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

Percentage Accurate: 90.3% → 93.3%

Time: 10.8s

Alternatives: 16

Speedup: 1.1×

• 47.6% 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: 90.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 93.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := i \cdot \left(\left(c \cdot b + a\right) \cdot c\right)\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{+270}:\\ \;\;\;\;\left(\left(t \cdot z + y \cdot x\right) - t\_1\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t, \mathsf{fma}\left(y, x, \left(-c\right) \cdot \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right)\right)\right) \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* i (* (+ (* c b) a) c))))
   (if (<= t_1 5e+270)
     (* (- (+ (* t z) (* y x)) t_1) 2.0)
     (* (fma z t (fma y x (* (- c) (* (fma c b a) i)))) 2.0))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = i * (((c * b) + a) * c);
	double tmp;
	if (t_1 <= 5e+270) {
		tmp = (((t * z) + (y * x)) - t_1) * 2.0;
	} else {
		tmp = fma(z, t, fma(y, x, (-c * (fma(c, b, a) * i)))) * 2.0;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(i * Float64(Float64(Float64(c * b) + a) * c))
	tmp = 0.0
	if (t_1 <= 5e+270)
		tmp = Float64(Float64(Float64(Float64(t * z) + Float64(y * x)) - t_1) * 2.0);
	else
		tmp = Float64(fma(z, t, fma(y, x, Float64(Float64(-c) * Float64(fma(c, b, a) * i)))) * 2.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(i * N[(N[(N[(c * b), $MachinePrecision] + a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 5e+270], N[(N[(N[(N[(t * z), $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(z * t + N[(y * x + N[((-c) * N[(N[(c * b + a), $MachinePrecision] * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 4.99999999999999976e270

   1. Initial program 97.6%

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

   if 4.99999999999999976e270 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 72.7%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. sub-neg N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

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

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

      16. lower-*.f64 N/A

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

   4. Applied rewrites 98.1%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq 5 \cdot 10^{+270}:\\ \;\;\;\;\left(\left(t \cdot z + y \cdot x\right) - i \cdot \left(\left(c \cdot b + a\right) \cdot c\right)\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t, \mathsf{fma}\left(y, x, \left(-c\right) \cdot \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right)\right)\right) \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-i, \mathsf{fma}\left(c, b, a\right) \cdot c, t \cdot z\right) \cdot 2\\ t_2 := i \cdot \left(\left(c \cdot b + a\right) \cdot c\right)\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+85}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{-25}:\\ \;\;\;\;\mathsf{fma}\left(\left(-a\right) \cdot c, i, \mathsf{fma}\left(t, z, y \cdot x\right)\right) \cdot 2\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+298}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-c, \mathsf{fma}\left(b, c, a\right) \cdot i, y \cdot x\right) \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (fma (- i) (* (fma c b a) c) (* t z)) 2.0))
        (t_2 (* i (* (+ (* c b) a) c))))
   (if (<= t_2 -2e+85)
     t_1
     (if (<= t_2 1e-25)
       (* (fma (* (- a) c) i (fma t z (* y x))) 2.0)
       (if (<= t_2 5e+298)
         t_1
         (* (fma (- c) (* (fma b c a) i) (* y x)) 2.0))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(-i, (fma(c, b, a) * c), (t * z)) * 2.0;
	double t_2 = i * (((c * b) + a) * c);
	double tmp;
	if (t_2 <= -2e+85) {
		tmp = t_1;
	} else if (t_2 <= 1e-25) {
		tmp = fma((-a * c), i, fma(t, z, (y * x))) * 2.0;
	} else if (t_2 <= 5e+298) {
		tmp = t_1;
	} else {
		tmp = fma(-c, (fma(b, c, a) * i), (y * x)) * 2.0;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(fma(Float64(-i), Float64(fma(c, b, a) * c), Float64(t * z)) * 2.0)
	t_2 = Float64(i * Float64(Float64(Float64(c * b) + a) * c))
	tmp = 0.0
	if (t_2 <= -2e+85)
		tmp = t_1;
	elseif (t_2 <= 1e-25)
		tmp = Float64(fma(Float64(Float64(-a) * c), i, fma(t, z, Float64(y * x))) * 2.0);
	elseif (t_2 <= 5e+298)
		tmp = t_1;
	else
		tmp = Float64(fma(Float64(-c), Float64(fma(b, c, a) * i), Float64(y * x)) * 2.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[((-i) * N[(N[(c * b + a), $MachinePrecision] * c), $MachinePrecision] + N[(t * z), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]}, Block[{t$95$2 = N[(i * N[(N[(N[(c * b), $MachinePrecision] + a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+85], t$95$1, If[LessEqual[t$95$2, 1e-25], N[(N[(N[((-a) * c), $MachinePrecision] * i + N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], If[LessEqual[t$95$2, 5e+298], t$95$1, N[(N[((-c) * N[(N[(b * c + a), $MachinePrecision] * i), $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -2e85 or 1.00000000000000004e-25 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 5.0000000000000003e298

   1. Initial program 95.8%

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

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

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

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

      7. mul-1-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-*.f64 91.9

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

   5. Applied rewrites 91.9%

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

   if -2e85 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 1.00000000000000004e-25

   1. Initial program 99.1%

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

   3. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. neg-mul-1 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 98.0

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

   5. Applied rewrites 98.0%

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

   if 5.0000000000000003e298 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 70.5%

