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

Percentage Accurate: 90.2% → 95.7%

Time: 13.8s

Alternatives: 15

Speedup: 0.5×

• 47.5% 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: 95.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z, t, \left(-c\right) \cdot \left(\mathsf{fma}\left(b, c, a\right) \cdot i\right)\right) \cdot 2\\ t_2 := i \cdot \left(\left(c \cdot b + a\right) \cdot c\right)\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+228}:\\ \;\;\;\;\left(\left(t \cdot z + x \cdot y\right) - t\_2\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (fma z t (* (- c) (* (fma b c a) i))) 2.0))
        (t_2 (* i (* (+ (* c b) a) c))))
   (if (<= t_2 (- INFINITY))
     t_1
     (if (<= t_2 2e+228) (* (- (+ (* t z) (* x y)) t_2) 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, (-c * (fma(b, c, a) * i))) * 2.0;
	double t_2 = i * (((c * b) + a) * c);
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_2 <= 2e+228) {
		tmp = (((t * z) + (x * y)) - t_2) * 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, Float64(Float64(-c) * Float64(fma(b, c, a) * i))) * 2.0)
	t_2 = Float64(i * Float64(Float64(Float64(c * b) + a) * c))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_2 <= 2e+228)
		tmp = Float64(Float64(Float64(Float64(t * z) + Float64(x * y)) - t_2) * 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[((-c) * N[(N[(b * c + a), $MachinePrecision] * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]}, Block[{t$95$2 = N[(i * N[(N[(N[(c * b), $MachinePrecision] + a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$1, If[LessEqual[t$95$2, 2e+228], N[(N[(N[(N[(t * z), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision] - t$95$2), $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) < -inf.0 or 1.9999999999999998e228 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 76.9%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. sub-neg N/A

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

      6. associate-+l+ N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

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

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

      14. lower-fma.f64 N/A

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

   4. Applied rewrites 90.6%

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

   5. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 94.7

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

   7. Applied rewrites 94.7%

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

   if -inf.0 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 1.9999999999999998e228

   1. Initial program 97.9%

      \[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(z, t, \left(-c\right) \cdot \left(\mathsf{fma}\left(b, c, a\right) \cdot i\right)\right) \cdot 2\\ \mathbf{elif}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq 2 \cdot 10^{+228}:\\ \;\;\;\;\left(\left(t \cdot z + x \cdot y\right) - i \cdot \left(\left(c \cdot b + a\right) \cdot c\right)\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t, \left(-c\right) \cdot \left(\mathsf{fma}\left(b, c, a\right) \cdot i\right)\right) \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z, t, \left(-c\right) \cdot \left(\mathsf{fma}\left(b, c, a\right) \cdot i\right)\right) \cdot 2\\ t_2 := i \cdot \left(\left(c \cdot b + a\right) \cdot c\right)\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+138}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{+35}:\\ \;\;\;\;\mathsf{fma}\left(t, z, x \cdot y\right) \cdot 2\\ \mathbf{elif}\;t\_2 \leq 3.2 \cdot 10^{+177}:\\ \;\;\;\;\mathsf{fma}\left(-i, \mathsf{fma}\left(c, b, a\right) \cdot c, x \cdot y\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (fma z t (* (- c) (* (fma b c a) i))) 2.0))
        (t_2 (* i (* (+ (* c b) a) c))))
   (if (<= t_2 -1e+138)
     t_1
     (if (<= t_2 1e+35)
       (* (fma t z (* x y)) 2.0)
       (if (<= t_2 3.2e+177)
         (* (fma (- i) (* (fma c b a) c) (* 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 = fma(z, t, (-c * (fma(b, c, a) * i))) * 2.0;
	double t_2 = i * (((c * b) + a) * c);
	double tmp;
	if (t_2 <= -1e+138) {
		tmp = t_1;
	} else if (t_2 <= 1e+35) {
		tmp = fma(t, z, (x * y)) * 2.0;
	} else if (t_2 <= 3.2e+177) {
		tmp = fma(-i, (fma(c, b, a) * c), (x * y)) * 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, Float64(Float64(-c) * Float64(fma(b, c, a) * i))) * 2.0)
	t_2 = Float64(i * Float64(Float64(Float64(c * b) + a) * c))
	tmp = 0.0
	if (t_2 <= -1e+138)
		tmp = t_1;
	elseif (t_2 <= 1e+35)
		tmp = Float64(fma(t, z, Float64(x * y)) * 2.0);
	elseif (t_2 <= 3.2e+177)
		tmp = Float64(fma(Float64(-i), Float64(fma(c, b, a) * c), Float64(x * y)) * 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[((-c) * N[(N[(b * c + a), $MachinePrecision] * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]}, Block[{t$95$2 = N[(i * N[(N[(N[(c * b), $MachinePrecision] + a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+138], t$95$1, If[LessEqual[t$95$2, 1e+35], N[(N[(t * z + N[(x * y), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], If[LessEqual[t$95$2, 3.2e+177], N[(N[((-i) * N[(N[(c * b + a), $MachinePrecision] * c), $MachinePrecision] + N[(x * y), $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) < -1e138 or 3.2e177 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 80.8%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. sub-neg N/A

