}:\\ \;\;\;\;\left(t \cdot z\right) \cdot 2\\ \mathbf{elif}\;\left(\left(b \cdot c + a\right) \cdot c\right) \cdot i \leq 1000000000:\\ \;\;\;\;\left(x \cdot y\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a \cdot c\right) \cdot i\right) \cdot -2\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 39.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(a \cdot c\right) \cdot i\right) \cdot -2\\ t_2 := \left(\left(b \cdot c + a\right) \cdot c\right) \cdot i\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+88}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{-163}:\\ \;\;\;\;\left(t \cdot z\right) \cdot 2\\ \mathbf{elif}\;t\_2 \leq 1000000000:\\ \;\;\;\;\left(x \cdot y\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* (* a c) i) -2.0)) (t_2 (* (* (+ (* b c) a) c) i)))
   (if (<= t_2 -2e+88)
     t_1
     (if (<= t_2 -5e-163)
       (* (* t z) 2.0)
       (if (<= t_2 1000000000.0) (* (* x y) 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 = ((a * c) * i) * -2.0;
	double t_2 = (((b * c) + a) * c) * i;
	double tmp;
	if (t_2 <= -2e+88) {
		tmp = t_1;
	} else if (t_2 <= -5e-163) {
		tmp = (t * z) * 2.0;
	} else if (t_2 <= 1000000000.0) {
		tmp = (x * y) * 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) :: t_2
    real(8) :: tmp
    t_1 = ((a * c) * i) * (-2.0d0)
    t_2 = (((b * c) + a) * c) * i
    if (t_2 <= (-2d+88)) then
        tmp = t_1
    else if (t_2 <= (-5d-163)) then
        tmp = (t * z) * 2.0d0
    else if (t_2 <= 1000000000.0d0) then
        tmp = (x * y) * 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 = ((a * c) * i) * -2.0;
	double t_2 = (((b * c) + a) * c) * i;
	double tmp;
	if (t_2 <= -2e+88) {
		tmp = t_1;
	} else if (t_2 <= -5e-163) {
		tmp = (t * z) * 2.0;
	} else if (t_2 <= 1000000000.0) {
		tmp = (x * y) * 2.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = ((a * c) * i) * -2.0
	t_2 = (((b * c) + a) * c) * i
	tmp = 0
	if t_2 <= -2e+88:
		tmp = t_1
	elif t_2 <= -5e-163:
		tmp = (t * z) * 2.0
	elif t_2 <= 1000000000.0:
		tmp = (x * y) * 2.0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(a * c) * i) * -2.0)
	t_2 = Float64(Float64(Float64(Float64(b * c) + a) * c) * i)
	tmp = 0.0
	if (t_2 <= -2e+88)
		tmp = t_1;
	elseif (t_2 <= -5e-163)
		tmp = Float64(Float64(t * z) * 2.0);
	elseif (t_2 <= 1000000000.0)
		tmp = Float64(Float64(x * y) * 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 = ((a * c) * i) * -2.0;
	t_2 = (((b * c) + a) * c) * i;
	tmp = 0.0;
	if (t_2 <= -2e+88)
		tmp = t_1;
	elseif (t_2 <= -5e-163)
		tmp = (t * z) * 2.0;
	elseif (t_2 <= 1000000000.0)
		tmp = (x * y) * 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[(N[(N[(a * c), $MachinePrecision] * i), $MachinePrecision] * -2.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(b * c), $MachinePrecision] + a), $MachinePrecision] * c), $MachinePrecision] * i), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+88], t$95$1, If[LessEqual[t$95$2, -5e-163], N[(N[(t * z), $MachinePrecision] * 2.0), $MachinePrecision], If[LessEqual[t$95$2, 1000000000.0], N[(N[(x * y), $MachinePrecision] * 2.0), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -1.99999999999999992e88 or 1e9 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 86.3%

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

   3. Taylor expanded in 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 47.3

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

   5. Applied rewrites 47.3%

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

   7. Applied rewrites 44.1%

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

   if -1.99999999999999992e88 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -4.99999999999999977e-163

   1. Initial program 96.8%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 51.4

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

   5. Applied rewrites 51.4%

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

   if -4.99999999999999977e-163 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 1e9