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

   3. Taylor expanded in a around inf

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

      1. associate-*r* N/A

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

      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 32.8

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

   5. Applied rewrites 32.8%

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

   6. Taylor expanded in z around 0

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. neg-mul-1 N/A

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

      7. lower-neg.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 88.1

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

   8. Applied rewrites 88.1%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq -2 \cdot 10^{+85}:\\ \;\;\;\;\mathsf{fma}\left(-i, \mathsf{fma}\left(c, b, a\right) \cdot c, t \cdot z\right) \cdot 2\\ \mathbf{elif}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq 10^{-25}:\\ \;\;\;\;\mathsf{fma}\left(\left(-a\right) \cdot c, i, \mathsf{fma}\left(t, z, y \cdot x\right)\right) \cdot 2\\ \mathbf{elif}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq 5 \cdot 10^{+298}:\\ \;\;\;\;\mathsf{fma}\left(-i, \mathsf{fma}\left(c, b, a\right) \cdot c, t \cdot z\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-c, \mathsf{fma}\left(b, c, a\right) \cdot i, y \cdot x\right) \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-i, \mathsf{fma}\left(c, b, a\right) \cdot c, t \cdot z\right) \cdot 2\\ t_2 := i \cdot \left(\left(c \cdot b + a\right) \cdot c\right)\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+85}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{-25}:\\ \;\;\;\;\mathsf{fma}\left(t, z, y \cdot x\right) \cdot 2\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+298}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-c, \mathsf{fma}\left(b, c, a\right) \cdot i, y \cdot x\right) \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (fma (- i) (* (fma c b a) c) (* t z)) 2.0))
        (t_2 (* i (* (+ (* c b) a) c))))
   (if (<= t_2 -2e+85)
     t_1
     (if (<= t_2 1e-25)
       (* (fma t z (* y x)) 2.0)
       (if (<= t_2 5e+298)
         t_1
         (* (fma (- c) (* (fma b c a) i) (* y x)) 2.0))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(-i, (fma(c, b, a) * c), (t * z)) * 2.0;
	double t_2 = i * (((c * b) + a) * c);
	double tmp;
	if (t_2 <= -2e+85) {
		tmp = t_1;
	} else if (t_2 <= 1e-25) {
		tmp = fma(t, z, (y * x)) * 2.0;
	} else if (t_2 <= 5e+298) {
		tmp = t_1;
	} else {
		tmp = fma(-c, (fma(b, c, a) * i), (y * x)) * 2.0;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(fma(Float64(-i), Float64(fma(c, b, a) * c), Float64(t * z)) * 2.0)
	t_2 = Float64(i * Float64(Float64(Float64(c * b) + a) * c))
	tmp = 0.0
	if (t_2 <= -2e+85)
		tmp = t_1;
	elseif (t_2 <= 1e-25)
		tmp = Float64(fma(t, z, Float64(y * x)) * 2.0);
	elseif (t_2 <= 5e+298)
		tmp = t_1;
	else
		tmp = Float64(fma(Float64(-c), Float64(fma(b, c, a) * i), Float64(y * x)) * 2.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[((-i) * N[(N[(c * b + a), $MachinePrecision] * c), $MachinePrecision] + N[(t * z), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]}, Block[{t$95$2 = N[(i * N[(N[(N[(c * b), $MachinePrecision] + a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+85], t$95$1, If[LessEqual[t$95$2, 1e-25], N[(N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], If[LessEqual[t$95$2, 5e+298], t$95$1, N[(N[((-c) * N[(N[(b * c + a), $MachinePrecision] * i), $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -2e85 or 1.00000000000000004e-25 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 5.0000000000000003e298

   1. Initial program 95.8%

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

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

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

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

      7. mul-1-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-*.f64 91.9

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

   5. Applied rewrites 91.9%

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

   if -2e85 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 1.00000000000000004e-25

   1. Initial program 99.1%

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

   3. Taylor expanded in c around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 94.0

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

   5. Applied rewrites 94.0%

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

   if 5.0000000000000003e298 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 70.5%

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

   3. Taylor expanded in a around inf

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

      1. associate-*r* N/A

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

      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 32.8

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

   5. Applied rewrites 32.8%

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

   6. Taylor expanded in z around 0

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. neg-mul-1 N/A

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

      7. lower-neg.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 88.1

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

   8. Applied rewrites 88.1%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq -2 \cdot 10^{+85}:\\ \;\;\;\;\mathsf{fma}\left(-i, \mathsf{fma}\left(c, b, a\right) \cdot c, t \cdot z\right) \cdot 2\\ \mathbf{elif}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq 10^{-25}:\\ \;\;\;\;\mathsf{fma}\left(t, z, y \cdot x\right) \cdot 2\\ \mathbf{elif}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq 5 \cdot 10^{+298}:\\ \;\;\;\;\mathsf{fma}\left(-i, \mathsf{fma}\left(c, b, a\right) \cdot c, t \cdot z\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-c, \mathsf{fma}\left(b, c, a\right) \cdot i, y \cdot x\right) \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(c \cdot b + a\right) \cdot c\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(\left(\left(c \cdot c\right) \cdot b\right) \cdot -2\right) \cdot i\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{+185}:\\ \;\;\;\;\left(\left(i \cdot c\right) \cdot a\right) \cdot -2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+179}:\\ \;\;\;\;\mathsf{fma}\left(t, z, y \cdot x\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(c \cdot c\right) \cdot i\right) \cdot b\right) \cdot -2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (+ (* c b) a) c)))
   (if (<= t_1 (- INFINITY))
     (* (* (* (* c c) b) -2.0) i)
     (if (<= t_1 -2e+185)
       (* (* (* i c) a) -2.0)
       (if (<= t_1 2e+179)
         (* (fma t z (* y x)) 2.0)
         (* (* (* (* c c) i) b) -2.0))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = ((c * b) + a) * c;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (((c * c) * b) * -2.0) * i;
	} else if (t_1 <= -2e+185) {
		tmp = ((i * c) * a) * -2.0;
	} else if (t_1 <= 2e+179) {
		tmp = fma(t, z, (y * x)) * 2.0;
	} else {
		tmp = (((c * c) * i) * b) * -2.0;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(c * b) + a) * c)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(Float64(c * c) * b) * -2.0) * i);
	elseif (t_1 <= -2e+185)
		tmp = Float64(Float64(Float64(i * c) * a) * -2.0);
	elseif (t_1 <= 2e+179)
		tmp = Float64(fma(t, z, Float64(y * x)) * 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(c * c) * i) * b) * -2.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(c * b), $MachinePrecision] + a), $MachinePrecision] * c), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(N[(c * c), $MachinePrecision] * b), $MachinePrecision] * -2.0), $MachinePrecision] * i), $MachinePrecision], If[LessEqual[t$95$1, -2e+185], N[(N[(N[(i * c), $MachinePrecision] * a), $MachinePrecision] * -2.0), $MachinePrecision], If[LessEqual[t$95$1, 2e+179], N[(N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(N[(N[(c * c), $MachinePrecision] * i), $MachinePrecision] * b), $MachinePrecision] * -2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 70.3%