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

      6. associate-+l+ N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

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

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

      14. lower-fma.f64 N/A

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

   4. Applied rewrites 90.7%

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

   5. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 93.0

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

   7. Applied rewrites 93.0%

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

   if -1e138 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 9.9999999999999997e34

   1. Initial program 97.2%

      \[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]
   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. lower-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 93.0

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

   5. Applied rewrites 93.0%

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

   if 9.9999999999999997e34 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 3.2e177

   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 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 83.7

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

   5. Applied rewrites 83.7%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq -1 \cdot 10^{+138}:\\ \;\;\;\;\mathsf{fma}\left(z, t, \left(-c\right) \cdot \left(\mathsf{fma}\left(b, c, a\right) \cdot i\right)\right) \cdot 2\\ \mathbf{elif}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq 10^{+35}:\\ \;\;\;\;\mathsf{fma}\left(t, z, x \cdot y\right) \cdot 2\\ \mathbf{elif}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq 3.2 \cdot 10^{+177}:\\ \;\;\;\;\mathsf{fma}\left(-i, \mathsf{fma}\left(c, b, a\right) \cdot c, x \cdot y\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t, \left(-c\right) \cdot \left(\mathsf{fma}\left(b, c, a\right) \cdot i\right)\right) \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -2 \cdot \left(\left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right)\\ t_2 := i \cdot \left(\left(c \cdot b + a\right) \cdot c\right)\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+138}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{+73}:\\ \;\;\;\;\mathsf{fma}\left(t, z, x \cdot y\right) \cdot 2\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+228}:\\ \;\;\;\;\mathsf{fma}\left(-i, \mathsf{fma}\left(c, b, a\right) \cdot c, t \cdot z\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 (* i (* (+ (* c b) a) c))))
   (if (<= t_2 -1e+138)
     t_1
     (if (<= t_2 4e+73)
       (* (fma t z (* x y)) 2.0)
       (if (<= t_2 2e+228)
         (* (fma (- i) (* (fma c b a) c) (* 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 = i * (((c * b) + a) * c);
	double tmp;
	if (t_2 <= -1e+138) {
		tmp = t_1;
	} else if (t_2 <= 4e+73) {
		tmp = fma(t, z, (x * y)) * 2.0;
	} else if (t_2 <= 2e+228) {
		tmp = fma(-i, (fma(c, b, a) * c), (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(i * Float64(Float64(Float64(c * b) + a) * c))
	tmp = 0.0
	if (t_2 <= -1e+138)
		tmp = t_1;
	elseif (t_2 <= 4e+73)
		tmp = Float64(fma(t, z, Float64(x * y)) * 2.0);
	elseif (t_2 <= 2e+228)
		tmp = Float64(fma(Float64(-i), Float64(fma(c, b, a) * c), 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[(i * N[(N[(N[(c * b), $MachinePrecision] + a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+138], t$95$1, If[LessEqual[t$95$2, 4e+73], N[(N[(t * z + N[(x * y), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], If[LessEqual[t$95$2, 2e+228], N[(N[((-i) * N[(N[(c * b + a), $MachinePrecision] * c), $MachinePrecision] + 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) < -1e138 or 1.9999999999999998e228 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 79.5%

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

   3. Taylor expanded in i around inf

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

      1. *-commutative N/A

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

      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. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 89.3

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

   5. Applied rewrites 89.3%

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

   if -1e138 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 3.99999999999999993e73

   1. Initial program 97.3%

      \[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]
   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. lower-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 92.3

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

   5. Applied rewrites 92.3%

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

   if 3.99999999999999993e73 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 1.9999999999999998e228

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. 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. lower-*.f64 79.3

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

   5. Applied rewrites 79.3%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq -1 \cdot 10^{+138}:\\ \;\;\;\;-2 \cdot \left(\left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right)\\ \mathbf{elif}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq 4 \cdot 10^{+73}:\\ \;\;\;\;\mathsf{fma}\left(t, z, x \cdot y\right) \cdot 2\\ \mathbf{elif}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq 2 \cdot 10^{+228}:\\ \;\;\;\;\mathsf{fma}\left(-i, \mathsf{fma}\left(c, b, a\right) \cdot c, t \cdot z\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -2 \cdot \left(\left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right)\\ t_2 := i \cdot \left(\left(c \cdot b + a\right) \cdot c\right)\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+138}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{+73}:\\ \;\;\;\;\mathsf{fma}\left(t, z, x \cdot y\right) \cdot 2\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+228}:\\ \;\;\;\;\mathsf{fma}\left(z, t, \left(\left(-a\right) \cdot c\right) \cdot i\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 (* i (* (+ (* c b) a) c))))
   (if (<= t_2 -1e+138)
     t_1
     (if (<= t_2 4e+73)
       (* (fma t z (* x y)) 2.0)
       (if (<= t_2 2e+228) (* (fma z t (* (* (- a) c) i)) 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 = i * (((c * b) + a) * c);
	double tmp;
	if (t_2 <= -1e+138) {
		tmp = t_1;
	} else if (t_2 <= 4e+73) {
		tmp = fma(t, z, (x * y)) * 2.0;
	} else if (t_2 <= 2e+228) {
		tmp = fma(z, t, ((-a * c) * i)) * 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(i * Float64(Float64(Float64(c * b) + a) * c))
	tmp = 0.0
	if (t_2 <= -1e+138)
		tmp = t_1;
	elseif (t_2 <= 4e+73)
		tmp = Float64(fma(t, z, Float64(x * y)) * 2.0);
	elseif (t_2 <= 2e+228)
		tmp = Float64(fma(z, t, Float64(Float64(Float64(-a) * c) * i)) * 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[(i * N[(N[(N[(c * b), $MachinePrecision] + a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+138], t$95$1, If[LessEqual[t$95$2, 4e+73], N[(N[(t * z + N[(x * y), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], If[LessEqual[t$95$2, 2e+228], N[(N[(z * t + N[(N[((-a) * c), $MachinePrecision] * i), $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) < -1e138 or 1.9999999999999998e228 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 79.5%

