Graphics.Rendering.Plot.Render.Plot.Legend:renderLegendOutside from plot-0.2.3.4, B

Percentage Accurate: 99.9% → 99.9%

Time: 8.0s

Alternatives: 13

Speedup: 1.2×

• 26.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+ (* x (+ (+ (+ (+ y z) z) y) t)) (* y 5.0)))

C

double code(double x, double y, double z, double t) {
	return (x * ((((y + z) + z) + y) + t)) + (y * 5.0);
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (x * ((((y + z) + z) + y) + t)) + (y * 5.0d0)
end function

Java

public static double code(double x, double y, double z, double t) {
	return (x * ((((y + z) + z) + y) + t)) + (y * 5.0);
}

Python

def code(x, y, z, t):
	return (x * ((((y + z) + z) + y) + t)) + (y * 5.0)

Julia

function code(x, y, z, t)
	return Float64(Float64(x * Float64(Float64(Float64(Float64(y + z) + z) + y) + t)) + Float64(y * 5.0))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (x * ((((y + z) + z) + y) + t)) + (y * 5.0);
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(x * N[(N[(N[(N[(y + z), $MachinePrecision] + z), $MachinePrecision] + y), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision] + N[(y * 5.0), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+ (* x (+ (+ (+ (+ y z) z) y) t)) (* y 5.0)))

C

double code(double x, double y, double z, double t) {
	return (x * ((((y + z) + z) + y) + t)) + (y * 5.0);
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (x * ((((y + z) + z) + y) + t)) + (y * 5.0d0)
end function

Java

public static double code(double x, double y, double z, double t) {
	return (x * ((((y + z) + z) + y) + t)) + (y * 5.0);
}

Python

def code(x, y, z, t):
	return (x * ((((y + z) + z) + y) + t)) + (y * 5.0)

Julia

function code(x, y, z, t)
	return Float64(Float64(x * Float64(Float64(Float64(Float64(y + z) + z) + y) + t)) + Float64(y * 5.0))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (x * ((((y + z) + z) + y) + t)) + (y * 5.0);
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(x * N[(N[(N[(N[(y + z), $MachinePrecision] + z), $MachinePrecision] + y), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision] + N[(y * 5.0), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, \mathsf{fma}\left(2, y + z, t\right), y \cdot 5\right) \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (fma x (fma 2.0 (+ y z) t) (* y 5.0)))

C

double code(double x, double y, double z, double t) {
	return fma(x, fma(2.0, (y + z), t), (y * 5.0));
}

Julia

function code(x, y, z, t)
	return fma(x, fma(2.0, Float64(y + z), t), Float64(y * 5.0))
end

Wolfram

code[x_, y_, z_, t_] := N[(x * N[(2.0 * N[(y + z), $MachinePrecision] + t), $MachinePrecision] + N[(y * 5.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{5 \cdot y + x \cdot \left(t + \left(2 \cdot y + 2 \cdot z\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{x \cdot \left(t + \left(2 \cdot y + 2 \cdot z\right)\right) + 5 \cdot y} \]

   2. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, t + \left(2 \cdot y + 2 \cdot z\right), 5 \cdot y\right)} \]

   3. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{\left(2 \cdot y + 2 \cdot z\right) + t}, 5 \cdot y\right) \]

   4. distribute-lft-out N/A

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{2 \cdot \left(y + z\right)} + t, 5 \cdot y\right) \]

   5. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(2, y + z, t\right)}, 5 \cdot y\right) \]

   6. lower-+.f64 N/A

      \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(2, \color{blue}{y + z}, t\right), 5 \cdot y\right) \]

   7. lower-*.f64 99.9

      \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(2, y + z, t\right), \color{blue}{5 \cdot y}\right) \]

5. Simplified 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(2, y + z, t\right), 5 \cdot y\right)} \]

6. Final simplification 99.9%

   \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(2, y + z, t\right), y \cdot 5\right) \]
7. Add Preprocessing

Alternative 2: 88.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \mathsf{fma}\left(2, y + z, t\right)\\ \mathbf{if}\;x \leq -5.5 \cdot 10^{-25}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -2.1 \cdot 10^{-213}:\\ \;\;\;\;\mathsf{fma}\left(y, 5, x \cdot t\right)\\ \mathbf{elif}\;x \leq 4.45 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(5, y, x \cdot \left(2 \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (fma 2.0 (+ y z) t))))
   (if (<= x -5.5e-25)
     t_1
     (if (<= x -2.1e-213)
       (fma y 5.0 (* x t))
       (if (<= x 4.45e-7) (fma 5.0 y (* x (* 2.0 z))) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * fma(2.0, (y + z), t);
	double tmp;
	if (x <= -5.5e-25) {
		tmp = t_1;
	} else if (x <= -2.1e-213) {
		tmp = fma(y, 5.0, (x * t));
	} else if (x <= 4.45e-7) {
		tmp = fma(5.0, y, (x * (2.0 * z)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(x * fma(2.0, Float64(y + z), t))
	tmp = 0.0
	if (x <= -5.5e-25)
		tmp = t_1;
	elseif (x <= -2.1e-213)
		tmp = fma(y, 5.0, Float64(x * t));
	elseif (x <= 4.45e-7)
		tmp = fma(5.0, y, Float64(x * Float64(2.0 * z)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[(2.0 * N[(y + z), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5.5e-25], t$95$1, If[LessEqual[x, -2.1e-213], N[(y * 5.0 + N[(x * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.45e-7], N[(5.0 * y + N[(x * N[(2.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.50000000000000004e-25 or 4.45e-7 < x

   1. Initial program 99.9%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-out N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-+.f64 96.6

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

   5. Simplified 96.6%

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

   if -5.50000000000000004e-25 < x < -2.0999999999999998e-213

   1. Initial program 99.9%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{5 \cdot y + x \cdot \left(t + \left(2 \cdot y + 2 \cdot z\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(t + \left(2 \cdot y + 2 \cdot z\right)\right) + 5 \cdot y} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, t + \left(2 \cdot y + 2 \cdot z\right), 5 \cdot y\right)} \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\left(2 \cdot y + 2 \cdot z\right) + t}, 5 \cdot y\right) \]