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

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*r* N/A

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

      5. distribute-lft-out N/A

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

      6. lower-*.f64 N/A

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

      7. distribute-lft-out N/A

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

      8. associate-*r* N/A

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

      9. distribute-rgt-in N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 84.4

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

   5. Applied rewrites 84.4%

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

   7. Applied rewrites 84.2%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 70.3%

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

   if -inf.0 < (*.f64 (+.f64 a (*.f64 b c)) c) < -2e185

   1. Initial program 99.7%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 58.6

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

   5. Applied rewrites 58.6%

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

   if -2e185 < (*.f64 (+.f64 a (*.f64 b c)) c) < 1.99999999999999996e179

   1. Initial program 98.8%

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

   3. Taylor expanded in c around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 76.1

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

   5. Applied rewrites 76.1%

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

   if 1.99999999999999996e179 < (*.f64 (+.f64 a (*.f64 b c)) c)

   1. Initial program 82.3%

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

   3. Taylor expanded in i around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*r* N/A

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

      5. distribute-lft-out N/A

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

      6. lower-*.f64 N/A

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

      7. distribute-lft-out N/A

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

      8. associate-*r* N/A

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

      9. distribute-rgt-in N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 84.6

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

   5. Applied rewrites 84.6%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 71.5%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(c \cdot b + a\right) \cdot c \leq -\infty:\\ \;\;\;\;\left(\left(\left(c \cdot c\right) \cdot b\right) \cdot -2\right) \cdot i\\ \mathbf{elif}\;\left(c \cdot b + a\right) \cdot c \leq -2 \cdot 10^{+185}:\\ \;\;\;\;\left(\left(i \cdot c\right) \cdot a\right) \cdot -2\\ \mathbf{elif}\;\left(c \cdot b + a\right) \cdot c \leq 2 \cdot 10^{+179}:\\ \;\;\;\;\mathsf{fma}\left(t, z, y \cdot x\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(c \cdot c\right) \cdot i\right) \cdot b\right) \cdot -2\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 71.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\left(c \cdot c\right) \cdot i\right) \cdot b\right) \cdot -2\\ t_2 := \left(c \cdot b + a\right) \cdot c\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq -2 \cdot 10^{+185}:\\ \;\;\;\;\left(\left(i \cdot c\right) \cdot a\right) \cdot -2\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+179}:\\ \;\;\;\;\mathsf{fma}\left(t, z, y \cdot x\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* (* (* c c) i) b) -2.0)) (t_2 (* (+ (* c b) a) c)))
   (if (<= t_2 (- INFINITY))
     t_1
     (if (<= t_2 -2e+185)
       (* (* (* i c) a) -2.0)
       (if (<= t_2 2e+179) (* (fma t z (* y x)) 2.0) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (((c * c) * i) * b) * -2.0;
	double t_2 = ((c * b) + a) * c;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_2 <= -2e+185) {
		tmp = ((i * c) * a) * -2.0;
	} else if (t_2 <= 2e+179) {
		tmp = fma(t, z, (y * x)) * 2.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(Float64(c * c) * i) * b) * -2.0)
	t_2 = Float64(Float64(Float64(c * b) + a) * c)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_2 <= -2e+185)
		tmp = Float64(Float64(Float64(i * c) * a) * -2.0);
	elseif (t_2 <= 2e+179)
		tmp = Float64(fma(t, z, Float64(y * x)) * 2.0);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(N[(c * c), $MachinePrecision] * i), $MachinePrecision] * b), $MachinePrecision] * -2.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(c * b), $MachinePrecision] + a), $MachinePrecision] * c), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$1, If[LessEqual[t$95$2, -2e+185], N[(N[(N[(i * c), $MachinePrecision] * a), $MachinePrecision] * -2.0), $MachinePrecision], If[LessEqual[t$95$2, 2e+179], N[(N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (+.f64 a (*.f64 b c)) c) < -inf.0 or 1.99999999999999996e179 < (*.f64 (+.f64 a (*.f64 b c)) c)

