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

Percentage Accurate: 90.7% → 93.7%

Time: 13.2s

Alternatives: 12

Speedup: 0.7×

• 46.9% 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: 90.7% 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: 93.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot z + y \cdot x\\ \mathbf{if}\;c \leq -1.5 \cdot 10^{-19}:\\ \;\;\;\;\left(t\_1 - \left(\left(\left(\frac{a}{c} + b\right) \cdot i\right) \cdot c\right) \cdot c\right) \cdot 2\\ \mathbf{elif}\;c \leq 3.05 \cdot 10^{+122}:\\ \;\;\;\;\left(t\_1 - \left(\left(b \cdot c + a\right) \cdot c\right) \cdot i\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ (* t z) (* y x))))
   (if (<= c -1.5e-19)
     (* (- t_1 (* (* (* (+ (/ a c) b) i) c) c)) 2.0)
     (if (<= c 3.05e+122)
       (* (- t_1 (* (* (+ (* b c) a) c) i)) 2.0)
       (* -2.0 (* (* (fma c b a) i) c))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (t * z) + (y * x);
	double tmp;
	if (c <= -1.5e-19) {
		tmp = (t_1 - (((((a / c) + b) * i) * c) * c)) * 2.0;
	} else if (c <= 3.05e+122) {
		tmp = (t_1 - ((((b * c) + a) * c) * i)) * 2.0;
	} else {
		tmp = -2.0 * ((fma(c, b, a) * i) * c);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(t * z) + Float64(y * x))
	tmp = 0.0
	if (c <= -1.5e-19)
		tmp = Float64(Float64(t_1 - Float64(Float64(Float64(Float64(Float64(a / c) + b) * i) * c) * c)) * 2.0);
	elseif (c <= 3.05e+122)
		tmp = Float64(Float64(t_1 - Float64(Float64(Float64(Float64(b * c) + a) * c) * i)) * 2.0);
	else
		tmp = Float64(-2.0 * Float64(Float64(fma(c, b, a) * i) * c));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(t * z), $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -1.5e-19], N[(N[(t$95$1 - N[(N[(N[(N[(N[(a / c), $MachinePrecision] + b), $MachinePrecision] * i), $MachinePrecision] * c), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], If[LessEqual[c, 3.05e+122], N[(N[(t$95$1 - N[(N[(N[(N[(b * c), $MachinePrecision] + a), $MachinePrecision] * c), $MachinePrecision] * i), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(-2.0 * N[(N[(N[(c * b + a), $MachinePrecision] * i), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if c < -1.49999999999999996e-19

   1. Initial program 79.4%

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

   3. Taylor expanded in c around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. distribute-lft-out N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 92.3

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

   5. Applied rewrites 92.3%

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

   if -1.49999999999999996e-19 < c < 3.0499999999999999e122

   1. Initial program 99.3%

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

   if 3.0499999999999999e122 < c

   1. Initial program 77.7%

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

   3. Taylor expanded in i around inf

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

      1. *-commutative N/A

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

      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 90.2

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

   5. Applied rewrites 90.2%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.5 \cdot 10^{-19}:\\ \;\;\;\;\left(\left(t \cdot z + y \cdot x\right) - \left(\left(\left(\frac{a}{c} + b\right) \cdot i\right) \cdot c\right) \cdot c\right) \cdot 2\\ \mathbf{elif}\;c \leq 3.05 \cdot 10^{+122}:\\ \;\;\;\;\left(\left(t \cdot z + y \cdot x\right) - \left(\left(b \cdot c + a\right) \cdot c\right) \cdot i\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 72.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(b \cdot c + a\right) \cdot c\right) \cdot i\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+195}:\\ \;\;\;\;\left(\left(-2 \cdot b\right) \cdot \left(i \cdot c\right)\right) \cdot c\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{+44}:\\ \;\;\;\;\left(\left(i \cdot c\right) \cdot a\right) \cdot -2\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+292}:\\ \;\;\;\;\mathsf{fma}\left(y, x, t \cdot z\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(i \cdot c\right) \cdot c\right) \cdot -2\right) \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* (+ (* b c) a) c) i)))
   (if (<= t_1 -2e+195)
     (* (* (* -2.0 b) (* i c)) c)
     (if (<= t_1 -2e+44)
       (* (* (* i c) a) -2.0)
       (if (<= t_1 4e+292)
         (* (fma y x (* t z)) 2.0)
         (* (* (* (* i c) c) -2.0) b))))))