      4. distribute-lft-out N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{2 \cdot \left(y + z\right)} + t, 5 \cdot y\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(2, y + z, t\right)}, 5 \cdot y\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(2, \color{blue}{y + z}, t\right), 5 \cdot y\right) \]

      7. lower-*.f64 99.9

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(2, y + z, t\right), \color{blue}{5 \cdot y}\right) \]

   5. Simplified 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(2, y + z, t\right), 5 \cdot y\right)} \]

   6. Taylor expanded in z around 0

      \[\leadsto \color{blue}{5 \cdot y + x \cdot \left(t + 2 \cdot y\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 5 \cdot y + x \cdot \color{blue}{\left(2 \cdot y + t\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto 5 \cdot y + \color{blue}{\left(x \cdot \left(2 \cdot y\right) + x \cdot t\right)} \]

      3. *-commutative N/A

         \[\leadsto 5 \cdot y + \left(x \cdot \color{blue}{\left(y \cdot 2\right)} + x \cdot t\right) \]

      4. associate-*r* N/A

         \[\leadsto 5 \cdot y + \left(\color{blue}{\left(x \cdot y\right) \cdot 2} + x \cdot t\right) \]

      5. *-commutative N/A

         \[\leadsto 5 \cdot y + \left(\left(x \cdot y\right) \cdot 2 + \color{blue}{t \cdot x}\right) \]

      6. *-commutative N/A

         \[\leadsto 5 \cdot y + \left(\color{blue}{2 \cdot \left(x \cdot y\right)} + t \cdot x\right) \]

      7. associate-*r* N/A

         \[\leadsto 5 \cdot y + \left(\color{blue}{\left(2 \cdot x\right) \cdot y} + t \cdot x\right) \]

      8. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(5 \cdot y + \left(2 \cdot x\right) \cdot y\right) + t \cdot x} \]

      9. distribute-rgt-in N/A

         \[\leadsto \color{blue}{y \cdot \left(5 + 2 \cdot x\right)} + t \cdot x \]

      10. +-commutative N/A

          \[\leadsto y \cdot \color{blue}{\left(2 \cdot x + 5\right)} + t \cdot x \]

      11. metadata-eval N/A

          \[\leadsto y \cdot \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot x + 5\right) + t \cdot x \]

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

          \[\leadsto y \cdot \left(\color{blue}{\left(\mathsf{neg}\left(-2 \cdot x\right)\right)} + 5\right) + t \cdot x \]

      13. metadata-eval N/A

          \[\leadsto y \cdot \left(\left(\mathsf{neg}\left(-2 \cdot x\right)\right) + \color{blue}{\left(\mathsf{neg}\left(-5\right)\right)}\right) + t \cdot x \]

      14. distribute-neg-in N/A

          \[\leadsto y \cdot \color{blue}{\left(\mathsf{neg}\left(\left(-2 \cdot x + -5\right)\right)\right)} + t \cdot x \]

      15. metadata-eval N/A

          \[\leadsto y \cdot \left(\mathsf{neg}\left(\left(-2 \cdot x + \color{blue}{\left(\mathsf{neg}\left(5\right)\right)}\right)\right)\right) + t \cdot x \]

      16. sub-neg N/A

          \[\leadsto y \cdot \left(\mathsf{neg}\left(\color{blue}{\left(-2 \cdot x - 5\right)}\right)\right) + t \cdot x \]

      17. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(y, \mathsf{neg}\left(\left(-2 \cdot x - 5\right)\right), t \cdot x\right)} \]

   8. Simplified 85.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(x, 2, 5\right), x \cdot t\right)} \]

   9. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{5}, x \cdot t\right) \]
   10. Step-by-step derivation

   11. Simplified 85.2%

       \[\leadsto \mathsf{fma}\left(y, \color{blue}{5}, x \cdot t\right) \]

   if -2.0999999999999998e-213 < x < 4.45e-7

   1. Initial program 99.9%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto x \cdot \left(\left(\left(\color{blue}{\left(y + z\right)} + z\right) + y\right) + t\right) + y \cdot 5 \]

      2. lift-+.f64 N/A

         \[\leadsto x \cdot \left(\left(\color{blue}{\left(\left(y + z\right) + z\right)} + y\right) + t\right) + y \cdot 5 \]

      3. lift-+.f64 N/A

         \[\leadsto x \cdot \left(\color{blue}{\left(\left(\left(y + z\right) + z\right) + y\right)} + t\right) + y \cdot 5 \]

      4. lift-+.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)} + y \cdot 5 \]

      5. lift-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)} + y \cdot 5 \]

      6. lift-*.f64 N/A

         \[\leadsto x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + \color{blue}{y \cdot 5} \]

      7. +-commutative N/A

         \[\leadsto \color{blue}{y \cdot 5 + x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto y \cdot 5 + \color{blue}{x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)} \]

      9. lift-+.f64 N/A

         \[\leadsto y \cdot 5 + x \cdot \color{blue}{\left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)} \]

      10. distribute-rgt-in N/A

          \[\leadsto y \cdot 5 + \color{blue}{\left(\left(\left(\left(y + z\right) + z\right) + y\right) \cdot x + t \cdot x\right)} \]

      11. associate-+r+ N/A

          \[\leadsto \color{blue}{\left(y \cdot 5 + \left(\left(\left(y + z\right) + z\right) + y\right) \cdot x\right) + t \cdot x} \]

      12. lower-+.f64 N/A

          \[\leadsto \color{blue}{\left(y \cdot 5 + \left(\left(\left(y + z\right) + z\right) + y\right) \cdot x\right) + t \cdot x} \]

   4. Applied egg-rr 99.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, 5, x \cdot \left(z + z\right)\right) + x \cdot t} \]

   5. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \left(x \cdot z\right) + 5 \cdot y} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{5 \cdot y + 2 \cdot \left(x \cdot z\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(5, y, 2 \cdot \left(x \cdot z\right)\right)} \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(5, y, \color{blue}{\left(2 \cdot x\right) \cdot z}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(5, y, \color{blue}{\left(x \cdot 2\right)} \cdot z\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(5, y, \color{blue}{x \cdot \left(2 \cdot z\right)}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(5, y, \color{blue}{x \cdot \left(2 \cdot z\right)}\right) \]