   1. Initial program 76.4%

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

   3. Taylor expanded in i around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*r* N/A

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

      5. distribute-lft-out N/A

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

      6. lower-*.f64 N/A

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

      7. distribute-lft-out N/A

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

      8. associate-*r* N/A

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

      9. distribute-rgt-in N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 84.5

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

   5. Applied rewrites 84.5%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 69.6%

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

   if -inf.0 < (*.f64 (+.f64 a (*.f64 b c)) c) < -2e185

   1. Initial program 99.7%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 58.6

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

   5. Applied rewrites 58.6%

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

   if -2e185 < (*.f64 (+.f64 a (*.f64 b c)) c) < 1.99999999999999996e179

   1. Initial program 98.8%

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

   3. Taylor expanded in c around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 76.1

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

   5. Applied rewrites 76.1%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(c \cdot b + a\right) \cdot c \leq -\infty:\\ \;\;\;\;\left(\left(\left(c \cdot c\right) \cdot i\right) \cdot b\right) \cdot -2\\ \mathbf{elif}\;\left(c \cdot b + a\right) \cdot c \leq -2 \cdot 10^{+185}:\\ \;\;\;\;\left(\left(i \cdot c\right) \cdot a\right) \cdot -2\\ \mathbf{elif}\;\left(c \cdot b + a\right) \cdot c \leq 2 \cdot 10^{+179}:\\ \;\;\;\;\mathsf{fma}\left(t, z, y \cdot x\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(c \cdot c\right) \cdot i\right) \cdot b\right) \cdot -2\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 82.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := i \cdot \left(\left(c \cdot b + a\right) \cdot c\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+85}:\\ \;\;\;\;\left(\left(-2 \cdot \mathsf{fma}\left(b, c, a\right)\right) \cdot c\right) \cdot i\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-42}:\\ \;\;\;\;\mathsf{fma}\left(t, z, y \cdot x\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-c, \mathsf{fma}\left(b, c, a\right) \cdot i, y \cdot x\right) \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* i (* (+ (* c b) a) c))))
   (if (<= t_1 -2e+85)
     (* (* (* -2.0 (fma b c a)) c) i)
     (if (<= t_1 2e-42)
       (* (fma t z (* y x)) 2.0)
       (* (fma (- c) (* (fma b c a) i) (* y x)) 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+85) {
		tmp = ((-2.0 * fma(b, c, a)) * c) * i;
	} else if (t_1 <= 2e-42) {
		tmp = fma(t, z, (y * x)) * 2.0;
	} else {
		tmp = fma(-c, (fma(b, c, a) * i), (y * x)) * 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+85)
		tmp = Float64(Float64(Float64(-2.0 * fma(b, c, a)) * c) * i);
	elseif (t_1 <= 2e-42)
		tmp = Float64(fma(t, z, Float64(y * x)) * 2.0);
	else
		tmp = Float64(fma(Float64(-c), Float64(fma(b, c, a) * i), Float64(y * x)) * 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+85], N[(N[(N[(-2.0 * N[(b * c + a), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * i), $MachinePrecision], If[LessEqual[t$95$1, 2e-42], N[(N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[((-c) * N[(N[(b * c + a), $MachinePrecision] * i), $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -2e85

   1. Initial program 93.3%

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

   3. Taylor expanded in i around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*r* N/A

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

      5. distribute-lft-out N/A

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

      6. lower-*.f64 N/A

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

      7. distribute-lft-out N/A

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

      8. associate-*r* N/A

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

      9. distribute-rgt-in N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 73.0

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

   5. Applied rewrites 73.0%

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

   7. Applied rewrites 81.8%

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

   if -2e85 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 2.00000000000000008e-42

   1. Initial program 99.1%

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

   3. Taylor expanded in c around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 94.8

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

   5. Applied rewrites 94.8%

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

   if 2.00000000000000008e-42 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 83.1%

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

   3. Taylor expanded in a around inf

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

      1. associate-*r* N/A

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

      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 33.8

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

   5. Applied rewrites 33.8%

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

   6. Taylor expanded in z around 0

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. neg-mul-1 N/A

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

      7. lower-neg.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 79.0

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

   8. Applied rewrites 79.0%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq -2 \cdot 10^{+85}:\\ \;\;\;\;\left(\left(-2 \cdot \mathsf{fma}\left(b, c, a\right)\right) \cdot c\right) \cdot i\\ \mathbf{elif}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq 2 \cdot 10^{-42}:\\ \;\;\;\;\mathsf{fma}\left(t, z, y \cdot x\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-c, \mathsf{fma}\left(b, c, a\right) \cdot i, y \cdot x\right) \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := i \cdot \left(\left(c \cdot b + a\right) \cdot c\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+85}:\\ \;\;\;\;\left(\left(-2 \cdot \mathsf{fma}\left(b, c, a\right)\right) \cdot c\right) \cdot i\\ \mathbf{elif}\;t\_1 \leq 10^{-25}:\\ \;\;\;\;\mathsf{fma}\left(t, z, y \cdot x\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right)\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* i (* (+ (* c b) a) c))))
   (if (<= t_1 -2e+85)
     (* (* (* -2.0 (fma b c a)) c) i)
     (if (<= t_1 1e-25)
       (* (fma t z (* y x)) 2.0)
       (* (* -2.0 (* (fma c b a) i)) c)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = i * (((c * b) + a) * c);
	double tmp;
	if (t_1 <= -2e+85) {
		tmp = ((-2.0 * fma(b, c, a)) * c) * i;
	} else if (t_1 <= 1e-25) {
		tmp = fma(t, z, (y * x)) * 2.0;
	} else {
		tmp = (-2.0 * (fma(c, b, a) * i)) * c;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(i * Float64(Float64(Float64(c * b) + a) * c))
	tmp = 0.0
	if (t_1 <= -2e+85)
		tmp = Float64(Float64(Float64(-2.0 * fma(b, c, a)) * c) * i);
	elseif (t_1 <= 1e-25)
		tmp = Float64(fma(t, z, Float64(y * x)) * 2.0);
	else
		tmp = Float64(Float64(-2.0 * Float64(fma(c, b, a) * i)) * c);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(i * N[(N[(N[(c * b), $MachinePrecision] + a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+85], N[(N[(N[(-2.0 * N[(b * c + a), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * i), $MachinePrecision], If[LessEqual[t$95$1, 1e-25], N[(N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(-2.0 * N[(N[(c * b + a), $MachinePrecision] * i), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -2e85

   1. Initial program 93.3%

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

   3. Taylor expanded in i around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*r* N/A

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

      5. distribute-lft-out N/A

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

      6. lower-*.f64 N/A

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

      7. distribute-lft-out N/A

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

      8. associate-*r* N/A

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

      9. distribute-rgt-in N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 73.0

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

   5. Applied rewrites 73.0%

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

   7. Applied rewrites 81.8%

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

   if -2e85 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 1.00000000000000004e-25

   1. Initial program 99.1%

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

   3. Taylor expanded in c around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 94.0

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

   5. Applied rewrites 94.0%

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

   if 1.00000000000000004e-25 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 82.7%