   1. Initial program 98.7%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 61.8

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

   5. Applied rewrites 61.8%

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

4. Final simplification 50.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(b \cdot c + a\right) \cdot c\right) \cdot i \leq -2 \cdot 10^{+88}:\\ \;\;\;\;\left(\left(a \cdot c\right) \cdot i\right) \cdot -2\\ \mathbf{elif}\;\left(\left(b \cdot c + a\right) \cdot c\right) \cdot i \leq -5 \cdot 10^{-163}:\\ \;\;\;\;\left(t \cdot z\right) \cdot 2\\ \mathbf{elif}\;\left(\left(b \cdot c + a\right) \cdot c\right) \cdot i \leq 1000000000:\\ \;\;\;\;\left(x \cdot y\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a \cdot c\right) \cdot i\right) \cdot -2\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\mathsf{fma}\left(b, c, a\right) \cdot i, -c, x \cdot y\right) \cdot 2\\ t_2 := \left(\left(b \cdot c + a\right) \cdot c\right) \cdot i\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+132}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+118}:\\ \;\;\;\;\mathsf{fma}\left(\left(-a\right) \cdot c, i, \mathsf{fma}\left(y, x, t \cdot z\right)\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (fma (* (fma b c a) i) (- c) (* x y)) 2.0))
        (t_2 (* (* (+ (* b c) a) c) i)))
   (if (<= t_2 -5e+132)
     t_1
     (if (<= t_2 2e+118) (* (fma (* (- a) c) i (fma y x (* t z))) 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((fma(b, c, a) * i), -c, (x * y)) * 2.0;
	double t_2 = (((b * c) + a) * c) * i;
	double tmp;
	if (t_2 <= -5e+132) {
		tmp = t_1;
	} else if (t_2 <= 2e+118) {
		tmp = fma((-a * c), i, fma(y, x, (t * z))) * 2.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(fma(Float64(fma(b, c, a) * i), Float64(-c), Float64(x * y)) * 2.0)
	t_2 = Float64(Float64(Float64(Float64(b * c) + a) * c) * i)
	tmp = 0.0
	if (t_2 <= -5e+132)
		tmp = t_1;
	elseif (t_2 <= 2e+118)
		tmp = Float64(fma(Float64(Float64(-a) * c), i, fma(y, x, Float64(t * z))) * 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[(b * c + a), $MachinePrecision] * i), $MachinePrecision] * (-c) + N[(x * y), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(b * c), $MachinePrecision] + a), $MachinePrecision] * c), $MachinePrecision] * i), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+132], t$95$1, If[LessEqual[t$95$2, 2e+118], N[(N[(N[((-a) * c), $MachinePrecision] * i + N[(y * x + N[(t * z), $MachinePrecision]), $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) < -5.0000000000000001e132 or 1.99999999999999993e118 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 85.1%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 13.8

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

   5. Applied rewrites 13.8%

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

   6. Taylor expanded in t around 0

      \[\leadsto 2 \cdot \color{blue}{\left(x \cdot y - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]
   7. 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. mul-1-neg N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 86.8

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

   8. Applied rewrites 86.8%

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

   if -5.0000000000000001e132 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 1.99999999999999993e118

   1. Initial program 97.5%

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

   3. Taylor expanded in c around 0

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. neg-mul-1 N/A

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

      7. lower-neg.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 94.1

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

   5. Applied rewrites 94.1%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(b \cdot c + a\right) \cdot c\right) \cdot i \leq -5 \cdot 10^{+132}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, c, a\right) \cdot i, -c, x \cdot y\right) \cdot 2\\ \mathbf{elif}\;\left(\left(b \cdot c + a\right) \cdot c\right) \cdot i \leq 2 \cdot 10^{+118}:\\ \;\;\;\;\mathsf{fma}\left(\left(-a\right) \cdot c, i, \mathsf{fma}\left(y, x, t \cdot z\right)\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, c, a\right) \cdot i, -c, x \cdot y\right) \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 82.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-i, \mathsf{fma}\left(c, b, a\right) \cdot c, t \cdot z\right) \cdot 2\\ t_2 := \left(\left(b \cdot c + a\right) \cdot c\right) \cdot i\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{-107}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{-23}:\\ \;\;\;\;\mathsf{fma}\left(y, x, t \cdot z\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (fma (- i) (* (fma c b a) c) (* t z)) 2.0))
        (t_2 (* (* (+ (* b c) a) c) i)))
   (if (<= t_2 -1e-107)
     t_1
     (if (<= t_2 1e-23) (* (fma y x (* t z)) 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(-i, (fma(c, b, a) * c), (t * z)) * 2.0;
	double t_2 = (((b * c) + a) * c) * i;
	double tmp;
	if (t_2 <= -1e-107) {
		tmp = t_1;
	} else if (t_2 <= 1e-23) {
		tmp = fma(y, x, (t * z)) * 2.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(fma(Float64(-i), Float64(fma(c, b, a) * c), Float64(t * z)) * 2.0)
	t_2 = Float64(Float64(Float64(Float64(b * c) + a) * c) * i)
	tmp = 0.0
	if (t_2 <= -1e-107)
		tmp = t_1;
	elseif (t_2 <= 1e-23)
		tmp = Float64(fma(y, x, Float64(t * z)) * 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[((-i) * N[(N[(c * b + a), $MachinePrecision] * c), $MachinePrecision] + N[(t * z), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(b * c), $MachinePrecision] + a), $MachinePrecision] * c), $MachinePrecision] * i), $MachinePrecision]}, If[LessEqual[t$95$2, -1e-107], t$95$1, If[LessEqual[t$95$2, 1e-23], N[(N[(y * x + N[(t * z), $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) < -1e-107 or 9.9999999999999996e-24 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 88.6%