      7. lower-*.f64 85.3

         \[\leadsto \mathsf{fma}\left(5, y, x \cdot \color{blue}{\left(2 \cdot z\right)}\right) \]

   7. Simplified 85.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(5, y, x \cdot \left(2 \cdot z\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 47.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \left(y + y\right)\\ \mathbf{if}\;x \leq -2 \cdot 10^{+225}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -2.55 \cdot 10^{-33}:\\ \;\;\;\;x \cdot t\\ \mathbf{elif}\;x \leq 4.45 \cdot 10^{-7}:\\ \;\;\;\;y \cdot 5\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{+93}:\\ \;\;\;\;x \cdot t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (+ y y))))
   (if (<= x -2e+225)
     t_1
     (if (<= x -2.55e-33)
       (* x t)
       (if (<= x 4.45e-7) (* y 5.0) (if (<= x 4.2e+93) (* x t) t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * (y + y);
	double tmp;
	if (x <= -2e+225) {
		tmp = t_1;
	} else if (x <= -2.55e-33) {
		tmp = x * t;
	} else if (x <= 4.45e-7) {
		tmp = y * 5.0;
	} else if (x <= 4.2e+93) {
		tmp = x * t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x * (y + y)
    if (x <= (-2d+225)) then
        tmp = t_1
    else if (x <= (-2.55d-33)) then
        tmp = x * t
    else if (x <= 4.45d-7) then
        tmp = y * 5.0d0
    else if (x <= 4.2d+93) then
        tmp = x * t
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x * (y + y);
	double tmp;
	if (x <= -2e+225) {
		tmp = t_1;
	} else if (x <= -2.55e-33) {
		tmp = x * t;
	} else if (x <= 4.45e-7) {
		tmp = y * 5.0;
	} else if (x <= 4.2e+93) {
		tmp = x * t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * (y + y)
	tmp = 0
	if x <= -2e+225:
		tmp = t_1
	elif x <= -2.55e-33:
		tmp = x * t
	elif x <= 4.45e-7:
		tmp = y * 5.0
	elif x <= 4.2e+93:
		tmp = x * t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * Float64(y + y))
	tmp = 0.0
	if (x <= -2e+225)
		tmp = t_1;
	elseif (x <= -2.55e-33)
		tmp = Float64(x * t);
	elseif (x <= 4.45e-7)
		tmp = Float64(y * 5.0);
	elseif (x <= 4.2e+93)
		tmp = Float64(x * t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * (y + y);
	tmp = 0.0;
	if (x <= -2e+225)
		tmp = t_1;
	elseif (x <= -2.55e-33)
		tmp = x * t;
	elseif (x <= 4.45e-7)
		tmp = y * 5.0;
	elseif (x <= 4.2e+93)
		tmp = x * t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[(y + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2e+225], t$95$1, If[LessEqual[x, -2.55e-33], N[(x * t), $MachinePrecision], If[LessEqual[x, 4.45e-7], N[(y * 5.0), $MachinePrecision], If[LessEqual[x, 4.2e+93], N[(x * t), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.99999999999999986e225 or 4.1999999999999996e93 < x

   1. Initial program 100.0%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-out N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-+.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

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

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

      5. lower-*.f64 N/A

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

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

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 51.1

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

   8. Simplified 51.1%

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

   9. Taylor expanded in y around 0

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. lower-*.f64 N/A

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

       5. count-2 N/A

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

       6. lower-+.f64 51.1

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

   11. Simplified 51.1%

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

   if -1.99999999999999986e225 < x < -2.55000000000000004e-33 or 4.45e-7 < x < 4.1999999999999996e93

   1. Initial program 99.9%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot x} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{x \cdot t} \]

      2. lower-*.f64 43.2

         \[\leadsto \color{blue}{x \cdot t} \]

   5. Simplified 43.2%

      \[\leadsto \color{blue}{x \cdot t} \]

   if -2.55000000000000004e-33 < x < 4.45e-7

   1. Initial program 99.9%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{5 \cdot y} \]
   4. Step-by-step derivation

      1. lower-*.f64 57.6

         \[\leadsto \color{blue}{5 \cdot y} \]

   5. Simplified 57.6%

      \[\leadsto \color{blue}{5 \cdot y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 52.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+225}:\\ \;\;\;\;x \cdot \left(y + y\right)\\ \mathbf{elif}\;x \leq -2.55 \cdot 10^{-33}:\\ \;\;\;\;x \cdot t\\ \mathbf{elif}\;x \leq 4.45 \cdot 10^{-7}:\\ \;\;\;\;y \cdot 5\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{+93}:\\ \;\;\;\;x \cdot t\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \mathsf{fma}\left(x, 2, 5\right)\\ \mathbf{if}\;y \leq -5.5 \cdot 10^{+53}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 55:\\ \;\;\;\;x \cdot \mathsf{fma}\left(2, z, t\right)\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{+118}:\\ \;\;\;\;\mathsf{fma}\left(x, t, y \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* y (fma x 2.0 5.0))))
   (if (<= y -5.5e+53)
     t_1
     (if (<= y 55.0)
       (* x (fma 2.0 z t))
       (if (<= y 1.4e+118) (fma x t (* y 5.0)) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = y * fma(x, 2.0, 5.0);
	double tmp;
	if (y <= -5.5e+53) {
		tmp = t_1;
	} else if (y <= 55.0) {
		tmp = x * fma(2.0, z, t);
	} else if (y <= 1.4e+118) {
		tmp = fma(x, t, (y * 5.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(y * fma(x, 2.0, 5.0))
	tmp = 0.0
	if (y <= -5.5e+53)
		tmp = t_1;
	elseif (y <= 55.0)
		tmp = Float64(x * fma(2.0, z, t));
	elseif (y <= 1.4e+118)
		tmp = fma(x, t, Float64(y * 5.0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y * N[(x * 2.0 + 5.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -5.5e+53], t$95$1, If[LessEqual[y, 55.0], N[(x * N[(2.0 * z + t), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.4e+118], N[(x * t + N[(y * 5.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.49999999999999975e53 or 1.39999999999999993e118 < y