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+195) {
		tmp = ((-2.0 * b) * (i * c)) * c;
	} else if (t_1 <= -2e+44) {
		tmp = ((i * c) * a) * -2.0;
	} else if (t_1 <= 4e+292) {
		tmp = fma(y, x, (t * z)) * 2.0;
	} else {
		tmp = (((i * c) * c) * -2.0) * b;
	}
	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+195)
		tmp = Float64(Float64(Float64(-2.0 * b) * Float64(i * c)) * c);
	elseif (t_1 <= -2e+44)
		tmp = Float64(Float64(Float64(i * c) * a) * -2.0);
	elseif (t_1 <= 4e+292)
		tmp = Float64(fma(y, x, Float64(t * z)) * 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(i * c) * c) * -2.0) * b);
	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, -2e+195], N[(N[(N[(-2.0 * b), $MachinePrecision] * N[(i * c), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision], If[LessEqual[t$95$1, -2e+44], N[(N[(N[(i * c), $MachinePrecision] * a), $MachinePrecision] * -2.0), $MachinePrecision], If[LessEqual[t$95$1, 4e+292], N[(N[(y * x + N[(t * z), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(N[(N[(i * c), $MachinePrecision] * c), $MachinePrecision] * -2.0), $MachinePrecision] * b), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -1.99999999999999995e195

   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 69.5

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

   5. Applied rewrites 69.5%

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

   7. Applied rewrites 73.3%

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

   if -1.99999999999999995e195 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -2.0000000000000002e44

   1. Initial program 99.2%

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

   3. Taylor expanded in 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 87.6

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

   5. Applied rewrites 87.6%

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

   if -2.0000000000000002e44 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 4.0000000000000001e292

   1. Initial program 99.9%

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

   3. Taylor expanded in 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 80.8

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

   5. Applied rewrites 80.8%

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

   if 4.0000000000000001e292 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 69.2%

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

   3. Taylor expanded in c around 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 72.0

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

   5. Applied rewrites 72.0%

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

   7. Applied rewrites 74.7%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(b \cdot c + a\right) \cdot c\right) \cdot i \leq -2 \cdot 10^{+195}:\\ \;\;\;\;\left(\left(-2 \cdot b\right) \cdot \left(i \cdot c\right)\right) \cdot c\\ \mathbf{elif}\;\left(\left(b \cdot c + a\right) \cdot c\right) \cdot i \leq -2 \cdot 10^{+44}:\\ \;\;\;\;\left(\left(i \cdot c\right) \cdot a\right) \cdot -2\\ \mathbf{elif}\;\left(\left(b \cdot c + a\right) \cdot c\right) \cdot i \leq 4 \cdot 10^{+292}:\\ \;\;\;\;\mathsf{fma}\left(y, x, t \cdot z\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(i \cdot c\right) \cdot c\right) \cdot -2\right) \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 72.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(-2 \cdot b\right) \cdot \left(i \cdot c\right)\right) \cdot c\\ t_2 := \left(\left(b \cdot c + a\right) \cdot c\right) \cdot i\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+195}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq -2 \cdot 10^{+44}:\\ \;\;\;\;\left(\left(i \cdot c\right) \cdot a\right) \cdot -2\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{+292}:\\ \;\;\;\;\mathsf{fma}\left(y, x, t \cdot z\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 -2e+195)
     t_1
     (if (<= t_2 -2e+44)
       (* (* (* i c) a) -2.0)
       (if (<= t_2 4e+292) (* (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 <= -2e+195) {
		tmp = t_1;
	} else if (t_2 <= -2e+44) {
		tmp = ((i * c) * a) * -2.0;
	} else if (t_2 <= 4e+292) {
		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 <= -2e+195)
		tmp = t_1;
	elseif (t_2 <= -2e+44)
		tmp = Float64(Float64(Float64(i * c) * a) * -2.0);
	elseif (t_2 <= 4e+292)
		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, -2e+195], t$95$1, If[LessEqual[t$95$2, -2e+44], N[(N[(N[(i * c), $MachinePrecision] * a), $MachinePrecision] * -2.0), $MachinePrecision], If[LessEqual[t$95$2, 4e+292], N[(N[(y * x + N[(t * z), $MachinePrecision]), $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) < -1.99999999999999995e195 or 4.0000000000000001e292 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 78.3%