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

   3. Taylor expanded in i around inf

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

      1. *-commutative N/A

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

      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. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 89.3

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

   5. Applied rewrites 89.3%

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

   if -1e138 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 3.99999999999999993e73

   1. Initial program 97.3%

      \[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]
   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. lower-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 92.3

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

   5. Applied rewrites 92.3%

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

   if 3.99999999999999993e73 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 1.9999999999999998e228

   1. Initial program 99.8%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. sub-neg N/A

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

      6. associate-+l+ N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

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

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

      14. lower-fma.f64 N/A

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

   4. Applied rewrites 77.5%

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

   5. Taylor expanded in a around inf

      \[\leadsto 2 \cdot \mathsf{fma}\left(z, t, \color{blue}{-1 \cdot \left(a \cdot \left(c \cdot i\right)\right)}\right) \]
   6. Step-by-step derivation

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. neg-mul-1 N/A

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

      7. lower-neg.f64 71.8

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

   7. Applied rewrites 71.8%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq -1 \cdot 10^{+138}:\\ \;\;\;\;-2 \cdot \left(\left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right)\\ \mathbf{elif}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq 4 \cdot 10^{+73}:\\ \;\;\;\;\mathsf{fma}\left(t, z, x \cdot y\right) \cdot 2\\ \mathbf{elif}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq 2 \cdot 10^{+228}:\\ \;\;\;\;\mathsf{fma}\left(z, t, \left(\left(-a\right) \cdot c\right) \cdot i\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z, t, \left(-c\right) \cdot \left(\mathsf{fma}\left(b, c, a\right) \cdot i\right)\right) \cdot 2\\ t_2 := i \cdot \left(\left(c \cdot b + a\right) \cdot c\right)\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+138}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+228}:\\ \;\;\;\;\left(\left(t \cdot z + x \cdot y\right) - \left(i \cdot c\right) \cdot a\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (fma z t (* (- c) (* (fma b c a) i))) 2.0))
        (t_2 (* i (* (+ (* c b) a) c))))
   (if (<= t_2 -1e+138)
     t_1
     (if (<= t_2 2e+228) (* (- (+ (* t z) (* x y)) (* (* i c) a)) 2.0) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(z, t, (-c * (fma(b, c, a) * i))) * 2.0;
	double t_2 = i * (((c * b) + a) * c);
	double tmp;
	if (t_2 <= -1e+138) {
		tmp = t_1;
	} else if (t_2 <= 2e+228) {
		tmp = (((t * z) + (x * y)) - ((i * c) * a)) * 2.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(fma(z, t, Float64(Float64(-c) * Float64(fma(b, c, a) * i))) * 2.0)
	t_2 = Float64(i * Float64(Float64(Float64(c * b) + a) * c))
	tmp = 0.0
	if (t_2 <= -1e+138)
		tmp = t_1;
	elseif (t_2 <= 2e+228)
		tmp = Float64(Float64(Float64(Float64(t * z) + Float64(x * y)) - Float64(Float64(i * c) * a)) * 2.0);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(z * t + N[((-c) * N[(N[(b * c + a), $MachinePrecision] * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]}, Block[{t$95$2 = N[(i * N[(N[(N[(c * b), $MachinePrecision] + a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+138], t$95$1, If[LessEqual[t$95$2, 2e+228], N[(N[(N[(N[(t * z), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision] - N[(N[(i * c), $MachinePrecision] * a), $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) < -1e138 or 1.9999999999999998e228 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 79.5%

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

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

      4. +-commutative N/A

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

      5. sub-neg N/A

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

      6. associate-+l+ N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

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

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

      14. lower-fma.f64 N/A

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

   4. Applied rewrites 91.0%

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

   5. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 93.8

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

   7. Applied rewrites 93.8%

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

   if -1e138 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 1.9999999999999998e228