   1. Initial program 99.9%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(5 + 2 \cdot x\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto y \cdot \color{blue}{\left(2 \cdot x + 5\right)} \]

      2. metadata-eval N/A

         \[\leadsto y \cdot \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot x + 5\right) \]

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

         \[\leadsto y \cdot \left(\color{blue}{\left(\mathsf{neg}\left(-2 \cdot x\right)\right)} + 5\right) \]

      4. neg-sub0 N/A

         \[\leadsto y \cdot \left(\color{blue}{\left(0 - -2 \cdot x\right)} + 5\right) \]

      5. associate--r- N/A

         \[\leadsto y \cdot \color{blue}{\left(0 - \left(-2 \cdot x - 5\right)\right)} \]

      6. neg-sub0 N/A

         \[\leadsto y \cdot \color{blue}{\left(\mathsf{neg}\left(\left(-2 \cdot x - 5\right)\right)\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot \left(\mathsf{neg}\left(\left(-2 \cdot x - 5\right)\right)\right)} \]

      8. neg-sub0 N/A

         \[\leadsto y \cdot \color{blue}{\left(0 - \left(-2 \cdot x - 5\right)\right)} \]

      9. associate--r- N/A

         \[\leadsto y \cdot \color{blue}{\left(\left(0 - -2 \cdot x\right) + 5\right)} \]

      10. neg-sub0 N/A

          \[\leadsto y \cdot \left(\color{blue}{\left(\mathsf{neg}\left(-2 \cdot x\right)\right)} + 5\right) \]

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

          \[\leadsto y \cdot \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right) \cdot x} + 5\right) \]

      12. metadata-eval N/A

          \[\leadsto y \cdot \left(\color{blue}{2} \cdot x + 5\right) \]

      13. *-commutative N/A

          \[\leadsto y \cdot \left(\color{blue}{x \cdot 2} + 5\right) \]

      14. lower-fma.f64 81.3

          \[\leadsto y \cdot \color{blue}{\mathsf{fma}\left(x, 2, 5\right)} \]

   5. Simplified 81.3%

      \[\leadsto \color{blue}{y \cdot \mathsf{fma}\left(x, 2, 5\right)} \]

   if -5.49999999999999975e53 < y < 55

   1. Initial program 99.9%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 82.7

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

   5. Simplified 82.7%

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

   if 55 < y < 1.39999999999999993e118

   1. Initial program 99.9%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto x \cdot \left(\left(\left(\color{blue}{\left(y + z\right)} + z\right) + y\right) + t\right) + y \cdot 5 \]

      2. lift-+.f64 N/A

         \[\leadsto x \cdot \left(\left(\color{blue}{\left(\left(y + z\right) + z\right)} + y\right) + t\right) + y \cdot 5 \]

      3. lift-+.f64 N/A

         \[\leadsto x \cdot \left(\color{blue}{\left(\left(\left(y + z\right) + z\right) + y\right)} + t\right) + y \cdot 5 \]

      4. lift-+.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)} + y \cdot 5 \]

      5. lift-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)} + y \cdot 5 \]

      6. lift-*.f64 N/A

         \[\leadsto x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + \color{blue}{y \cdot 5} \]

      7. +-commutative N/A

         \[\leadsto \color{blue}{y \cdot 5 + x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto y \cdot 5 + \color{blue}{x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)} \]

      9. lift-+.f64 N/A

         \[\leadsto y \cdot 5 + x \cdot \color{blue}{\left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)} \]

      10. distribute-rgt-in N/A

          \[\leadsto y \cdot 5 + \color{blue}{\left(\left(\left(\left(y + z\right) + z\right) + y\right) \cdot x + t \cdot x\right)} \]

      11. associate-+r+ N/A

          \[\leadsto \color{blue}{\left(y \cdot 5 + \left(\left(\left(y + z\right) + z\right) + y\right) \cdot x\right) + t \cdot x} \]

      12. lower-+.f64 N/A

          \[\leadsto \color{blue}{\left(y \cdot 5 + \left(\left(\left(y + z\right) + z\right) + y\right) \cdot x\right) + t \cdot x} \]

   4. Applied egg-rr 96.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, 5, x \cdot \left(z + z\right)\right) + x \cdot t} \]

   5. Taylor expanded in z around 0

      \[\leadsto \color{blue}{5 \cdot y + t \cdot x} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{t \cdot x + 5 \cdot y} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{x \cdot t} + 5 \cdot y \]

      3. metadata-eval N/A

         \[\leadsto x \cdot t + \color{blue}{\left(\mathsf{neg}\left(-5\right)\right)} \cdot y \]

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

         \[\leadsto x \cdot t + \color{blue}{\left(\mathsf{neg}\left(-5 \cdot y\right)\right)} \]

      5. *-rgt-identity N/A

         \[\leadsto x \cdot t + \left(\mathsf{neg}\left(-5 \cdot \color{blue}{\left(y \cdot 1\right)}\right)\right) \]

      6. *-inverses N/A

         \[\leadsto x \cdot t + \left(\mathsf{neg}\left(-5 \cdot \left(y \cdot \color{blue}{\frac{z}{z}}\right)\right)\right) \]

      7. associate-/l* N/A

         \[\leadsto x \cdot t + \left(\mathsf{neg}\left(-5 \cdot \color{blue}{\frac{y \cdot z}{z}}\right)\right) \]

      8. associate-*l/ N/A

         \[\leadsto x \cdot t + \left(\mathsf{neg}\left(-5 \cdot \color{blue}{\left(\frac{y}{z} \cdot z\right)}\right)\right) \]

      9. associate-*l* N/A

         \[\leadsto x \cdot t + \left(\mathsf{neg}\left(\color{blue}{\left(-5 \cdot \frac{y}{z}\right) \cdot z}\right)\right) \]

      10. associate-*r/ N/A

          \[\leadsto x \cdot t + \left(\mathsf{neg}\left(\color{blue}{\frac{-5 \cdot y}{z}} \cdot z\right)\right) \]

      11. associate-*l/ N/A

          \[\leadsto x \cdot t + \left(\mathsf{neg}\left(\color{blue}{\frac{\left(-5 \cdot y\right) \cdot z}{z}}\right)\right) \]