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

   3. Taylor expanded in c around 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 70.3

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

   5. Applied rewrites 70.3%

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

   7. Applied rewrites 72.9%

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

   if -1.99999999999999995e195 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -2.0000000000000002e44

   1. Initial program 99.2%

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

   3. Taylor expanded in 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 87.6

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

   5. Applied rewrites 87.6%

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

   if -2.0000000000000002e44 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 4.0000000000000001e292

   1. Initial program 99.9%

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

   3. Taylor expanded in 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 80.8

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

   5. Applied rewrites 80.8%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(b \cdot c + a\right) \cdot c\right) \cdot i \leq -2 \cdot 10^{+195}:\\ \;\;\;\;\left(\left(-2 \cdot b\right) \cdot \left(i \cdot c\right)\right) \cdot c\\ \mathbf{elif}\;\left(\left(b \cdot c + a\right) \cdot c\right) \cdot i \leq -2 \cdot 10^{+44}:\\ \;\;\;\;\left(\left(i \cdot c\right) \cdot a\right) \cdot -2\\ \mathbf{elif}\;\left(\left(b \cdot c + a\right) \cdot c\right) \cdot i \leq 4 \cdot 10^{+292}:\\ \;\;\;\;\mathsf{fma}\left(y, x, t \cdot z\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-2 \cdot b\right) \cdot \left(i \cdot c\right)\right) \cdot c\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

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) + (y * x)) - ((((b * c) + a) * c) * i);
	double tmp;
	if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = t_1 * 2.0;
	} else {
		tmp = (((i * c) * c) * -2.0) * b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = ((t * z) + (y * x)) - ((((b * c) + a) * c) * i)
	tmp = 0
	if t_1 <= math.inf:
		tmp = t_1 * 2.0
	else:
		tmp = (((i * c) * c) * -2.0) * b
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	t_1 = ((t * z) + (y * x)) - ((((b * c) + a) * c) * i);
	tmp = 0.0;
	if (t_1 <= Inf)
		tmp = t_1 * 2.0;
	else
		tmp = (((i * c) * c) * -2.0) * b;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 95.2%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in 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 66.7

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

   5. Applied rewrites 66.7%

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

   7. Applied rewrites 66.7%

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

4. Final simplification 93.9%

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

Alternative 5: 69.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(b \cdot c + a\right) \cdot c\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+231}:\\ \;\;\;\;\left(\left(-2 \cdot b\right) \cdot \left(i \cdot c\right)\right) \cdot c\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{+67}:\\ \;\;\;\;\left(\left(i \cdot c\right) \cdot a\right) \cdot -2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+300}:\\ \;\;\;\;\mathsf{fma}\left(y, x, t \cdot z\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot i\right) \cdot \left(\left(c \cdot c\right) \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (+ (* b c) a) c)))
   (if (<= t_1 -4e+231)
     (* (* (* -2.0 b) (* i c)) c)
     (if (<= t_1 -2e+67)
       (* (* (* i c) a) -2.0)
       (if (<= t_1 5e+300)
         (* (fma y x (* t z)) 2.0)
         (* (* -2.0 i) (* (* c c) b)))))))

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 <= -4e+231) {
		tmp = ((-2.0 * b) * (i * c)) * c;
	} else if (t_1 <= -2e+67) {
		tmp = ((i * c) * a) * -2.0;
	} else if (t_1 <= 5e+300) {
		tmp = fma(y, x, (t * z)) * 2.0;
	} else {
		tmp = (-2.0 * i) * ((c * c) * b);
	}
	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 <= -4e+231)
		tmp = Float64(Float64(Float64(-2.0 * b) * Float64(i * c)) * c);
	elseif (t_1 <= -2e+67)
		tmp = Float64(Float64(Float64(i * c) * a) * -2.0);
	elseif (t_1 <= 5e+300)
		tmp = Float64(fma(y, x, Float64(t * z)) * 2.0);
	else
		tmp = Float64(Float64(-2.0 * i) * Float64(Float64(c * c) * b));
	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, -4e+231], N[(N[(N[(-2.0 * b), $MachinePrecision] * N[(i * c), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision], If[LessEqual[t$95$1, -2e+67], N[(N[(N[(i * c), $MachinePrecision] * a), $MachinePrecision] * -2.0), $MachinePrecision], If[LessEqual[t$95$1, 5e+300], N[(N[(y * x + N[(t * z), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(-2.0 * i), $MachinePrecision] * N[(N[(c * c), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (+.f64 a (*.f64 b c)) c) < -4.0000000000000002e231