   1. Initial program 97.7%

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

   3. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 92.2

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

   5. Applied rewrites 92.2%

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

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq -1 \cdot 10^{+138}:\\ \;\;\;\;\mathsf{fma}\left(z, t, \left(-c\right) \cdot \left(\mathsf{fma}\left(b, c, a\right) \cdot i\right)\right) \cdot 2\\ \mathbf{elif}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq 2 \cdot 10^{+228}:\\ \;\;\;\;\left(\left(t \cdot z + x \cdot y\right) - \left(i \cdot c\right) \cdot a\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t, \left(-c\right) \cdot \left(\mathsf{fma}\left(b, c, a\right) \cdot i\right)\right) \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 93.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(N[(N[(t * z), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision] - N[(i * N[(N[(N[(c * b), $MachinePrecision] + a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(z * t + N[(N[((-i) * N[(c * b + a), $MachinePrecision]), $MachinePrecision] * c + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(z * t + N[((-c) * N[(N[(b * c + a), $MachinePrecision] * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $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 94.4%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. sub-neg N/A

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

      6. associate-+l+ N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

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

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

      14. lower-fma.f64 N/A

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

   4. Applied rewrites 96.0%

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

   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. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. sub-neg N/A

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

      6. associate-+l+ N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

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

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

      14. lower-fma.f64 N/A

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

   4. Applied rewrites 28.6%

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

   5. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 71.4

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

   7. Applied rewrites 71.4%

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

4. Final simplification 94.7%

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

Alternative 7: 82.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -2 \cdot \left(\left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right)\\ t_2 := i \cdot \left(\left(c \cdot b + a\right) \cdot c\right)\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+138}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+188}:\\ \;\;\;\;\mathsf{fma}\left(t, z, x \cdot y\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* -2.0 (* (* (fma c b a) i) c))) (t_2 (* i (* (+ (* c b) a) c))))
   (if (<= t_2 -1e+138)
     t_1
     (if (<= t_2 2e+188) (* (fma t z (* 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 = -2.0 * ((fma(c, b, a) * i) * c);
	double t_2 = i * (((c * b) + a) * c);
	double tmp;
	if (t_2 <= -1e+138) {
		tmp = t_1;
	} else if (t_2 <= 2e+188) {
		tmp = fma(t, z, (x * y)) * 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(i * Float64(Float64(Float64(c * b) + a) * c))
	tmp = 0.0
	if (t_2 <= -1e+138)
		tmp = t_1;
	elseif (t_2 <= 2e+188)
		tmp = Float64(fma(t, z, Float64(x * y)) * 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[(i * N[(N[(N[(c * b), $MachinePrecision] + a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+138], t$95$1, If[LessEqual[t$95$2, 2e+188], N[(N[(t * z + N[(x * y), $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) < -1e138 or 2e188 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 80.2%

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

   3. Taylor expanded in 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. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 88.5

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

   5. Applied rewrites 88.5%

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

   if -1e138 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 2e188

   1. Initial program 97.7%

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

   3. Taylor expanded in c around 0

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

      1. lower-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 85.6

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

   5. Applied rewrites 85.6%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq -1 \cdot 10^{+138}:\\ \;\;\;\;-2 \cdot \left(\left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right)\\ \mathbf{elif}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq 2 \cdot 10^{+188}:\\ \;\;\;\;\mathsf{fma}\left(t, z, x \cdot y\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := i \cdot \left(\left(c \cdot b + a\right) \cdot c\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+223}:\\ \;\;\;\;\left(\left(\left(c \cdot b\right) \cdot i\right) \cdot c\right) \cdot -2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+228}:\\ \;\;\;\;\mathsf{fma}\left(t, z, x \cdot y\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(i \cdot c\right) \cdot b\right) \cdot c\right) \cdot -2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* i (* (+ (* c b) a) c))))
   (if (<= t_1 -2e+223)
     (* (* (* (* c b) i) c) -2.0)
     (if (<= t_1 2e+228)
       (* (fma t z (* x y)) 2.0)
       (* (* (* (* i c) b) c) -2.0)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = i * (((c * b) + a) * c);
	double tmp;
	if (t_1 <= -2e+223) {
		tmp = (((c * b) * i) * c) * -2.0;
	} else if (t_1 <= 2e+228) {
		tmp = fma(t, z, (x * y)) * 2.0;
	} else {
		tmp = (((i * c) * b) * c) * -2.0;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(i * Float64(Float64(Float64(c * b) + a) * c))
	tmp = 0.0
	if (t_1 <= -2e+223)
		tmp = Float64(Float64(Float64(Float64(c * b) * i) * c) * -2.0);
	elseif (t_1 <= 2e+228)
		tmp = Float64(fma(t, z, Float64(x * y)) * 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(i * c) * b) * c) * -2.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(i * N[(N[(N[(c * b), $MachinePrecision] + a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+223], N[(N[(N[(N[(c * b), $MachinePrecision] * i), $MachinePrecision] * c), $MachinePrecision] * -2.0), $MachinePrecision], If[LessEqual[t$95$1, 2e+228], N[(N[(t * z + N[(x * y), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(N[(N[(i * c), $MachinePrecision] * b), $MachinePrecision] * c), $MachinePrecision] * -2.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -2.00000000000000009e223