      12. associate-/l* N/A

          \[\leadsto x \cdot t + \left(\mathsf{neg}\left(\color{blue}{\left(-5 \cdot y\right) \cdot \frac{z}{z}}\right)\right) \]

      13. *-inverses N/A

          \[\leadsto x \cdot t + \left(\mathsf{neg}\left(\left(-5 \cdot y\right) \cdot \color{blue}{1}\right)\right) \]

      14. *-inverses N/A

          \[\leadsto x \cdot t + \left(\mathsf{neg}\left(\left(-5 \cdot y\right) \cdot \color{blue}{\frac{x}{x}}\right)\right) \]

      15. associate-/l* N/A

          \[\leadsto x \cdot t + \left(\mathsf{neg}\left(\color{blue}{\frac{\left(-5 \cdot y\right) \cdot x}{x}}\right)\right) \]

      16. associate-*l/ N/A

          \[\leadsto x \cdot t + \left(\mathsf{neg}\left(\color{blue}{\frac{-5 \cdot y}{x} \cdot x}\right)\right) \]

      17. associate-*r/ N/A

          \[\leadsto x \cdot t + \left(\mathsf{neg}\left(\color{blue}{\left(-5 \cdot \frac{y}{x}\right)} \cdot x\right)\right) \]

      18. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(x, t, \mathsf{neg}\left(\left(-5 \cdot \frac{y}{x}\right) \cdot x\right)\right)} \]

      19. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(x, t, \mathsf{neg}\left(\color{blue}{\frac{-5 \cdot y}{x}} \cdot x\right)\right) \]

      20. associate-*l/ N/A

          \[\leadsto \mathsf{fma}\left(x, t, \mathsf{neg}\left(\color{blue}{\frac{\left(-5 \cdot y\right) \cdot x}{x}}\right)\right) \]

      21. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(x, t, \mathsf{neg}\left(\color{blue}{\left(-5 \cdot y\right) \cdot \frac{x}{x}}\right)\right) \]

      22. *-inverses N/A

          \[\leadsto \mathsf{fma}\left(x, t, \mathsf{neg}\left(\left(-5 \cdot y\right) \cdot \color{blue}{1}\right)\right) \]

      23. *-inverses N/A

          \[\leadsto \mathsf{fma}\left(x, t, \mathsf{neg}\left(\left(-5 \cdot y\right) \cdot \color{blue}{\frac{z}{z}}\right)\right) \]

      24. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(x, t, \mathsf{neg}\left(\color{blue}{\frac{\left(-5 \cdot y\right) \cdot z}{z}}\right)\right) \]

   7. Simplified 80.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, t, 5 \cdot y\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{+53}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(x, 2, 5\right)\\ \mathbf{elif}\;y \leq 55:\\ \;\;\;\;x \cdot \mathsf{fma}\left(2, z, t\right)\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{+118}:\\ \;\;\;\;\mathsf{fma}\left(x, t, y \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(x, 2, 5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 66.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \mathsf{fma}\left(2, z, t\right)\\ \mathbf{if}\;x \leq -1.05 \cdot 10^{-37}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-132}:\\ \;\;\;\;y \cdot 5\\ \mathbf{elif}\;x \leq 8 \cdot 10^{-10}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y, 2, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (fma 2.0 z t))))
   (if (<= x -1.05e-37)
     t_1
     (if (<= x 7e-132) (* y 5.0) (if (<= x 8e-10) t_1 (* x (fma y 2.0 t)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * fma(2.0, z, t);
	double tmp;
	if (x <= -1.05e-37) {
		tmp = t_1;
	} else if (x <= 7e-132) {
		tmp = y * 5.0;
	} else if (x <= 8e-10) {
		tmp = t_1;
	} else {
		tmp = x * fma(y, 2.0, t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(x * fma(2.0, z, t))
	tmp = 0.0
	if (x <= -1.05e-37)
		tmp = t_1;
	elseif (x <= 7e-132)
		tmp = Float64(y * 5.0);
	elseif (x <= 8e-10)
		tmp = t_1;
	else
		tmp = Float64(x * fma(y, 2.0, t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[(2.0 * z + t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.05e-37], t$95$1, If[LessEqual[x, 7e-132], N[(y * 5.0), $MachinePrecision], If[LessEqual[x, 8e-10], t$95$1, N[(x * N[(y * 2.0 + t), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.05e-37 or 6.9999999999999999e-132 < x < 8.00000000000000029e-10

   1. Initial program 99.9%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 67.1

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

   5. Simplified 67.1%

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

   if -1.05e-37 < x < 6.9999999999999999e-132

   1. Initial program 99.9%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{5 \cdot y} \]
   4. Step-by-step derivation

      1. lower-*.f64 64.2

         \[\leadsto \color{blue}{5 \cdot y} \]

   5. Simplified 64.2%

      \[\leadsto \color{blue}{5 \cdot y} \]

   if 8.00000000000000029e-10 < x

   1. Initial program 100.0%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-out N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-+.f64 97.3

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

   5. Simplified 97.3%

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

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 78.9

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

   8. Simplified 78.9%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{-37}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(2, z, t\right)\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-132}:\\ \;\;\;\;y \cdot 5\\ \mathbf{elif}\;x \leq 8 \cdot 10^{-10}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(2, z, t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y, 2, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 64.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.3 \cdot 10^{+250}:\\ \;\;\;\;x \cdot \left(y + y\right)\\ \mathbf{elif}\;y \leq -5.5 \cdot 10^{+73}:\\ \;\;\;\;y \cdot 5\\ \mathbf{elif}\;y \leq 10^{+53}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(2, z, t\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -3.3e+250)
   (* x (+ y y))
   (if (<= y -5.5e+73)
     (* y 5.0)
     (if (<= y 1e+53) (* x (fma 2.0 z t)) (* y 5.0)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -3.3e+250) {
		tmp = x * (y + y);
	} else if (y <= -5.5e+73) {
		tmp = y * 5.0;
	} else if (y <= 1e+53) {
		tmp = x * fma(2.0, z, t);
	} else {
		tmp = y * 5.0;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -3.3e+250)
		tmp = Float64(x * Float64(y + y));
	elseif (y <= -5.5e+73)
		tmp = Float64(y * 5.0);
	elseif (y <= 1e+53)
		tmp = Float64(x * fma(2.0, z, t));
	else
		tmp = Float64(y * 5.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -3.3e+250], N[(x * N[(y + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -5.5e+73], N[(y * 5.0), $MachinePrecision], If[LessEqual[y, 1e+53], N[(x * N[(2.0 * z + t), $MachinePrecision]), $MachinePrecision], N[(y * 5.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.30000000000000006e250