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

   3. Taylor expanded in x around 0

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

      1. 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. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      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. *-commutative N/A

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

      17. lower-*.f64 82.0

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

   5. Applied rewrites 82.0%

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

   if -1e-107 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 9.9999999999999996e-24

   1. Initial program 97.5%

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

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(b \cdot c + a\right) \cdot c\right) \cdot i \leq -1 \cdot 10^{-107}:\\ \;\;\;\;\mathsf{fma}\left(-i, \mathsf{fma}\left(c, b, a\right) \cdot c, t \cdot z\right) \cdot 2\\ \mathbf{elif}\;\left(\left(b \cdot c + a\right) \cdot c\right) \cdot i \leq 10^{-23}:\\ \;\;\;\;\mathsf{fma}\left(y, x, t \cdot z\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-i, \mathsf{fma}\left(c, b, a\right) \cdot c, t \cdot z\right) \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 83.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-i, \mathsf{fma}\left(c, b, a\right) \cdot c, x \cdot y\right) \cdot 2\\ t_2 := \left(\left(b \cdot c + a\right) \cdot c\right) \cdot i\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+41}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{+19}:\\ \;\;\;\;\mathsf{fma}\left(y, x, t \cdot z\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (fma (- i) (* (fma c b a) c) (* x y)) 2.0))
        (t_2 (* (* (+ (* b c) a) c) i)))
   (if (<= t_2 -1e+41) t_1 (if (<= t_2 1e+19) (* (fma y x (* t z)) 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(-i, (fma(c, b, a) * c), (x * y)) * 2.0;
	double t_2 = (((b * c) + a) * c) * i;
	double tmp;
	if (t_2 <= -1e+41) {
		tmp = t_1;
	} else if (t_2 <= 1e+19) {
		tmp = fma(y, x, (t * z)) * 2.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(fma(Float64(-i), Float64(fma(c, b, a) * c), Float64(x * y)) * 2.0)
	t_2 = Float64(Float64(Float64(Float64(b * c) + a) * c) * i)
	tmp = 0.0
	if (t_2 <= -1e+41)
		tmp = t_1;
	elseif (t_2 <= 1e+19)
		tmp = Float64(fma(y, x, Float64(t * z)) * 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[((-i) * N[(N[(c * b + a), $MachinePrecision] * c), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(b * c), $MachinePrecision] + a), $MachinePrecision] * c), $MachinePrecision] * i), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+41], t$95$1, If[LessEqual[t$95$2, 1e+19], N[(N[(y * x + N[(t * z), $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) < -1.00000000000000001e41 or 1e19 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 87.1%

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

   3. Taylor expanded in t 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. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      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 80.7

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

   5. Applied rewrites 80.7%

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

   if -1.00000000000000001e41 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 1e19