   1. Initial program 73.2%

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

   3. Taylor expanded in c around 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 73.3

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

   5. Applied rewrites 73.3%

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

   7. Applied rewrites 80.6%

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

   if -4.0000000000000002e231 < (*.f64 (+.f64 a (*.f64 b c)) c) < -1.99999999999999997e67

   1. Initial program 99.6%

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

   3. Taylor expanded in 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 58.1

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

   5. Applied rewrites 58.1%

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

   if -1.99999999999999997e67 < (*.f64 (+.f64 a (*.f64 b c)) c) < 5.00000000000000026e300

   1. Initial program 99.2%

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

   3. Taylor expanded in c around 0

      \[\leadsto 2 \cdot \color{blue}{\left(t \cdot z + x \cdot y\right)} \]
   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 78.3

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

   5. Applied rewrites 78.3%

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

   if 5.00000000000000026e300 < (*.f64 (+.f64 a (*.f64 b c)) c)

   1. Initial program 78.7%

      \[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]
   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 81.1

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

   5. Applied rewrites 81.1%

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

   7. Applied rewrites 83.5%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(b \cdot c + a\right) \cdot c \leq -4 \cdot 10^{+231}:\\ \;\;\;\;\left(\left(-2 \cdot b\right) \cdot \left(i \cdot c\right)\right) \cdot c\\ \mathbf{elif}\;\left(b \cdot c + a\right) \cdot c \leq -2 \cdot 10^{+67}:\\ \;\;\;\;\left(\left(i \cdot c\right) \cdot a\right) \cdot -2\\ \mathbf{elif}\;\left(b \cdot c + a\right) \cdot c \leq 5 \cdot 10^{+300}:\\ \;\;\;\;\mathsf{fma}\left(y, x, t \cdot z\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot i\right) \cdot \left(\left(c \cdot c\right) \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(b \cdot c + a\right) \cdot c\right) \cdot i\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+151}:\\ \;\;\;\;-2 \cdot \left(\left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right)\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+148}:\\ \;\;\;\;\mathsf{fma}\left(\left(-a\right) \cdot c, i, \mathsf{fma}\left(y, x, t \cdot z\right)\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(c, b, a\right) \cdot c\right) \cdot \left(-i\right)\right) \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* (+ (* b c) a) c) i)))
   (if (<= t_1 -5e+151)
     (* -2.0 (* (* (fma c b a) i) c))
     (if (<= t_1 4e+148)
       (* (fma (* (- a) c) i (fma y x (* t z))) 2.0)
       (* (* (* (fma c b 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 <= -5e+151) {
		tmp = -2.0 * ((fma(c, b, a) * i) * c);
	} else if (t_1 <= 4e+148) {
		tmp = fma((-a * c), i, fma(y, x, (t * z))) * 2.0;
	} else {
		tmp = ((fma(c, b, 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 <= -5e+151)
		tmp = Float64(-2.0 * Float64(Float64(fma(c, b, a) * i) * c));
	elseif (t_1 <= 4e+148)
		tmp = Float64(fma(Float64(Float64(-a) * c), i, fma(y, x, Float64(t * z))) * 2.0);
	else
		tmp = Float64(Float64(Float64(fma(c, b, a) * c) * Float64(-i)) * 2.0);
	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+151], N[(-2.0 * N[(N[(N[(c * b + a), $MachinePrecision] * i), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 4e+148], N[(N[(N[((-a) * c), $MachinePrecision] * i + N[(y * x + N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(N[(N[(c * b + a), $MachinePrecision] * c), $MachinePrecision] * (-i)), $MachinePrecision] * 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -5.0000000000000002e151

   1. Initial program 82.9%

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

   3. Taylor expanded in i around inf

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

      1. *-commutative N/A

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

      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 89.7

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

   5. Applied rewrites 89.7%

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

   if -5.0000000000000002e151 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 4.0000000000000002e148

   1. Initial program 99.9%

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

   3. Taylor expanded in 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

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      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 4.0000000000000002e148 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 80.0%