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

   3. Taylor expanded in i around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*r* N/A

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

      5. distribute-lft-out N/A

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

      6. lower-*.f64 N/A

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

      7. distribute-lft-out N/A

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

      8. associate-*r* N/A

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

      9. distribute-rgt-in N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 76.3

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

   5. Applied rewrites 76.3%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq -2 \cdot 10^{+85}:\\ \;\;\;\;\left(\left(-2 \cdot \mathsf{fma}\left(b, c, a\right)\right) \cdot c\right) \cdot i\\ \mathbf{elif}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq 10^{-25}:\\ \;\;\;\;\mathsf{fma}\left(t, z, y \cdot x\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right)\right) \cdot c\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 79.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-2 \cdot \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right)\right) \cdot c\\ t_2 := i \cdot \left(\left(c \cdot b + a\right) \cdot c\right)\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+85}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{-25}:\\ \;\;\;\;\mathsf{fma}\left(t, z, y \cdot x\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* -2.0 (* (fma c b a) i)) c)) (t_2 (* i (* (+ (* c b) a) c))))
   (if (<= t_2 -2e+85) t_1 (if (<= t_2 1e-25) (* (fma t z (* y x)) 2.0) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (-2.0 * (fma(c, b, a) * i)) * c;
	double t_2 = i * (((c * b) + a) * c);
	double tmp;
	if (t_2 <= -2e+85) {
		tmp = t_1;
	} else if (t_2 <= 1e-25) {
		tmp = fma(t, z, (y * x)) * 2.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(-2.0 * Float64(fma(c, b, a) * i)) * c)
	t_2 = Float64(i * Float64(Float64(Float64(c * b) + a) * c))
	tmp = 0.0
	if (t_2 <= -2e+85)
		tmp = t_1;
	elseif (t_2 <= 1e-25)
		tmp = Float64(fma(t, z, Float64(y * x)) * 2.0);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(-2.0 * N[(N[(c * b + a), $MachinePrecision] * i), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]}, Block[{t$95$2 = N[(i * N[(N[(N[(c * b), $MachinePrecision] + a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+85], t$95$1, If[LessEqual[t$95$2, 1e-25], N[(N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -2e85 or 1.00000000000000004e-25 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 87.0%

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

   3. Taylor expanded in i around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*r* N/A

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

      5. distribute-lft-out N/A

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

      6. lower-*.f64 N/A

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

      7. distribute-lft-out N/A

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

      8. associate-*r* N/A

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

      9. distribute-rgt-in N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 75.0

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

   5. Applied rewrites 75.0%

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

   if -2e85 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 1.00000000000000004e-25

   1. Initial program 99.1%

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

   3. Taylor expanded in c around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 94.0

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

   5. Applied rewrites 94.0%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq -2 \cdot 10^{+85}:\\ \;\;\;\;\left(-2 \cdot \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right)\right) \cdot c\\ \mathbf{elif}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq 10^{-25}:\\ \;\;\;\;\mathsf{fma}\left(t, z, y \cdot x\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right)\right) \cdot c\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 74.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(-2 \cdot \left(c \cdot b\right)\right) \cdot i\right) \cdot c\\ t_2 := i \cdot \left(\left(c \cdot b + a\right) \cdot c\right)\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+233}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{+254}:\\ \;\;\;\;\mathsf{fma}\left(t, z, y \cdot x\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* (* -2.0 (* c b)) i) c)) (t_2 (* i (* (+ (* c b) a) c))))
   (if (<= t_2 -2e+233)
     t_1
     (if (<= t_2 1e+254) (* (fma t z (* y x)) 2.0) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = ((-2.0 * (c * b)) * i) * c;
	double t_2 = i * (((c * b) + a) * c);
	double tmp;
	if (t_2 <= -2e+233) {
		tmp = t_1;
	} else if (t_2 <= 1e+254) {
		tmp = fma(t, z, (y * x)) * 2.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(-2.0 * Float64(c * b)) * i) * c)
	t_2 = Float64(i * Float64(Float64(Float64(c * b) + a) * c))
	tmp = 0.0
	if (t_2 <= -2e+233)
		tmp = t_1;
	elseif (t_2 <= 1e+254)
		tmp = Float64(fma(t, z, Float64(y * x)) * 2.0);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(-2.0 * N[(c * b), $MachinePrecision]), $MachinePrecision] * i), $MachinePrecision] * c), $MachinePrecision]}, Block[{t$95$2 = N[(i * N[(N[(N[(c * b), $MachinePrecision] + a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+233], t$95$1, If[LessEqual[t$95$2, 1e+254], N[(N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -1.99999999999999995e233 or 9.9999999999999994e253 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 81.1%

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

   3. Taylor expanded in i around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*r* N/A

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

      5. distribute-lft-out N/A

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

      6. lower-*.f64 N/A

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

      7. distribute-lft-out N/A

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

      8. associate-*r* N/A

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

      9. distribute-rgt-in N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 84.7

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

   5. Applied rewrites 84.7%

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

   7. Applied rewrites 83.7%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 60.3%

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

   12. Applied rewrites 61.3%

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

   if -1.99999999999999995e233 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 9.9999999999999994e253