   1. Initial program 98.0%

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

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 93.8

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

   5. Applied rewrites 93.8%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(b \cdot c + a\right) \cdot c\right) \cdot i \leq -1 \cdot 10^{+41}:\\ \;\;\;\;\mathsf{fma}\left(-i, \mathsf{fma}\left(c, b, a\right) \cdot c, x \cdot y\right) \cdot 2\\ \mathbf{elif}\;\left(\left(b \cdot c + a\right) \cdot c\right) \cdot i \leq 10^{+19}:\\ \;\;\;\;\mathsf{fma}\left(y, x, t \cdot z\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-i, \mathsf{fma}\left(c, b, a\right) \cdot c, x \cdot y\right) \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 81.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -2 \cdot \left(\left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right)\\ t_2 := \left(\left(b \cdot c + a\right) \cdot c\right) \cdot i\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+125}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{+112}:\\ \;\;\;\;\mathsf{fma}\left(y, x, t \cdot z\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* -2.0 (* (* (fma c b a) i) c))) (t_2 (* (* (+ (* b c) a) c) i)))
   (if (<= t_2 -5e+125)
     t_1
     (if (<= t_2 1e+112) (* (fma y x (* t z)) 2.0) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = -2.0 * ((fma(c, b, a) * i) * c);
	double t_2 = (((b * c) + a) * c) * i;
	double tmp;
	if (t_2 <= -5e+125) {
		tmp = t_1;
	} else if (t_2 <= 1e+112) {
		tmp = fma(y, x, (t * z)) * 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(Float64(fma(c, b, a) * i) * c))
	t_2 = Float64(Float64(Float64(Float64(b * c) + a) * c) * i)
	tmp = 0.0
	if (t_2 <= -5e+125)
		tmp = t_1;
	elseif (t_2 <= 1e+112)
		tmp = Float64(fma(y, x, Float64(t * z)) * 2.0);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(-2.0 * N[(N[(N[(c * b + a), $MachinePrecision] * i), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(b * c), $MachinePrecision] + a), $MachinePrecision] * c), $MachinePrecision] * i), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+125], t$95$1, If[LessEqual[t$95$2, 1e+112], N[(N[(y * x + N[(t * z), $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) < -4.99999999999999962e125 or 9.9999999999999993e111 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 84.8%

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

   3. Taylor expanded in 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

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 81.3

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

   5. Applied rewrites 81.3%

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

   if -4.99999999999999962e125 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 9.9999999999999993e111

   1. Initial program 98.3%

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

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 85.3

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

   5. Applied rewrites 85.3%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(b \cdot c + a\right) \cdot c\right) \cdot i \leq -5 \cdot 10^{+125}:\\ \;\;\;\;-2 \cdot \left(\left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right)\\ \mathbf{elif}\;\left(\left(b \cdot c + a\right) \cdot c\right) \cdot i \leq 10^{+112}:\\ \;\;\;\;\mathsf{fma}\left(y, x, t \cdot z\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 73.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(b \cdot c + a\right) \cdot c\right) \cdot i\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+132}:\\ \;\;\;\;\left(\left(-2 \cdot b\right) \cdot \left(i \cdot c\right)\right) \cdot c\\ \mathbf{elif}\;t\_1 \leq 10^{+298}:\\ \;\;\;\;\mathsf{fma}\left(y, x, t \cdot z\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(-2 \cdot b\right) \cdot i\right) \cdot c\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* (+ (* b c) a) c) i)))
   (if (<= t_1 -5e+132)
     (* (* (* -2.0 b) (* i c)) c)
     (if (<= t_1 1e+298)
       (* (fma y x (* t z)) 2.0)
       (* (* (* (* -2.0 b) i) c) c)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (((b * c) + a) * c) * i;
	double tmp;
	if (t_1 <= -5e+132) {
		tmp = ((-2.0 * b) * (i * c)) * c;
	} else if (t_1 <= 1e+298) {
		tmp = fma(y, x, (t * z)) * 2.0;
	} else {
		tmp = (((-2.0 * b) * i) * c) * c;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(Float64(b * c) + a) * c) * i)
	tmp = 0.0
	if (t_1 <= -5e+132)
		tmp = Float64(Float64(Float64(-2.0 * b) * Float64(i * c)) * c);
	elseif (t_1 <= 1e+298)
		tmp = Float64(fma(y, x, Float64(t * z)) * 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(-2.0 * b) * i) * c) * c);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(N[(b * c), $MachinePrecision] + a), $MachinePrecision] * c), $MachinePrecision] * i), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+132], N[(N[(N[(-2.0 * b), $MachinePrecision] * N[(i * c), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision], If[LessEqual[t$95$1, 1e+298], N[(N[(y * x + N[(t * z), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(N[(N[(-2.0 * b), $MachinePrecision] * i), $MachinePrecision] * c), $MachinePrecision] * c), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -5.0000000000000001e132

   1. Initial program 88.1%

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

   3. Taylor expanded in c around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 60.9

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

   5. Applied rewrites 60.9%

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

   7. Applied rewrites 59.3%

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

   if -5.0000000000000001e132 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 9.9999999999999996e297

   1. Initial program 97.8%

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

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 78.7

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

   5. Applied rewrites 78.7%

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

   if 9.9999999999999996e297 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 76.4%