   1. Initial program 100.0%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-out N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-+.f64 64.8

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

   5. Simplified 64.8%

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

   6. Taylor expanded in y around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

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

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

      5. lower-*.f64 N/A

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

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

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 63.8

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

   8. Simplified 63.8%

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

   9. Taylor expanded in y around 0

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. lower-*.f64 N/A

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

       5. count-2 N/A

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

       6. lower-+.f64 63.8

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

   11. Simplified 63.8%

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

   if -3.30000000000000006e250 < y < -5.5000000000000003e73 or 9.9999999999999999e52 < y

   1. Initial program 99.9%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{5 \cdot y} \]
   4. Step-by-step derivation

      1. lower-*.f64 50.3

         \[\leadsto \color{blue}{5 \cdot y} \]

   5. Simplified 50.3%

      \[\leadsto \color{blue}{5 \cdot y} \]

   if -5.5000000000000003e73 < y < 9.9999999999999999e52

   1. Initial program 99.9%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 78.8

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

   5. Simplified 78.8%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.3 \cdot 10^{+250}:\\ \;\;\;\;x \cdot \left(y + y\right)\\ \mathbf{elif}\;y \leq -5.5 \cdot 10^{+73}:\\ \;\;\;\;y \cdot 5\\ \mathbf{elif}\;y \leq 10^{+53}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(2, z, t\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 5\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \mathsf{fma}\left(2, y + z, t\right)\\ \mathbf{if}\;x \leq -48000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 2.5:\\ \;\;\;\;\mathsf{fma}\left(y, 5, x \cdot \left(t + \left(z + z\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (fma 2.0 (+ y z) t))))
   (if (<= x -48000.0)
     t_1
     (if (<= x 2.5) (fma y 5.0 (* x (+ t (+ z z)))) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * fma(2.0, (y + z), t);
	double tmp;
	if (x <= -48000.0) {
		tmp = t_1;
	} else if (x <= 2.5) {
		tmp = fma(y, 5.0, (x * (t + (z + z))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(x * fma(2.0, Float64(y + z), t))
	tmp = 0.0
	if (x <= -48000.0)
		tmp = t_1;
	elseif (x <= 2.5)
		tmp = fma(y, 5.0, Float64(x * Float64(t + Float64(z + z))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[(2.0 * N[(y + z), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -48000.0], t$95$1, If[LessEqual[x, 2.5], N[(y * 5.0 + N[(x * N[(t + N[(z + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -48000 or 2.5 < x

   1. Initial program 100.0%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-out N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-+.f64 98.9

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

   5. Simplified 98.9%

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

   if -48000 < x < 2.5

   1. Initial program 99.9%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto x \cdot \left(\left(\left(\color{blue}{\left(y + z\right)} + z\right) + y\right) + t\right) + y \cdot 5 \]

      2. lift-+.f64 N/A

         \[\leadsto x \cdot \left(\left(\color{blue}{\left(\left(y + z\right) + z\right)} + y\right) + t\right) + y \cdot 5 \]

      3. lift-+.f64 N/A

         \[\leadsto x \cdot \left(\color{blue}{\left(\left(\left(y + z\right) + z\right) + y\right)} + t\right) + y \cdot 5 \]

      4. lift-+.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)} + y \cdot 5 \]

      5. lift-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)} + y \cdot 5 \]

      6. lift-*.f64 N/A

         \[\leadsto x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + \color{blue}{y \cdot 5} \]

      7. +-commutative N/A

         \[\leadsto \color{blue}{y \cdot 5 + x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot 5} + x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) \]

      9. lower-fma.f64 100.0

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, 5, x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)\right)} \]

   4. Applied egg-rr 98.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, 5, x \cdot \left(\left(z + z\right) + t\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -48000:\\ \;\;\;\;x \cdot \mathsf{fma}\left(2, y + z, t\right)\\ \mathbf{elif}\;x \leq 2.5:\\ \;\;\;\;\mathsf{fma}\left(y, 5, x \cdot \left(t + \left(z + z\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(2, y + z, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 86.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.7 \cdot 10^{+109}:\\ \;\;\;\;\mathsf{fma}\left(5, y, x \cdot \left(2 \cdot z\right)\right)\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{+112}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(y, 2, t\right), y \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(2, y + z, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -5.7e+109)
   (fma 5.0 y (* x (* 2.0 z)))
   (if (<= z 2.9e+112)
     (fma x (fma y 2.0 t) (* y 5.0))
     (* x (fma 2.0 (+ y z) t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -5.7e+109) {
		tmp = fma(5.0, y, (x * (2.0 * z)));
	} else if (z <= 2.9e+112) {
		tmp = fma(x, fma(y, 2.0, t), (y * 5.0));
	} else {
		tmp = x * fma(2.0, (y + z), t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -5.7e+109)
		tmp = fma(5.0, y, Float64(x * Float64(2.0 * z)));
	elseif (z <= 2.9e+112)
		tmp = fma(x, fma(y, 2.0, t), Float64(y * 5.0));
	else
		tmp = Float64(x * fma(2.0, Float64(y + z), t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -5.7e+109], N[(5.0 * y + N[(x * N[(2.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.9e+112], N[(x * N[(y * 2.0 + t), $MachinePrecision] + N[(y * 5.0), $MachinePrecision]), $MachinePrecision], N[(x * N[(2.0 * N[(y + z), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.7000000000000002e109

   1. Initial program 100.0%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto x \cdot \left(\left(\left(\color{blue}{\left(y + z\right)} + z\right) + y\right) + t\right) + y \cdot 5 \]