   1. Initial program 79.6%

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

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 89.0

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

   5. Applied rewrites 89.0%

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

   6. Taylor expanded in c around inf

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

   8. Applied rewrites 61.9%

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

   10. Applied rewrites 61.9%

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

   if -2.00000000000000009e223 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 1.9999999999999998e228

   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. lower-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 80.5

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

   5. Applied rewrites 80.5%

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

   if 1.9999999999999998e228 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 75.3%

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

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 93.3

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

   5. Applied rewrites 93.3%

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

   6. Taylor expanded in c around inf

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

   8. Applied rewrites 84.6%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq -2 \cdot 10^{+223}:\\ \;\;\;\;\left(\left(\left(c \cdot b\right) \cdot i\right) \cdot c\right) \cdot -2\\ \mathbf{elif}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq 2 \cdot 10^{+228}:\\ \;\;\;\;\mathsf{fma}\left(t, z, x \cdot y\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(i \cdot c\right) \cdot b\right) \cdot c\right) \cdot -2\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 74.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\left(c \cdot b\right) \cdot i\right) \cdot c\right) \cdot -2\\ t_2 := i \cdot \left(\left(c \cdot b + a\right) \cdot c\right)\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+223}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+228}:\\ \;\;\;\;\mathsf{fma}\left(t, z, x \cdot y\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* (* (* c b) i) c) -2.0)) (t_2 (* i (* (+ (* c b) a) c))))
   (if (<= t_2 -2e+223)
     t_1
     (if (<= t_2 2e+228) (* (fma t z (* 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 = (((c * b) * i) * c) * -2.0;
	double t_2 = i * (((c * b) + a) * c);
	double tmp;
	if (t_2 <= -2e+223) {
		tmp = t_1;
	} else if (t_2 <= 2e+228) {
		tmp = fma(t, z, (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(Float64(c * b) * i) * c) * -2.0)
	t_2 = Float64(i * Float64(Float64(Float64(c * b) + a) * c))
	tmp = 0.0
	if (t_2 <= -2e+223)
		tmp = t_1;
	elseif (t_2 <= 2e+228)
		tmp = Float64(fma(t, z, Float64(x * y)) * 2.0);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(N[(c * b), $MachinePrecision] * i), $MachinePrecision] * c), $MachinePrecision] * -2.0), $MachinePrecision]}, Block[{t$95$2 = N[(i * N[(N[(N[(c * b), $MachinePrecision] + a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+223], t$95$1, If[LessEqual[t$95$2, 2e+228], N[(N[(t * z + N[(x * y), $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) < -2.00000000000000009e223 or 1.9999999999999998e228 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 77.9%

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

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 90.7

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

   5. Applied rewrites 90.7%

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

   6. Taylor expanded in c around inf

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

   8. Applied rewrites 70.7%

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

   10. Applied rewrites 69.1%

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

   if -2.00000000000000009e223 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 1.9999999999999998e228

   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. lower-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 80.5

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

   5. Applied rewrites 80.5%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq -2 \cdot 10^{+223}:\\ \;\;\;\;\left(\left(\left(c \cdot b\right) \cdot i\right) \cdot c\right) \cdot -2\\ \mathbf{elif}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq 2 \cdot 10^{+228}:\\ \;\;\;\;\mathsf{fma}\left(t, z, x \cdot y\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(c \cdot b\right) \cdot i\right) \cdot c\right) \cdot -2\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 62.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := i \cdot \left(\left(c \cdot b + a\right) \cdot c\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+138}:\\ \;\;\;\;\left(\left(\left(-a\right) \cdot c\right) \cdot i\right) \cdot 2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+188}:\\ \;\;\;\;\mathsf{fma}\left(t, z, x \cdot y\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\left(i \cdot c\right) \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* i (* (+ (* c b) a) c))))
   (if (<= t_1 -1e+138)
     (* (* (* (- a) c) i) 2.0)
     (if (<= t_1 2e+188) (* (fma t z (* x y)) 2.0) (* -2.0 (* (* i c) a))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = i * (((c * b) + a) * c);
	double tmp;
	if (t_1 <= -1e+138) {
		tmp = ((-a * c) * i) * 2.0;
	} else if (t_1 <= 2e+188) {
		tmp = fma(t, z, (x * y)) * 2.0;
	} else {
		tmp = -2.0 * ((i * c) * a);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(i * Float64(Float64(Float64(c * b) + a) * c))
	tmp = 0.0
	if (t_1 <= -1e+138)
		tmp = Float64(Float64(Float64(Float64(-a) * c) * i) * 2.0);
	elseif (t_1 <= 2e+188)
		tmp = Float64(fma(t, z, Float64(x * y)) * 2.0);
	else
		tmp = Float64(-2.0 * Float64(Float64(i * c) * a));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(i * N[(N[(N[(c * b), $MachinePrecision] + a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+138], N[(N[(N[((-a) * c), $MachinePrecision] * i), $MachinePrecision] * 2.0), $MachinePrecision], If[LessEqual[t$95$1, 2e+188], N[(N[(t * z + N[(x * y), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(-2.0 * N[(N[(i * c), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   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 t around inf