   1. Initial program 99.3%

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

   3. Taylor expanded in c around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 78.6

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

   5. Applied rewrites 78.6%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq -2 \cdot 10^{+233}:\\ \;\;\;\;\left(\left(-2 \cdot \left(c \cdot b\right)\right) \cdot i\right) \cdot c\\ \mathbf{elif}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq 10^{+254}:\\ \;\;\;\;\mathsf{fma}\left(t, z, y \cdot x\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-2 \cdot \left(c \cdot b\right)\right) \cdot i\right) \cdot c\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 73.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := i \cdot \left(\left(c \cdot b + a\right) \cdot c\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+233}:\\ \;\;\;\;\left(\left(-2 \cdot \left(c \cdot b\right)\right) \cdot c\right) \cdot i\\ \mathbf{elif}\;t\_1 \leq 10^{+254}:\\ \;\;\;\;\mathsf{fma}\left(t, z, y \cdot x\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(c \cdot c\right) \cdot i\right) \cdot b\right) \cdot -2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* i (* (+ (* c b) a) c))))
   (if (<= t_1 -2e+233)
     (* (* (* -2.0 (* c b)) c) i)
     (if (<= t_1 1e+254)
       (* (fma t z (* y x)) 2.0)
       (* (* (* (* c c) i) b) -2.0)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = i * (((c * b) + a) * c);
	double tmp;
	if (t_1 <= -2e+233) {
		tmp = ((-2.0 * (c * b)) * c) * i;
	} else if (t_1 <= 1e+254) {
		tmp = fma(t, z, (y * x)) * 2.0;
	} else {
		tmp = (((c * c) * i) * b) * -2.0;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(i * Float64(Float64(Float64(c * b) + a) * c))
	tmp = 0.0
	if (t_1 <= -2e+233)
		tmp = Float64(Float64(Float64(-2.0 * Float64(c * b)) * c) * i);
	elseif (t_1 <= 1e+254)
		tmp = Float64(fma(t, z, Float64(y * x)) * 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(c * c) * i) * b) * -2.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(i * N[(N[(N[(c * b), $MachinePrecision] + a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+233], N[(N[(N[(-2.0 * N[(c * b), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * i), $MachinePrecision], If[LessEqual[t$95$1, 1e+254], N[(N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(N[(N[(c * c), $MachinePrecision] * i), $MachinePrecision] * b), $MachinePrecision] * -2.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -1.99999999999999995e233

   1. Initial program 91.0%

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

   3. Taylor expanded in i around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*r* N/A

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

      5. distribute-lft-out N/A

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

      6. lower-*.f64 N/A

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

      7. distribute-lft-out N/A

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

      8. associate-*r* N/A

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

      9. distribute-rgt-in N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 84.2

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

   5. Applied rewrites 84.2%

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

   7. Applied rewrites 88.6%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 65.4%

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

   if -1.99999999999999995e233 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 9.9999999999999994e253

   1. Initial program 99.3%

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

   3. Taylor expanded in c around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 78.6

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

   5. Applied rewrites 78.6%

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

   if 9.9999999999999994e253 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 73.7%

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

   3. Taylor expanded in i around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*r* N/A

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

      5. distribute-lft-out N/A

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

      6. lower-*.f64 N/A

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

      7. distribute-lft-out N/A

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

      8. associate-*r* N/A

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

      9. distribute-rgt-in N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 85.1

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

   5. Applied rewrites 85.1%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 58.0%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq -2 \cdot 10^{+233}:\\ \;\;\;\;\left(\left(-2 \cdot \left(c \cdot b\right)\right) \cdot c\right) \cdot i\\ \mathbf{elif}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq 10^{+254}:\\ \;\;\;\;\mathsf{fma}\left(t, z, y \cdot x\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(c \cdot c\right) \cdot i\right) \cdot b\right) \cdot -2\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 62.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := i \cdot \left(\left(c \cdot b + a\right) \cdot c\right)\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+163}:\\ \;\;\;\;\left(\left(-2 \cdot a\right) \cdot c\right) \cdot i\\ \mathbf{elif}\;t\_1 \leq 3 \cdot 10^{+140}:\\ \;\;\;\;\mathsf{fma}\left(t, z, y \cdot x\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(i \cdot c\right) \cdot a\right) \cdot -2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* i (* (+ (* c b) a) c))))
   (if (<= t_1 -4e+163)
     (* (* (* -2.0 a) c) i)
     (if (<= t_1 3e+140) (* (fma t z (* y x)) 2.0) (* (* (* i c) a) -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 <= -4e+163) {
		tmp = ((-2.0 * a) * c) * i;
	} else if (t_1 <= 3e+140) {
		tmp = fma(t, z, (y * x)) * 2.0;
	} else {
		tmp = ((i * c) * a) * -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 <= -4e+163)
		tmp = Float64(Float64(Float64(-2.0 * a) * c) * i);
	elseif (t_1 <= 3e+140)
		tmp = Float64(fma(t, z, Float64(y * x)) * 2.0);
	else
		tmp = Float64(Float64(Float64(i * c) * a) * -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, -4e+163], N[(N[(N[(-2.0 * a), $MachinePrecision] * c), $MachinePrecision] * i), $MachinePrecision], If[LessEqual[t$95$1, 3e+140], N[(N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(N[(i * c), $MachinePrecision] * a), $MachinePrecision] * -2.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -3.9999999999999998e163

   1. Initial program 92.6%

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

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*r* N/A

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

      5. distribute-lft-out N/A

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

      6. lower-*.f64 N/A

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

      7. distribute-lft-out N/A

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

      8. associate-*r* N/A

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

      9. distribute-rgt-in N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 76.0

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

   5. Applied rewrites 76.0%

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

   7. Applied rewrites 83.2%

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

   8. Taylor expanded in a around inf

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

   10. Applied rewrites 37.9%

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

   if -3.9999999999999998e163 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 2.99999999999999997e140

   1. Initial program 99.2%

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

   3. Taylor expanded in c around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 85.6

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

   5. Applied rewrites 85.6%

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

   if 2.99999999999999997e140 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 79.0%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 41.2