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

   3. Taylor expanded in i around inf

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

      1. mul-1-neg N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 73.2

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

   5. Applied rewrites 73.2%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(b \cdot c + a\right) \cdot c\right) \cdot i \leq -5 \cdot 10^{+151}:\\ \;\;\;\;-2 \cdot \left(\left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right)\\ \mathbf{elif}\;\left(\left(b \cdot c + a\right) \cdot c\right) \cdot i \leq 4 \cdot 10^{+148}:\\ \;\;\;\;\mathsf{fma}\left(\left(-a\right) \cdot c, i, \mathsf{fma}\left(y, x, t \cdot z\right)\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(c, b, a\right) \cdot c\right) \cdot \left(-i\right)\right) \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(b \cdot c + a\right) \cdot c\right) \cdot i\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+44}:\\ \;\;\;\;-2 \cdot \left(\left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right)\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+148}:\\ \;\;\;\;\mathsf{fma}\left(y, x, t \cdot z\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(c, b, a\right) \cdot c\right) \cdot \left(-i\right)\right) \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* (+ (* b c) a) c) i)))
   (if (<= t_1 -2e+44)
     (* -2.0 (* (* (fma c b a) i) c))
     (if (<= t_1 4e+148)
       (* (fma y x (* t z)) 2.0)
       (* (* (* (fma c b 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+44) {
		tmp = -2.0 * ((fma(c, b, a) * i) * c);
	} else if (t_1 <= 4e+148) {
		tmp = fma(y, x, (t * z)) * 2.0;
	} else {
		tmp = ((fma(c, b, 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+44)
		tmp = Float64(-2.0 * Float64(Float64(fma(c, b, a) * i) * c));
	elseif (t_1 <= 4e+148)
		tmp = Float64(fma(y, x, Float64(t * z)) * 2.0);
	else
		tmp = Float64(Float64(Float64(fma(c, b, a) * c) * Float64(-i)) * 2.0);
	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, -2e+44], N[(-2.0 * N[(N[(N[(c * b + a), $MachinePrecision] * i), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 4e+148], N[(N[(y * x + N[(t * z), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(N[(N[(c * b + a), $MachinePrecision] * c), $MachinePrecision] * (-i)), $MachinePrecision] * 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -2.0000000000000002e44

   1. Initial program 84.3%

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

   3. Taylor expanded in 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 88.1

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

   5. Applied rewrites 88.1%

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

   if -2.0000000000000002e44 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 4.0000000000000002e148

   1. Initial program 99.9%

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

   3. Taylor expanded in 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 86.7

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

   5. Applied rewrites 86.7%

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

   if 4.0000000000000002e148 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 80.0%

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

   3. Taylor expanded in i around inf

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

      1. mul-1-neg N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 73.2

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

   5. Applied rewrites 73.2%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(b \cdot c + a\right) \cdot c\right) \cdot i \leq -2 \cdot 10^{+44}:\\ \;\;\;\;-2 \cdot \left(\left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right)\\ \mathbf{elif}\;\left(\left(b \cdot c + a\right) \cdot c\right) \cdot i \leq 4 \cdot 10^{+148}:\\ \;\;\;\;\mathsf{fma}\left(y, x, t \cdot z\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(c, b, a\right) \cdot c\right) \cdot \left(-i\right)\right) \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 81.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -2 \cdot \left(\left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right)\\ t_2 := \left(\left(b \cdot c + a\right) \cdot c\right) \cdot i\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+44}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{+148}:\\ \;\;\;\;\mathsf{fma}\left(y, x, t \cdot z\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 -2e+44)
     t_1
     (if (<= t_2 4e+148) (* (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 <= -2e+44) {
		tmp = t_1;
	} else if (t_2 <= 4e+148) {
		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 <= -2e+44)
		tmp = t_1;
	elseif (t_2 <= 4e+148)
		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, -2e+44], t$95$1, If[LessEqual[t$95$2, 4e+148], 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) < -2.0000000000000002e44 or 4.0000000000000002e148 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 82.6%

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

   3. Taylor expanded in 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 79.6

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

   5. Applied rewrites 79.6%

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

   if -2.0000000000000002e44 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 4.0000000000000002e148