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

      1. lower-*.f64 10.3

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

   5. Applied rewrites 10.3%

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

   6. Taylor expanded in a around inf

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. neg-mul-1 N/A

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

      7. lower-neg.f64 48.5

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

   8. Applied rewrites 48.5%

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

   if -1e138 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 2e188

   1. Initial program 97.7%

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

   3. Taylor expanded in c around 0

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

      1. lower-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 85.6

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

   5. Applied rewrites 85.6%

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

   if 2e188 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 77.4%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 30.4

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

   5. Applied rewrites 30.4%

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

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq -1 \cdot 10^{+138}:\\ \;\;\;\;\left(\left(\left(-a\right) \cdot c\right) \cdot i\right) \cdot 2\\ \mathbf{elif}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq 2 \cdot 10^{+188}:\\ \;\;\;\;\mathsf{fma}\left(t, z, x \cdot y\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\left(i \cdot c\right) \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 62.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := i \cdot \left(\left(c \cdot b + a\right) \cdot c\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+138}:\\ \;\;\;\;\left(\left(i \cdot a\right) \cdot c\right) \cdot -2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+188}:\\ \;\;\;\;\mathsf{fma}\left(t, z, x \cdot y\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\left(i \cdot c\right) \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* i (* (+ (* c b) a) c))))
   (if (<= t_1 -1e+138)
     (* (* (* i a) c) -2.0)
     (if (<= t_1 2e+188) (* (fma t z (* x y)) 2.0) (* -2.0 (* (* i c) a))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = i * (((c * b) + a) * c);
	double tmp;
	if (t_1 <= -1e+138) {
		tmp = ((i * a) * c) * -2.0;
	} else if (t_1 <= 2e+188) {
		tmp = fma(t, z, (x * y)) * 2.0;
	} else {
		tmp = -2.0 * ((i * c) * a);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(i * Float64(Float64(Float64(c * b) + a) * c))
	tmp = 0.0
	if (t_1 <= -1e+138)
		tmp = Float64(Float64(Float64(i * a) * c) * -2.0);
	elseif (t_1 <= 2e+188)
		tmp = Float64(fma(t, z, Float64(x * y)) * 2.0);
	else
		tmp = Float64(-2.0 * Float64(Float64(i * c) * a));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(i * N[(N[(N[(c * b), $MachinePrecision] + a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+138], N[(N[(N[(i * a), $MachinePrecision] * c), $MachinePrecision] * -2.0), $MachinePrecision], If[LessEqual[t$95$1, 2e+188], N[(N[(t * z + N[(x * y), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(-2.0 * N[(N[(i * c), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 82.0%

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

   3. Taylor expanded in i around inf

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

      1. *-commutative N/A

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

      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. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 87.1

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

   5. Applied rewrites 87.1%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 46.6%

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

   if -1e138 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 2e188

   1. Initial program 97.7%

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

   3. Taylor expanded in c around 0

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

      1. lower-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 85.6

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

   5. Applied rewrites 85.6%

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

   if 2e188 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 77.4%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 30.4