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

   5. Applied rewrites 41.2%

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

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq -4 \cdot 10^{+163}:\\ \;\;\;\;\left(\left(-2 \cdot a\right) \cdot c\right) \cdot i\\ \mathbf{elif}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq 3 \cdot 10^{+140}:\\ \;\;\;\;\mathsf{fma}\left(t, z, y \cdot x\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(i \cdot c\right) \cdot a\right) \cdot -2\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 44.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(y \cdot x\right) \cdot 2\\ \mathbf{if}\;y \cdot x \leq -4 \cdot 10^{+62}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \cdot x \leq 10^{-67}:\\ \;\;\;\;\left(t \cdot z\right) \cdot 2\\ \mathbf{elif}\;y \cdot x \leq 5 \cdot 10^{+49}:\\ \;\;\;\;\left(\left(i \cdot c\right) \cdot a\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* y x) 2.0)))
   (if (<= (* y x) -4e+62)
     t_1
     (if (<= (* y x) 1e-67)
       (* (* t z) 2.0)
       (if (<= (* y x) 5e+49) (* (* (* 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 = (y * x) * 2.0;
	double tmp;
	if ((y * x) <= -4e+62) {
		tmp = t_1;
	} else if ((y * x) <= 1e-67) {
		tmp = (t * z) * 2.0;
	} else if ((y * x) <= 5e+49) {
		tmp = ((i * c) * a) * -2.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (y * x) * 2.0d0
    if ((y * x) <= (-4d+62)) then
        tmp = t_1
    else if ((y * x) <= 1d-67) then
        tmp = (t * z) * 2.0d0
    else if ((y * x) <= 5d+49) then
        tmp = ((i * c) * a) * (-2.0d0)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (y * x) * 2.0;
	double tmp;
	if ((y * x) <= -4e+62) {
		tmp = t_1;
	} else if ((y * x) <= 1e-67) {
		tmp = (t * z) * 2.0;
	} else if ((y * x) <= 5e+49) {
		tmp = ((i * c) * a) * -2.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = (y * x) * 2.0
	tmp = 0
	if (y * x) <= -4e+62:
		tmp = t_1
	elif (y * x) <= 1e-67:
		tmp = (t * z) * 2.0
	elif (y * x) <= 5e+49:
		tmp = ((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(Float64(y * x) * 2.0)
	tmp = 0.0
	if (Float64(y * x) <= -4e+62)
		tmp = t_1;
	elseif (Float64(y * x) <= 1e-67)
		tmp = Float64(Float64(t * z) * 2.0);
	elseif (Float64(y * x) <= 5e+49)
		tmp = Float64(Float64(Float64(i * c) * a) * -2.0);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	t_1 = (y * x) * 2.0;
	tmp = 0.0;
	if ((y * x) <= -4e+62)
		tmp = t_1;
	elseif ((y * x) <= 1e-67)
		tmp = (t * z) * 2.0;
	elseif ((y * x) <= 5e+49)
		tmp = ((i * c) * a) * -2.0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(y * x), $MachinePrecision] * 2.0), $MachinePrecision]}, If[LessEqual[N[(y * x), $MachinePrecision], -4e+62], t$95$1, If[LessEqual[N[(y * x), $MachinePrecision], 1e-67], N[(N[(t * z), $MachinePrecision] * 2.0), $MachinePrecision], If[LessEqual[N[(y * x), $MachinePrecision], 5e+49], N[(N[(N[(i * c), $MachinePrecision] * a), $MachinePrecision] * -2.0), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -4.00000000000000014e62 or 5.0000000000000004e49 < (*.f64 x y)

   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 x around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 60.0

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

   5. Applied rewrites 60.0%

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

   if -4.00000000000000014e62 < (*.f64 x y) < 9.99999999999999943e-68

   1. Initial program 94.3%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 42.9

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

   5. Applied rewrites 42.9%

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

   if 9.99999999999999943e-68 < (*.f64 x y) < 5.0000000000000004e49

   1. Initial program 95.5%

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

   3. Taylor expanded in 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 48.8

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

   5. Applied rewrites 48.8%

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

4. Final simplification 50.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot x \leq -4 \cdot 10^{+62}:\\ \;\;\;\;\left(y \cdot x\right) \cdot 2\\ \mathbf{elif}\;y \cdot x \leq 10^{-67}:\\ \;\;\;\;\left(t \cdot z\right) \cdot 2\\ \mathbf{elif}\;y \cdot x \leq 5 \cdot 10^{+49}:\\ \;\;\;\;\left(\left(i \cdot c\right) \cdot a\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot x\right) \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 44.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(y \cdot x\right) \cdot 2\\ \mathbf{if}\;y \cdot x \leq -4 \cdot 10^{+62}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \cdot x \leq 10^{-67}:\\ \;\;\;\;\left(t \cdot z\right) \cdot 2\\ \mathbf{elif}\;y \cdot x \leq 10^{+41}:\\ \;\;\;\;\left(\left(-2 \cdot a\right) \cdot c\right) \cdot i\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* y x) 2.0)))
   (if (<= (* y x) -4e+62)
     t_1
     (if (<= (* y x) 1e-67)
       (* (* t z) 2.0)
       (if (<= (* y x) 1e+41) (* (* (* -2.0 a) c) i) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (y * x) * 2.0;
	double tmp;
	if ((y * x) <= -4e+62) {
		tmp = t_1;
	} else if ((y * x) <= 1e-67) {
		tmp = (t * z) * 2.0;
	} else if ((y * x) <= 1e+41) {
		tmp = ((-2.0 * a) * c) * i;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (y * x) * 2.0d0
    if ((y * x) <= (-4d+62)) then
        tmp = t_1
    else if ((y * x) <= 1d-67) then
        tmp = (t * z) * 2.0d0
    else if ((y * x) <= 1d+41) then
        tmp = (((-2.0d0) * a) * c) * i
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (y * x) * 2.0;
	double tmp;
	if ((y * x) <= -4e+62) {
		tmp = t_1;
	} else if ((y * x) <= 1e-67) {
		tmp = (t * z) * 2.0;
	} else if ((y * x) <= 1e+41) {
		tmp = ((-2.0 * a) * c) * i;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = (y * x) * 2.0
	tmp = 0
	if (y * x) <= -4e+62:
		tmp = t_1
	elif (y * x) <= 1e-67:
		tmp = (t * z) * 2.0
	elif (y * x) <= 1e+41:
		tmp = ((-2.0 * a) * c) * i
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(y * x) * 2.0)
	tmp = 0.0
	if (Float64(y * x) <= -4e+62)
		tmp = t_1;
	elseif (Float64(y * x) <= 1e-67)
		tmp = Float64(Float64(t * z) * 2.0);
	elseif (Float64(y * x) <= 1e+41)
		tmp = Float64(Float64(Float64(-2.0 * a) * c) * i);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	t_1 = (y * x) * 2.0;
	tmp = 0.0;
	if ((y * x) <= -4e+62)
		tmp = t_1;
	elseif ((y * x) <= 1e-67)
		tmp = (t * z) * 2.0;
	elseif ((y * x) <= 1e+41)
		tmp = ((-2.0 * a) * c) * i;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(y * x), $MachinePrecision] * 2.0), $MachinePrecision]}, If[LessEqual[N[(y * x), $MachinePrecision], -4e+62], t$95$1, If[LessEqual[N[(y * x), $MachinePrecision], 1e-67], N[(N[(t * z), $MachinePrecision] * 2.0), $MachinePrecision], If[LessEqual[N[(y * x), $MachinePrecision], 1e+41], N[(N[(N[(-2.0 * a), $MachinePrecision] * c), $MachinePrecision] * i), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -4.00000000000000014e62 or 1.00000000000000001e41 < (*.f64 x y)