      2. lift-+.f64 N/A

         \[\leadsto x \cdot \left(\left(\color{blue}{\left(\left(y + z\right) + z\right)} + y\right) + t\right) + y \cdot 5 \]

      3. lift-+.f64 N/A

         \[\leadsto x \cdot \left(\color{blue}{\left(\left(\left(y + z\right) + z\right) + y\right)} + t\right) + y \cdot 5 \]

      4. lift-+.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)} + y \cdot 5 \]

      5. lift-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)} + y \cdot 5 \]

      6. lift-*.f64 N/A

         \[\leadsto x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + \color{blue}{y \cdot 5} \]

      7. +-commutative N/A

         \[\leadsto \color{blue}{y \cdot 5 + x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto y \cdot 5 + \color{blue}{x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)} \]

      9. lift-+.f64 N/A

         \[\leadsto y \cdot 5 + x \cdot \color{blue}{\left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)} \]

      10. distribute-rgt-in N/A

          \[\leadsto y \cdot 5 + \color{blue}{\left(\left(\left(\left(y + z\right) + z\right) + y\right) \cdot x + t \cdot x\right)} \]

      11. associate-+r+ N/A

          \[\leadsto \color{blue}{\left(y \cdot 5 + \left(\left(\left(y + z\right) + z\right) + y\right) \cdot x\right) + t \cdot x} \]

      12. lower-+.f64 N/A

          \[\leadsto \color{blue}{\left(y \cdot 5 + \left(\left(\left(y + z\right) + z\right) + y\right) \cdot x\right) + t \cdot x} \]

   4. Applied egg-rr 93.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, 5, x \cdot \left(z + z\right)\right) + x \cdot t} \]

   5. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \left(x \cdot z\right) + 5 \cdot y} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{5 \cdot y + 2 \cdot \left(x \cdot z\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(5, y, 2 \cdot \left(x \cdot z\right)\right)} \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(5, y, \color{blue}{\left(2 \cdot x\right) \cdot z}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(5, y, \color{blue}{\left(x \cdot 2\right)} \cdot z\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(5, y, \color{blue}{x \cdot \left(2 \cdot z\right)}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(5, y, \color{blue}{x \cdot \left(2 \cdot z\right)}\right) \]

      7. lower-*.f64 87.4

         \[\leadsto \mathsf{fma}\left(5, y, x \cdot \color{blue}{\left(2 \cdot z\right)}\right) \]

   7. Simplified 87.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(5, y, x \cdot \left(2 \cdot z\right)\right)} \]

   if -5.7000000000000002e109 < z < 2.9000000000000002e112

   1. Initial program 99.9%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{5 \cdot y + x \cdot \left(t + 2 \cdot y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(t + 2 \cdot y\right) + 5 \cdot y} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, t + 2 \cdot y, 5 \cdot y\right)} \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{2 \cdot y + t}, 5 \cdot y\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{y \cdot 2} + t, 5 \cdot y\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(y, 2, t\right)}, 5 \cdot y\right) \]

      6. lower-*.f64 91.6

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(y, 2, t\right), \color{blue}{5 \cdot y}\right) \]

   5. Simplified 91.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(y, 2, t\right), 5 \cdot y\right)} \]

   if 2.9000000000000002e112 < z

   1. Initial program 100.0%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-out N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-+.f64 83.9

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

   5. Simplified 83.9%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.7 \cdot 10^{+109}:\\ \;\;\;\;\mathsf{fma}\left(5, y, x \cdot \left(2 \cdot z\right)\right)\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{+112}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(y, 2, t\right), y \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(2, y + z, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 47.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{-25}:\\ \;\;\;\;z \cdot \left(x + x\right)\\ \mathbf{elif}\;x \leq 4.45 \cdot 10^{-7}:\\ \;\;\;\;y \cdot 5\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{+93}:\\ \;\;\;\;x \cdot t\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -5.5e-25)
   (* z (+ x x))
   (if (<= x 4.45e-7) (* y 5.0) (if (<= x 4.2e+93) (* x t) (* x (+ y y))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -5.5e-25) {
		tmp = z * (x + x);
	} else if (x <= 4.45e-7) {
		tmp = y * 5.0;
	} else if (x <= 4.2e+93) {
		tmp = x * t;
	} else {
		tmp = x * (y + y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (x <= (-5.5d-25)) then
        tmp = z * (x + x)
    else if (x <= 4.45d-7) then
        tmp = y * 5.0d0
    else if (x <= 4.2d+93) then
        tmp = x * t
    else
        tmp = x * (y + y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -5.5e-25) {
		tmp = z * (x + x);
	} else if (x <= 4.45e-7) {
		tmp = y * 5.0;
	} else if (x <= 4.2e+93) {
		tmp = x * t;
	} else {
		tmp = x * (y + y);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -5.5e-25:
		tmp = z * (x + x)
	elif x <= 4.45e-7:
		tmp = y * 5.0
	elif x <= 4.2e+93:
		tmp = x * t
	else:
		tmp = x * (y + y)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -5.5e-25)
		tmp = Float64(z * Float64(x + x));
	elseif (x <= 4.45e-7)
		tmp = Float64(y * 5.0);
	elseif (x <= 4.2e+93)
		tmp = Float64(x * t);
	else
		tmp = Float64(x * Float64(y + y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -5.5e-25)
		tmp = z * (x + x);
	elseif (x <= 4.45e-7)
		tmp = y * 5.0;
	elseif (x <= 4.2e+93)
		tmp = x * t;
	else
		tmp = x * (y + y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -5.5e-25], N[(z * N[(x + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.45e-7], N[(y * 5.0), $MachinePrecision], If[LessEqual[x, 4.2e+93], N[(x * t), $MachinePrecision], N[(x * N[(y + y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -5.50000000000000004e-25

   1. Initial program 99.9%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-out N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-+.f64 94.9

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

   5. Simplified 94.9%

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

   6. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 68.6

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

   8. Simplified 68.6%

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

      1. distribute-lft-in N/A

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

      2. count-2 N/A

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

      3. associate-+r+ N/A

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

      4. lower-+.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 68.6

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

   10. Applied egg-rr 68.6%

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

   11. Taylor expanded in y around 0

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

       1. associate-*r* N/A

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

       2. lower-*.f64 N/A

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

       3. count-2 N/A

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

       4. lower-+.f64 41.6

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

   13. Simplified 41.6%

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

   if -5.50000000000000004e-25 < x < 4.45e-7

   1. Initial program 99.9%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{5 \cdot y} \]
   4. Step-by-step derivation