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

   3. Taylor expanded in c around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 58.6

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

   5. Applied rewrites 58.6%

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

   7. Applied rewrites 62.3%

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

   9. Applied rewrites 62.3%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(b \cdot c + a\right) \cdot c\right) \cdot i \leq -5 \cdot 10^{+132}:\\ \;\;\;\;\left(\left(-2 \cdot b\right) \cdot \left(i \cdot c\right)\right) \cdot c\\ \mathbf{elif}\;\left(\left(b \cdot c + a\right) \cdot c\right) \cdot i \leq 10^{+298}:\\ \;\;\;\;\mathsf{fma}\left(y, x, t \cdot z\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(-2 \cdot b\right) \cdot i\right) \cdot c\right) \cdot c\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 73.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(-2 \cdot b\right) \cdot \left(i \cdot c\right)\right) \cdot c\\ t_2 := \left(\left(b \cdot c + a\right) \cdot c\right) \cdot i\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+132}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{+298}:\\ \;\;\;\;\mathsf{fma}\left(y, x, t \cdot z\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* (* -2.0 b) (* i c)) c)) (t_2 (* (* (+ (* b c) a) c) i)))
   (if (<= t_2 -5e+132)
     t_1
     (if (<= t_2 1e+298) (* (fma y x (* t z)) 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 * b) * (i * c)) * c;
	double t_2 = (((b * c) + a) * c) * i;
	double tmp;
	if (t_2 <= -5e+132) {
		tmp = t_1;
	} else if (t_2 <= 1e+298) {
		tmp = fma(y, x, (t * z)) * 2.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(-2.0 * b) * Float64(i * c)) * c)
	t_2 = Float64(Float64(Float64(Float64(b * c) + a) * c) * i)
	tmp = 0.0
	if (t_2 <= -5e+132)
		tmp = t_1;
	elseif (t_2 <= 1e+298)
		tmp = Float64(fma(y, x, Float64(t * z)) * 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[(-2.0 * b), $MachinePrecision] * N[(i * c), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(b * c), $MachinePrecision] + a), $MachinePrecision] * c), $MachinePrecision] * i), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+132], t$95$1, If[LessEqual[t$95$2, 1e+298], N[(N[(y * x + N[(t * z), $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) < -5.0000000000000001e132 or 9.9999999999999996e297 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 82.7%

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

   3. Taylor expanded in c around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 59.8

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

   5. Applied rewrites 59.8%

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

   7. Applied rewrites 60.7%

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

   if -5.0000000000000001e132 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 9.9999999999999996e297

   1. Initial program 97.8%

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

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 78.7

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

   5. Applied rewrites 78.7%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(b \cdot c + a\right) \cdot c\right) \cdot i \leq -5 \cdot 10^{+132}:\\ \;\;\;\;\left(\left(-2 \cdot b\right) \cdot \left(i \cdot c\right)\right) \cdot c\\ \mathbf{elif}\;\left(\left(b \cdot c + a\right) \cdot c\right) \cdot i \leq 10^{+298}:\\ \;\;\;\;\mathsf{fma}\left(y, x, t \cdot z\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-2 \cdot b\right) \cdot \left(i \cdot c\right)\right) \cdot c\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 83.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(c, b, a\right) \cdot c\\ t_2 := \left(b \cdot c + a\right) \cdot c\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+22}:\\ \;\;\;\;\mathsf{fma}\left(-i, t\_1, t \cdot z\right) \cdot 2\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+111}:\\ \;\;\;\;\mathsf{fma}\left(\left(-a\right) \cdot c, i, \mathsf{fma}\left(y, x, t \cdot z\right)\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-i, t\_1, x \cdot y\right) \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (fma c b a) c)) (t_2 (* (+ (* b c) a) c)))
   (if (<= t_2 -2e+22)
     (* (fma (- i) t_1 (* t z)) 2.0)
     (if (<= t_2 5e+111)
       (* (fma (* (- a) c) i (fma y x (* t z))) 2.0)
       (* (fma (- i) t_1 (* x y)) 2.0)))))