   1. Initial program 99.9%

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

   3. Taylor expanded in 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 86.7

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

   5. Applied rewrites 86.7%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(b \cdot c + a\right) \cdot c\right) \cdot i \leq -2 \cdot 10^{+44}:\\ \;\;\;\;-2 \cdot \left(\left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right)\\ \mathbf{elif}\;\left(\left(b \cdot c + a\right) \cdot c\right) \cdot i \leq 4 \cdot 10^{+148}:\\ \;\;\;\;\mathsf{fma}\left(y, x, t \cdot z\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 44.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t \cdot z\right) \cdot 2\\ \mathbf{if}\;t \cdot z \leq -1.75 \cdot 10^{+89}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \cdot z \leq -3.1 \cdot 10^{-111}:\\ \;\;\;\;\left(\left(i \cdot c\right) \cdot a\right) \cdot -2\\ \mathbf{elif}\;t \cdot z \leq 1.08 \cdot 10^{+49}:\\ \;\;\;\;\left(y \cdot x\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* t z) 2.0)))
   (if (<= (* t z) -1.75e+89)
     t_1
     (if (<= (* t z) -3.1e-111)
       (* (* (* i c) a) -2.0)
       (if (<= (* t z) 1.08e+49) (* (* y x) 2.0) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (t * z) * 2.0;
	double tmp;
	if ((t * z) <= -1.75e+89) {
		tmp = t_1;
	} else if ((t * z) <= -3.1e-111) {
		tmp = ((i * c) * a) * -2.0;
	} else if ((t * z) <= 1.08e+49) {
		tmp = (y * x) * 2.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (t * z) * 2.0d0
    if ((t * z) <= (-1.75d+89)) then
        tmp = t_1
    else if ((t * z) <= (-3.1d-111)) then
        tmp = ((i * c) * a) * (-2.0d0)
    else if ((t * z) <= 1.08d+49) then
        tmp = (y * x) * 2.0d0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (t * z) * 2.0;
	double tmp;
	if ((t * z) <= -1.75e+89) {
		tmp = t_1;
	} else if ((t * z) <= -3.1e-111) {
		tmp = ((i * c) * a) * -2.0;
	} else if ((t * z) <= 1.08e+49) {
		tmp = (y * x) * 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) <= -1.75e+89:
		tmp = t_1
	elif (t * z) <= -3.1e-111:
		tmp = ((i * c) * a) * -2.0
	elif (t * z) <= 1.08e+49:
		tmp = (y * x) * 2.0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(t * z) * 2.0)
	tmp = 0.0
	if (Float64(t * z) <= -1.75e+89)
		tmp = t_1;
	elseif (Float64(t * z) <= -3.1e-111)
		tmp = Float64(Float64(Float64(i * c) * a) * -2.0);
	elseif (Float64(t * z) <= 1.08e+49)
		tmp = Float64(Float64(y * x) * 2.0);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	t_1 = (t * z) * 2.0;
	tmp = 0.0;
	if ((t * z) <= -1.75e+89)
		tmp = t_1;
	elseif ((t * z) <= -3.1e-111)
		tmp = ((i * c) * a) * -2.0;
	elseif ((t * z) <= 1.08e+49)
		tmp = (y * x) * 2.0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(t * z), $MachinePrecision] * 2.0), $MachinePrecision]}, If[LessEqual[N[(t * z), $MachinePrecision], -1.75e+89], t$95$1, If[LessEqual[N[(t * z), $MachinePrecision], -3.1e-111], N[(N[(N[(i * c), $MachinePrecision] * a), $MachinePrecision] * -2.0), $MachinePrecision], If[LessEqual[N[(t * z), $MachinePrecision], 1.08e+49], N[(N[(y * x), $MachinePrecision] * 2.0), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -1.75e89 or 1.08000000000000001e49 < (*.f64 z t)

   1. Initial program 90.6%

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

   3. Taylor expanded in 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 60.7

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

   5. Applied rewrites 60.7%

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

   if -1.75e89 < (*.f64 z t) < -3.10000000000000014e-111

   1. Initial program 88.3%

      \[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]
   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 37.4