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

   5. Applied rewrites 30.4%

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

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq -1 \cdot 10^{+138}:\\ \;\;\;\;\left(\left(i \cdot a\right) \cdot c\right) \cdot -2\\ \mathbf{elif}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq 2 \cdot 10^{+188}:\\ \;\;\;\;\mathsf{fma}\left(t, z, x \cdot y\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\left(i \cdot c\right) \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 42.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x \cdot y\right) \cdot 2\\ \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+171}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \cdot y \leq -5 \cdot 10^{-53}:\\ \;\;\;\;\left(\left(i \cdot a\right) \cdot c\right) \cdot -2\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{-125}:\\ \;\;\;\;\left(t \cdot z\right) \cdot 2\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{+15}:\\ \;\;\;\;-2 \cdot \left(\left(i \cdot c\right) \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* x y) 2.0)))
   (if (<= (* x y) -2e+171)
     t_1
     (if (<= (* x y) -5e-53)
       (* (* (* i a) c) -2.0)
       (if (<= (* x y) 5e-125)
         (* (* t z) 2.0)
         (if (<= (* x y) 2e+15) (* -2.0 (* (* i c) a)) t_1))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (x * y) * 2.0;
	double tmp;
	if ((x * y) <= -2e+171) {
		tmp = t_1;
	} else if ((x * y) <= -5e-53) {
		tmp = ((i * a) * c) * -2.0;
	} else if ((x * y) <= 5e-125) {
		tmp = (t * z) * 2.0;
	} else if ((x * y) <= 2e+15) {
		tmp = -2.0 * ((i * c) * a);
	} 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 = (x * y) * 2.0d0
    if ((x * y) <= (-2d+171)) then
        tmp = t_1
    else if ((x * y) <= (-5d-53)) then
        tmp = ((i * a) * c) * (-2.0d0)
    else if ((x * y) <= 5d-125) then
        tmp = (t * z) * 2.0d0
    else if ((x * y) <= 2d+15) then
        tmp = (-2.0d0) * ((i * c) * a)
    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 = (x * y) * 2.0;
	double tmp;
	if ((x * y) <= -2e+171) {
		tmp = t_1;
	} else if ((x * y) <= -5e-53) {
		tmp = ((i * a) * c) * -2.0;
	} else if ((x * y) <= 5e-125) {
		tmp = (t * z) * 2.0;
	} else if ((x * y) <= 2e+15) {
		tmp = -2.0 * ((i * c) * a);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = (x * y) * 2.0
	tmp = 0
	if (x * y) <= -2e+171:
		tmp = t_1
	elif (x * y) <= -5e-53:
		tmp = ((i * a) * c) * -2.0
	elif (x * y) <= 5e-125:
		tmp = (t * z) * 2.0
	elif (x * y) <= 2e+15:
		tmp = -2.0 * ((i * c) * a)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(x * y) * 2.0)
	tmp = 0.0
	if (Float64(x * y) <= -2e+171)
		tmp = t_1;
	elseif (Float64(x * y) <= -5e-53)
		tmp = Float64(Float64(Float64(i * a) * c) * -2.0);
	elseif (Float64(x * y) <= 5e-125)
		tmp = Float64(Float64(t * z) * 2.0);
	elseif (Float64(x * y) <= 2e+15)
		tmp = Float64(-2.0 * Float64(Float64(i * c) * a));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	t_1 = (x * y) * 2.0;
	tmp = 0.0;
	if ((x * y) <= -2e+171)
		tmp = t_1;
	elseif ((x * y) <= -5e-53)
		tmp = ((i * a) * c) * -2.0;
	elseif ((x * y) <= 5e-125)
		tmp = (t * z) * 2.0;
	elseif ((x * y) <= 2e+15)
		tmp = -2.0 * ((i * c) * a);
	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[(x * y), $MachinePrecision] * 2.0), $MachinePrecision]}, If[LessEqual[N[(x * y), $MachinePrecision], -2e+171], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], -5e-53], N[(N[(N[(i * a), $MachinePrecision] * c), $MachinePrecision] * -2.0), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 5e-125], N[(N[(t * z), $MachinePrecision] * 2.0), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 2e+15], N[(-2.0 * N[(N[(i * c), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 x y) < -1.99999999999999991e171 or 2e15 < (*.f64 x y)

   1. Initial program 88.5%

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

   3. Taylor expanded in 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 55.1

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

   5. Applied rewrites 55.1%

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

   if -1.99999999999999991e171 < (*.f64 x y) < -5e-53

   1. Initial program 97.0%

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

   3. Taylor expanded in i around inf

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

      1. *-commutative N/A

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

      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. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 57.2

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

   5. Applied rewrites 57.2%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 38.7%

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

   if -5e-53 < (*.f64 x y) < 4.99999999999999967e-125

   1. Initial program 89.0%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 44.3

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

   5. Applied rewrites 44.3%

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

   if 4.99999999999999967e-125 < (*.f64 x y) < 2e15

   1. Initial program 83.9%

      \[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]
   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 44.4