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) * c;
	double t_2 = ((b * c) + a) * c;
	double tmp;
	if (t_2 <= -2e+22) {
		tmp = fma(-i, t_1, (t * z)) * 2.0;
	} else if (t_2 <= 5e+111) {
		tmp = fma((-a * c), i, fma(y, x, (t * z))) * 2.0;
	} else {
		tmp = fma(-i, t_1, (x * y)) * 2.0;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(fma(c, b, a) * c)
	t_2 = Float64(Float64(Float64(b * c) + a) * c)
	tmp = 0.0
	if (t_2 <= -2e+22)
		tmp = Float64(fma(Float64(-i), t_1, Float64(t * z)) * 2.0);
	elseif (t_2 <= 5e+111)
		tmp = Float64(fma(Float64(Float64(-a) * c), i, fma(y, x, Float64(t * z))) * 2.0);
	else
		tmp = Float64(fma(Float64(-i), t_1, Float64(x * y)) * 2.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(c * b + a), $MachinePrecision] * c), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(b * c), $MachinePrecision] + a), $MachinePrecision] * c), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+22], N[(N[((-i) * t$95$1 + N[(t * z), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], If[LessEqual[t$95$2, 5e+111], N[(N[(N[((-a) * c), $MachinePrecision] * i + N[(y * x + N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[((-i) * t$95$1 + N[(x * y), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (+.f64 a (*.f64 b c)) c) < -2e22

   1. Initial program 86.6%

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

   3. Taylor expanded in x around 0

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

      1. 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. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      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. *-commutative N/A

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

      17. lower-*.f64 84.0

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

   5. Applied rewrites 84.0%

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

   if -2e22 < (*.f64 (+.f64 a (*.f64 b c)) c) < 4.9999999999999997e111

   1. Initial program 95.9%

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

   3. Taylor expanded in c around 0

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. neg-mul-1 N/A

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

      7. lower-neg.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 95.4

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

   5. Applied rewrites 95.4%

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

   if 4.9999999999999997e111 < (*.f64 (+.f64 a (*.f64 b c)) c)

   1. Initial program 87.8%

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

   3. Taylor expanded in t 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. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      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 84.5

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

   5. Applied rewrites 84.5%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(b \cdot c + a\right) \cdot c \leq -2 \cdot 10^{+22}:\\ \;\;\;\;\mathsf{fma}\left(-i, \mathsf{fma}\left(c, b, a\right) \cdot c, t \cdot z\right) \cdot 2\\ \mathbf{elif}\;\left(b \cdot c + a\right) \cdot c \leq 5 \cdot 10^{+111}:\\ \;\;\;\;\mathsf{fma}\left(\left(-a\right) \cdot c, i, \mathsf{fma}\left(y, x, t \cdot z\right)\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-i, \mathsf{fma}\left(c, b, a\right) \cdot c, x \cdot y\right) \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 62.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(i \cdot c\right) \cdot a\right) \cdot -2\\ t_2 := \left(\left(b \cdot c + a\right) \cdot c\right) \cdot i\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+153}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{+298}:\\ \;\;\;\;\mathsf{fma}\left(y, x, t \cdot z\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* (* i c) a) -2.0)) (t_2 (* (* (+ (* b c) a) c) i)))
   (if (<= t_2 -2e+153)
     t_1
     (if (<= t_2 1e+298) (* (fma y x (* t z)) 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 = (((b * c) + a) * c) * i;
	double tmp;
	if (t_2 <= -2e+153) {
		tmp = t_1;
	} else if (t_2 <= 1e+298) {
		tmp = fma(y, x, (t * z)) * 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(Float64(Float64(Float64(b * c) + a) * c) * i)
	tmp = 0.0
	if (t_2 <= -2e+153)
		tmp = t_1;
	elseif (t_2 <= 1e+298)
		tmp = Float64(fma(y, x, Float64(t * z)) * 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[(N[(N[(N[(b * c), $MachinePrecision] + a), $MachinePrecision] * c), $MachinePrecision] * i), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+153], t$95$1, If[LessEqual[t$95$2, 1e+298], N[(N[(y * x + N[(t * z), $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) < -2e153 or 9.9999999999999996e297 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 82.0%

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

   3. Taylor expanded in 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 54.4

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

   5. Applied rewrites 54.4%

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

   if -2e153 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 9.9999999999999996e297