   1. Initial program 89.6%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 59.2

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

   5. Applied rewrites 59.2%

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

   if -4.00000000000000014e62 < (*.f64 x y) < 9.99999999999999943e-68

   1. Initial program 94.3%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 42.9

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

   5. Applied rewrites 42.9%

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

   if 9.99999999999999943e-68 < (*.f64 x y) < 1.00000000000000001e41

   1. Initial program 94.7%

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

   3. Taylor expanded in i around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*r* N/A

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

      5. distribute-lft-out N/A

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

      6. lower-*.f64 N/A

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

      7. distribute-lft-out N/A

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

      8. associate-*r* N/A

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

      9. distribute-rgt-in N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 62.4

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

   5. Applied rewrites 62.4%

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

   7. Applied rewrites 73.2%

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

   8. Taylor expanded in a around inf

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

   10. Applied rewrites 45.9%

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

4. Final simplification 49.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot x \leq -4 \cdot 10^{+62}:\\ \;\;\;\;\left(y \cdot x\right) \cdot 2\\ \mathbf{elif}\;y \cdot x \leq 10^{-67}:\\ \;\;\;\;\left(t \cdot z\right) \cdot 2\\ \mathbf{elif}\;y \cdot x \leq 10^{+41}:\\ \;\;\;\;\left(\left(-2 \cdot a\right) \cdot c\right) \cdot i\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot x\right) \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 94.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 92.4%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. associate--l+ N/A

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

   5. lift-*.f64 N/A

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

   6. lower-fma.f64 N/A

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

   7. sub-neg N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lift-*.f64 N/A

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

   14. associate-*r* N/A

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

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

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

   16. lower-*.f64 N/A

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

4. Applied rewrites 95.8%

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

5. Final simplification 95.8%

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

Alternative 15: 44.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(y \cdot x\right) \cdot 2\\ \mathbf{if}\;y \cdot x \leq -4 \cdot 10^{+62}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \cdot x \leq 2 \cdot 10^{-47}:\\ \;\;\;\;\left(t \cdot z\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* y x) 2.0)))
   (if (<= (* y x) -4e+62) t_1 (if (<= (* y x) 2e-47) (* (* t z) 2.0) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (y * x) * 2.0;
	double tmp;
	if ((y * x) <= -4e+62) {
		tmp = t_1;
	} else if ((y * x) <= 2e-47) {
		tmp = (t * z) * 2.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (y * x) * 2.0d0
    if ((y * x) <= (-4d+62)) then
        tmp = t_1
    else if ((y * x) <= 2d-47) then
        tmp = (t * z) * 2.0d0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (y * x) * 2.0;
	double tmp;
	if ((y * x) <= -4e+62) {
		tmp = t_1;
	} else if ((y * x) <= 2e-47) {
		tmp = (t * z) * 2.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = (y * x) * 2.0
	tmp = 0
	if (y * x) <= -4e+62:
		tmp = t_1
	elif (y * x) <= 2e-47:
		tmp = (t * z) * 2.0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(y * x) * 2.0)
	tmp = 0.0
	if (Float64(y * x) <= -4e+62)
		tmp = t_1;
	elseif (Float64(y * x) <= 2e-47)
		tmp = Float64(Float64(t * z) * 2.0);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	t_1 = (y * x) * 2.0;
	tmp = 0.0;
	if ((y * x) <= -4e+62)
		tmp = t_1;
	elseif ((y * x) <= 2e-47)
		tmp = (t * z) * 2.0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(y * x), $MachinePrecision] * 2.0), $MachinePrecision]}, If[LessEqual[N[(y * x), $MachinePrecision], -4e+62], t$95$1, If[LessEqual[N[(y * x), $MachinePrecision], 2e-47], N[(N[(t * z), $MachinePrecision] * 2.0), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -4.00000000000000014e62 or 1.9999999999999999e-47 < (*.f64 x y)

   1. Initial program 90.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 54.8

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

   5. Applied rewrites 54.8%

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

   if -4.00000000000000014e62 < (*.f64 x y) < 1.9999999999999999e-47

   1. Initial program 94.5%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 41.8

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

   5. Applied rewrites 41.8%

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

4. Final simplification 47.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot x \leq -4 \cdot 10^{+62}:\\ \;\;\;\;\left(y \cdot x\right) \cdot 2\\ \mathbf{elif}\;y \cdot x \leq 2 \cdot 10^{-47}:\\ \;\;\;\;\left(t \cdot z\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot x\right) \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 29.0% 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 92.4%

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

3. Taylor expanded in z around inf

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 29.4

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

5. Applied rewrites 29.4%

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

6. Final simplification 29.4%

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

Developer Target 1: 94.2% 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 2024284 
(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))))