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

   5. Applied rewrites 44.4%

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

4. Final simplification 47.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+171}:\\ \;\;\;\;\left(x \cdot y\right) \cdot 2\\ \mathbf{elif}\;x \cdot y \leq -5 \cdot 10^{-53}:\\ \;\;\;\;\left(\left(i \cdot a\right) \cdot c\right) \cdot -2\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{-125}:\\ \;\;\;\;\left(t \cdot z\right) \cdot 2\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{+15}:\\ \;\;\;\;-2 \cdot \left(\left(i \cdot c\right) \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 43.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x \cdot y\right) \cdot 2\\ \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+171}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \cdot y \leq -5 \cdot 10^{-53}:\\ \;\;\;\;\left(\left(i \cdot a\right) \cdot c\right) \cdot -2\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{+36}:\\ \;\;\;\;\left(t \cdot z\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* x y) 2.0)))
   (if (<= (* x y) -2e+171)
     t_1
     (if (<= (* x y) -5e-53)
       (* (* (* i a) c) -2.0)
       (if (<= (* x y) 2e+36) (* (* 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 = (x * y) * 2.0;
	double tmp;
	if ((x * y) <= -2e+171) {
		tmp = t_1;
	} else if ((x * y) <= -5e-53) {
		tmp = ((i * a) * c) * -2.0;
	} else if ((x * y) <= 2e+36) {
		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 = (x * y) * 2.0d0
    if ((x * y) <= (-2d+171)) then
        tmp = t_1
    else if ((x * y) <= (-5d-53)) then
        tmp = ((i * a) * c) * (-2.0d0)
    else if ((x * y) <= 2d+36) 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 = (x * y) * 2.0;
	double tmp;
	if ((x * y) <= -2e+171) {
		tmp = t_1;
	} else if ((x * y) <= -5e-53) {
		tmp = ((i * a) * c) * -2.0;
	} else if ((x * y) <= 2e+36) {
		tmp = (t * z) * 2.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = (x * y) * 2.0
	tmp = 0
	if (x * y) <= -2e+171:
		tmp = t_1
	elif (x * y) <= -5e-53:
		tmp = ((i * a) * c) * -2.0
	elif (x * y) <= 2e+36:
		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(x * y) * 2.0)
	tmp = 0.0
	if (Float64(x * y) <= -2e+171)
		tmp = t_1;
	elseif (Float64(x * y) <= -5e-53)
		tmp = Float64(Float64(Float64(i * a) * c) * -2.0);
	elseif (Float64(x * y) <= 2e+36)
		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 = (x * y) * 2.0;
	tmp = 0.0;
	if ((x * y) <= -2e+171)
		tmp = t_1;
	elseif ((x * y) <= -5e-53)
		tmp = ((i * a) * c) * -2.0;
	elseif ((x * y) <= 2e+36)
		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[(x * y), $MachinePrecision] * 2.0), $MachinePrecision]}, If[LessEqual[N[(x * y), $MachinePrecision], -2e+171], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], -5e-53], N[(N[(N[(i * a), $MachinePrecision] * c), $MachinePrecision] * -2.0), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 2e+36], N[(N[(t * z), $MachinePrecision] * 2.0), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -1.99999999999999991e171 or 2.00000000000000008e36 < (*.f64 x y)

   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 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 57.7

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

   5. Applied rewrites 57.7%

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

   if -1.99999999999999991e171 < (*.f64 x y) < -5e-53

   1. Initial program 97.0%

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

   3. Taylor expanded in i around inf

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

      1. *-commutative N/A

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

      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. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 57.2

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

   5. Applied rewrites 57.2%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 38.7%

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

   if -5e-53 < (*.f64 x y) < 2.00000000000000008e36

   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 t around inf

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

      1. lower-*.f64 39.6

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

   5. Applied rewrites 39.6%

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

4. Final simplification 46.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+171}:\\ \;\;\;\;\left(x \cdot y\right) \cdot 2\\ \mathbf{elif}\;x \cdot y \leq -5 \cdot 10^{-53}:\\ \;\;\;\;\left(\left(i \cdot a\right) \cdot c\right) \cdot -2\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{+36}:\\ \;\;\;\;\left(t \cdot z\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 44.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x \cdot y\right) \cdot 2\\ \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+171}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{+36}:\\ \;\;\;\;\left(t \cdot z\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* x y) 2.0)))
   (if (<= (* x y) -2e+171) t_1 (if (<= (* x y) 2e+36) (* (* 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 = (x * y) * 2.0;
	double tmp;
	if ((x * y) <= -2e+171) {
		tmp = t_1;
	} else if ((x * y) <= 2e+36) {
		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 = (x * y) * 2.0d0
    if ((x * y) <= (-2d+171)) then
        tmp = t_1
    else if ((x * y) <= 2d+36) 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 = (x * y) * 2.0;
	double tmp;
	if ((x * y) <= -2e+171) {
		tmp = t_1;
	} else if ((x * y) <= 2e+36) {
		tmp = (t * z) * 2.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = (x * y) * 2.0
	tmp = 0
	if (x * y) <= -2e+171:
		tmp = t_1
	elif (x * y) <= 2e+36:
		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(x * y) * 2.0)
	tmp = 0.0
	if (Float64(x * y) <= -2e+171)
		tmp = t_1;
	elseif (Float64(x * y) <= 2e+36)
		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 = (x * y) * 2.0;
	tmp = 0.0;
	if ((x * y) <= -2e+171)
		tmp = t_1;
	elseif ((x * y) <= 2e+36)
		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[(x * y), $MachinePrecision] * 2.0), $MachinePrecision]}, If[LessEqual[N[(x * y), $MachinePrecision], -2e+171], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], 2e+36], N[(N[(t * z), $MachinePrecision] * 2.0), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -1.99999999999999991e171 or 2.00000000000000008e36 < (*.f64 x y)

   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 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 57.7

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

   5. Applied rewrites 57.7%

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

   if -1.99999999999999991e171 < (*.f64 x y) < 2.00000000000000008e36

   1. Initial program 89.7%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 36.8

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

   5. Applied rewrites 36.8%

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

4. Final simplification 44.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+171}:\\ \;\;\;\;\left(x \cdot y\right) \cdot 2\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{+36}:\\ \;\;\;\;\left(t \cdot z\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 30.1% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.3%

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

3. Taylor expanded in t around inf

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

   1. lower-*.f64 27.9

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

5. Applied rewrites 27.9%

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

6. Final simplification 27.9%

   \[\leadsto \left(t \cdot z\right) \cdot 2 \]
7. Add Preprocessing

Developer Target 1: 94.3% 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 2024276 
(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))))