   1. Initial program 97.8%

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

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 77.3

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

   5. Applied rewrites 77.3%

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

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(b \cdot c + a\right) \cdot c\right) \cdot i \leq -2 \cdot 10^{+153}:\\ \;\;\;\;\left(\left(i \cdot c\right) \cdot a\right) \cdot -2\\ \mathbf{elif}\;\left(\left(b \cdot c + a\right) \cdot c\right) \cdot i \leq 10^{+298}:\\ \;\;\;\;\mathsf{fma}\left(y, x, t \cdot z\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(i \cdot c\right) \cdot a\right) \cdot -2\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 69.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(b \cdot c + a\right) \cdot c\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+246}:\\ \;\;\;\;\left(\left(\left(-2 \cdot b\right) \cdot i\right) \cdot c\right) \cdot c\\ \mathbf{elif}\;t\_1 \leq 10^{+120}:\\ \;\;\;\;\mathsf{fma}\left(y, x, t \cdot z\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-2 \cdot i\right) \cdot b\right) \cdot \left(c \cdot c\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (+ (* b c) a) c)))
   (if (<= t_1 -1e+246)
     (* (* (* (* -2.0 b) i) c) c)
     (if (<= t_1 1e+120)
       (* (fma y x (* t z)) 2.0)
       (* (* (* -2.0 i) b) (* c c))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = ((b * c) + a) * c;
	double tmp;
	if (t_1 <= -1e+246) {
		tmp = (((-2.0 * b) * i) * c) * c;
	} else if (t_1 <= 1e+120) {
		tmp = fma(y, x, (t * z)) * 2.0;
	} else {
		tmp = ((-2.0 * i) * b) * (c * c);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(b * c) + a) * c)
	tmp = 0.0
	if (t_1 <= -1e+246)
		tmp = Float64(Float64(Float64(Float64(-2.0 * b) * i) * c) * c);
	elseif (t_1 <= 1e+120)
		tmp = Float64(fma(y, x, Float64(t * z)) * 2.0);
	else
		tmp = Float64(Float64(Float64(-2.0 * i) * b) * Float64(c * c));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(b * c), $MachinePrecision] + a), $MachinePrecision] * c), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+246], N[(N[(N[(N[(-2.0 * b), $MachinePrecision] * i), $MachinePrecision] * c), $MachinePrecision] * c), $MachinePrecision], If[LessEqual[t$95$1, 1e+120], N[(N[(y * x + N[(t * z), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(N[(-2.0 * i), $MachinePrecision] * b), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (+.f64 a (*.f64 b c)) c) < -1.00000000000000007e246

   1. Initial program 81.4%

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

   3. Taylor expanded in c around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 58.2

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

   5. Applied rewrites 58.2%

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

   7. Applied rewrites 64.0%

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

   9. Applied rewrites 64.1%

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

   if -1.00000000000000007e246 < (*.f64 (+.f64 a (*.f64 b c)) c) < 9.9999999999999998e119

   1. Initial program 95.9%

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

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 76.2

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

   5. Applied rewrites 76.2%

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

   if 9.9999999999999998e119 < (*.f64 (+.f64 a (*.f64 b c)) c)

   1. Initial program 87.5%

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

   3. Taylor expanded in c around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 63.5

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

   5. Applied rewrites 63.5%

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

   7. Applied rewrites 65.2%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(b \cdot c + a\right) \cdot c \leq -1 \cdot 10^{+246}:\\ \;\;\;\;\left(\left(\left(-2 \cdot b\right) \cdot i\right) \cdot c\right) \cdot c\\ \mathbf{elif}\;\left(b \cdot c + a\right) \cdot c \leq 10^{+120}:\\ \;\;\;\;\mathsf{fma}\left(y, x, t \cdot z\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-2 \cdot i\right) \cdot b\right) \cdot \left(c \cdot c\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 84.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\left(-c\right) \cdot b, i \cdot c, \mathsf{fma}\left(y, x, t \cdot z\right)\right) \cdot 2\\ \mathbf{if}\;b \leq -1.7 \cdot 10^{-58}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 1.06 \cdot 10^{-66}:\\ \;\;\;\;\mathsf{fma}\left(z, t, \mathsf{fma}\left(y, x, \left(a \cdot i\right) \cdot \left(-c\right)\right)\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (fma (* (- c) b) (* i c) (fma y x (* t z))) 2.0)))
   (if (<= b -1.7e-58)
     t_1
     (if (<= b 1.06e-66) (* (fma z t (fma y x (* (* a i) (- c)))) 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), (i * c), fma(y, x, (t * z))) * 2.0;
	double tmp;
	if (b <= -1.7e-58) {
		tmp = t_1;
	} else if (b <= 1.06e-66) {
		tmp = fma(z, t, fma(y, x, ((a * i) * -c))) * 2.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(fma(Float64(Float64(-c) * b), Float64(i * c), fma(y, x, Float64(t * z))) * 2.0)
	tmp = 0.0
	if (b <= -1.7e-58)
		tmp = t_1;
	elseif (b <= 1.06e-66)
		tmp = Float64(fma(z, t, fma(y, x, Float64(Float64(a * i) * Float64(-c)))) * 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[((-c) * b), $MachinePrecision] * N[(i * c), $MachinePrecision] + N[(y * x + N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]}, If[LessEqual[b, -1.7e-58], t$95$1, If[LessEqual[b, 1.06e-66], N[(N[(z * t + N[(y * x + N[(N[(a * i), $MachinePrecision] * (-c)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if b < -1.69999999999999987e-58 or 1.05999999999999994e-66 < b