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

   5. Applied rewrites 37.4%

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

   if -3.10000000000000014e-111 < (*.f64 z t) < 1.08000000000000001e49

   1. Initial program 91.8%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 44.7

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

   5. Applied rewrites 44.7%

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

4. Final simplification 49.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot z \leq -1.75 \cdot 10^{+89}:\\ \;\;\;\;\left(t \cdot z\right) \cdot 2\\ \mathbf{elif}\;t \cdot z \leq -3.1 \cdot 10^{-111}:\\ \;\;\;\;\left(\left(i \cdot c\right) \cdot a\right) \cdot -2\\ \mathbf{elif}\;t \cdot z \leq 1.08 \cdot 10^{+49}:\\ \;\;\;\;\left(y \cdot x\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(t \cdot z\right) \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 58.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(b \cdot c + a\right) \cdot c\right) \cdot i \leq -2 \cdot 10^{+44}:\\ \;\;\;\;\left(\left(i \cdot c\right) \cdot a\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, t \cdot z\right) \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* (* (+ (* b c) a) c) i) -2e+44)
   (* (* (* i c) a) -2.0)
   (* (fma y x (* t z)) 2.0)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (((((b * c) + a) * c) * i) <= -2e+44) {
		tmp = ((i * c) * a) * -2.0;
	} else {
		tmp = fma(y, x, (t * z)) * 2.0;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(b * c) + a) * c) * i) <= -2e+44)
		tmp = Float64(Float64(Float64(i * c) * a) * -2.0);
	else
		tmp = Float64(fma(y, x, Float64(t * z)) * 2.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(N[(N[(N[(b * c), $MachinePrecision] + a), $MachinePrecision] * c), $MachinePrecision] * i), $MachinePrecision], -2e+44], N[(N[(N[(i * c), $MachinePrecision] * a), $MachinePrecision] * -2.0), $MachinePrecision], N[(N[(y * x + N[(t * z), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -2.0000000000000002e44

   1. Initial program 84.3%

      \[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]
   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 37.0

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

   5. Applied rewrites 37.0%

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

   if -2.0000000000000002e44 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 93.8%

      \[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]
   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 70.1

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

   5. Applied rewrites 70.1%

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

4. Final simplification 59.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(b \cdot c + a\right) \cdot c\right) \cdot i \leq -2 \cdot 10^{+44}:\\ \;\;\;\;\left(\left(i \cdot c\right) \cdot a\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, t \cdot z\right) \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 44.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(y \cdot x\right) \cdot 2\\ \mathbf{if}\;y \cdot x \leq -1 \cdot 10^{+40}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \cdot x \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\left(t \cdot z\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* y x) 2.0)))
   (if (<= (* y x) -1e+40) t_1 (if (<= (* y x) 2e-12) (* (* 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 = (y * x) * 2.0;
	double tmp;
	if ((y * x) <= -1e+40) {
		tmp = t_1;
	} else if ((y * x) <= 2e-12) {
		tmp = (t * z) * 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 = (y * x) * 2.0d0
    if ((y * x) <= (-1d+40)) then
        tmp = t_1
    else if ((y * x) <= 2d-12) then
        tmp = (t * z) * 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 = (y * x) * 2.0;
	double tmp;
	if ((y * x) <= -1e+40) {
		tmp = t_1;
	} else if ((y * x) <= 2e-12) {
		tmp = (t * z) * 2.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = (y * x) * 2.0
	tmp = 0
	if (y * x) <= -1e+40:
		tmp = t_1
	elif (y * x) <= 2e-12:
		tmp = (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(y * x) * 2.0)
	tmp = 0.0
	if (Float64(y * x) <= -1e+40)
		tmp = t_1;
	elseif (Float64(y * x) <= 2e-12)
		tmp = Float64(Float64(t * z) * 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 = (y * x) * 2.0;
	tmp = 0.0;
	if ((y * x) <= -1e+40)
		tmp = t_1;
	elseif ((y * x) <= 2e-12)
		tmp = (t * z) * 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[(y * x), $MachinePrecision] * 2.0), $MachinePrecision]}, If[LessEqual[N[(y * x), $MachinePrecision], -1e+40], t$95$1, If[LessEqual[N[(y * x), $MachinePrecision], 2e-12], N[(N[(t * z), $MachinePrecision] * 2.0), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -1.00000000000000003e40 or 1.99999999999999996e-12 < (*.f64 x y)

   1. Initial program 87.7%

      \[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]
   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 55.2

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

   5. Applied rewrites 55.2%

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

   if -1.00000000000000003e40 < (*.f64 x y) < 1.99999999999999996e-12

   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 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 40.3

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

   5. Applied rewrites 40.3%

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

4. Final simplification 47.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot x \leq -1 \cdot 10^{+40}:\\ \;\;\;\;\left(y \cdot x\right) \cdot 2\\ \mathbf{elif}\;y \cdot x \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\left(t \cdot z\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot x\right) \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 29.8% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.8%

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

3. Taylor expanded in 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 30.2

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

5. Applied rewrites 30.2%

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

6. Final simplification 30.2%

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

Developer Target 1: 94.9% 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 2024235 
(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))))