   1. Initial program 91.2%

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

   3. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

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

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

      14. mul-1-neg N/A

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 89.2%

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

   if -1.69999999999999987e-58 < b < 1.05999999999999994e-66

   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. 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. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. sub-neg N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

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

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

      16. lower-*.f64 N/A

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

   4. Applied rewrites 97.2%

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

   5. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 93.3

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

   7. Applied rewrites 93.3%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.7 \cdot 10^{-58}:\\ \;\;\;\;\mathsf{fma}\left(\left(-c\right) \cdot b, i \cdot c, \mathsf{fma}\left(y, x, t \cdot z\right)\right) \cdot 2\\ \mathbf{elif}\;b \leq 1.06 \cdot 10^{-66}:\\ \;\;\;\;\mathsf{fma}\left(z, t, \mathsf{fma}\left(y, x, \left(a \cdot i\right) \cdot \left(-c\right)\right)\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(-c\right) \cdot b, i \cdot c, \mathsf{fma}\left(y, x, t \cdot z\right)\right) \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 44.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t \cdot z\right) \cdot 2\\ \mathbf{if}\;t \cdot z \leq -2.5 \cdot 10^{+37}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \cdot z \leq 2.3 \cdot 10^{+67}:\\ \;\;\;\;\left(x \cdot y\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* t z) 2.0)))
   (if (<= (* t z) -2.5e+37)
     t_1
     (if (<= (* t z) 2.3e+67) (* (* x y) 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 = (t * z) * 2.0;
	double tmp;
	if ((t * z) <= -2.5e+37) {
		tmp = t_1;
	} else if ((t * z) <= 2.3e+67) {
		tmp = (x * y) * 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 = (t * z) * 2.0d0
    if ((t * z) <= (-2.5d+37)) then
        tmp = t_1
    else if ((t * z) <= 2.3d+67) then
        tmp = (x * y) * 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 = (t * z) * 2.0;
	double tmp;
	if ((t * z) <= -2.5e+37) {
		tmp = t_1;
	} else if ((t * z) <= 2.3e+67) {
		tmp = (x * y) * 2.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = (t * z) * 2.0
	tmp = 0
	if (t * z) <= -2.5e+37:
		tmp = t_1
	elif (t * z) <= 2.3e+67:
		tmp = (x * y) * 2.0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(t * z) * 2.0)
	tmp = 0.0
	if (Float64(t * z) <= -2.5e+37)
		tmp = t_1;
	elseif (Float64(t * z) <= 2.3e+67)
		tmp = Float64(Float64(x * y) * 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 = (t * z) * 2.0;
	tmp = 0.0;
	if ((t * z) <= -2.5e+37)
		tmp = t_1;
	elseif ((t * z) <= 2.3e+67)
		tmp = (x * y) * 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[(N[(t * z), $MachinePrecision] * 2.0), $MachinePrecision]}, If[LessEqual[N[(t * z), $MachinePrecision], -2.5e+37], t$95$1, If[LessEqual[N[(t * z), $MachinePrecision], 2.3e+67], N[(N[(x * y), $MachinePrecision] * 2.0), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -2.49999999999999994e37 or 2.2999999999999999e67 < (*.f64 z t)

   1. Initial program 87.6%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 53.8

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

   5. Applied rewrites 53.8%

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

   if -2.49999999999999994e37 < (*.f64 z t) < 2.2999999999999999e67

   1. Initial program 94.0%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 40.7

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

   5. Applied rewrites 40.7%

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

4. Final simplification 46.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot z \leq -2.5 \cdot 10^{+37}:\\ \;\;\;\;\left(t \cdot z\right) \cdot 2\\ \mathbf{elif}\;t \cdot z \leq 2.3 \cdot 10^{+67}:\\ \;\;\;\;\left(x \cdot y\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(t \cdot z\right) \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 29.6% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \left(x \cdot y\right) \cdot 2 \end{array} \]

FPCore

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

C

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

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 = (x * y) * 2.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.4%

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

3. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 30.0

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

5. Applied rewrites 30.0%

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

6. Final simplification 30.0%

   \[\leadsto \left(x \cdot y\right) \cdot 2 \]
7. 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 2024248 
(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))))