      1. lower-*.f64 57.5

         \[\leadsto \color{blue}{5 \cdot y} \]

   5. Simplified 57.5%

      \[\leadsto \color{blue}{5 \cdot y} \]

   if 4.45e-7 < x < 4.1999999999999996e93

   1. Initial program 100.0%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot x} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{x \cdot t} \]

      2. lower-*.f64 69.9

         \[\leadsto \color{blue}{x \cdot t} \]

   5. Simplified 69.9%

      \[\leadsto \color{blue}{x \cdot t} \]

   if 4.1999999999999996e93 < x

   1. Initial program 100.0%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-out N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-+.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

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

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

      5. lower-*.f64 N/A

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

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

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 49.6

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

   8. Simplified 49.6%

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

   9. Taylor expanded in y around 0

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. lower-*.f64 N/A

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

       5. count-2 N/A

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

       6. lower-+.f64 49.6

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

   11. Simplified 49.6%

       \[\leadsto \color{blue}{x \cdot \left(y + y\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 52.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{-25}:\\ \;\;\;\;z \cdot \left(x + x\right)\\ \mathbf{elif}\;x \leq 4.45 \cdot 10^{-7}:\\ \;\;\;\;y \cdot 5\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{+93}:\\ \;\;\;\;x \cdot t\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 87.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \mathsf{fma}\left(2, y + z, t\right)\\ \mathbf{if}\;x \leq -5.5 \cdot 10^{-25}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-131}:\\ \;\;\;\;\mathsf{fma}\left(y, 5, x \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (fma 2.0 (+ y z) t))))
   (if (<= x -5.5e-25) t_1 (if (<= x 4.5e-131) (fma y 5.0 (* x t)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * fma(2.0, (y + z), t);
	double tmp;
	if (x <= -5.5e-25) {
		tmp = t_1;
	} else if (x <= 4.5e-131) {
		tmp = fma(y, 5.0, (x * t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(x * fma(2.0, Float64(y + z), t))
	tmp = 0.0
	if (x <= -5.5e-25)
		tmp = t_1;
	elseif (x <= 4.5e-131)
		tmp = fma(y, 5.0, Float64(x * t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[(2.0 * N[(y + z), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5.5e-25], t$95$1, If[LessEqual[x, 4.5e-131], N[(y * 5.0 + N[(x * t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -5.50000000000000004e-25 or 4.5000000000000002e-131 < x

   1. Initial program 99.9%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-out N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-+.f64 91.3

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

   5. Simplified 91.3%

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

   if -5.50000000000000004e-25 < x < 4.5000000000000002e-131

   1. Initial program 99.9%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{5 \cdot y + x \cdot \left(t + \left(2 \cdot y + 2 \cdot z\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(t + \left(2 \cdot y + 2 \cdot z\right)\right) + 5 \cdot y} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, t + \left(2 \cdot y + 2 \cdot z\right), 5 \cdot y\right)} \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\left(2 \cdot y + 2 \cdot z\right) + t}, 5 \cdot y\right) \]

      4. distribute-lft-out N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{2 \cdot \left(y + z\right)} + t, 5 \cdot y\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(2, y + z, t\right)}, 5 \cdot y\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(2, \color{blue}{y + z}, t\right), 5 \cdot y\right) \]

      7. lower-*.f64 99.9

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(2, y + z, t\right), \color{blue}{5 \cdot y}\right) \]

   5. Simplified 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(2, y + z, t\right), 5 \cdot y\right)} \]

   6. Taylor expanded in z around 0

      \[\leadsto \color{blue}{5 \cdot y + x \cdot \left(t + 2 \cdot y\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 5 \cdot y + x \cdot \color{blue}{\left(2 \cdot y + t\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto 5 \cdot y + \color{blue}{\left(x \cdot \left(2 \cdot y\right) + x \cdot t\right)} \]

      3. *-commutative N/A

         \[\leadsto 5 \cdot y + \left(x \cdot \color{blue}{\left(y \cdot 2\right)} + x \cdot t\right) \]

      4. associate-*r* N/A

         \[\leadsto 5 \cdot y + \left(\color{blue}{\left(x \cdot y\right) \cdot 2} + x \cdot t\right) \]

      5. *-commutative N/A

         \[\leadsto 5 \cdot y + \left(\left(x \cdot y\right) \cdot 2 + \color{blue}{t \cdot x}\right) \]

      6. *-commutative N/A

         \[\leadsto 5 \cdot y + \left(\color{blue}{2 \cdot \left(x \cdot y\right)} + t \cdot x\right) \]

      7. associate-*r* N/A

         \[\leadsto 5 \cdot y + \left(\color{blue}{\left(2 \cdot x\right) \cdot y} + t \cdot x\right) \]

      8. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(5 \cdot y + \left(2 \cdot x\right) \cdot y\right) + t \cdot x} \]

      9. distribute-rgt-in N/A

         \[\leadsto \color{blue}{y \cdot \left(5 + 2 \cdot x\right)} + t \cdot x \]

      10. +-commutative N/A

          \[\leadsto y \cdot \color{blue}{\left(2 \cdot x + 5\right)} + t \cdot x \]

      11. metadata-eval N/A

          \[\leadsto y \cdot \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot x + 5\right) + t \cdot x \]

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

          \[\leadsto y \cdot \left(\color{blue}{\left(\mathsf{neg}\left(-2 \cdot x\right)\right)} + 5\right) + t \cdot x \]

      13. metadata-eval N/A

          \[\leadsto y \cdot \left(\left(\mathsf{neg}\left(-2 \cdot x\right)\right) + \color{blue}{\left(\mathsf{neg}\left(-5\right)\right)}\right) + t \cdot x \]

      14. distribute-neg-in N/A

          \[\leadsto y \cdot \color{blue}{\left(\mathsf{neg}\left(\left(-2 \cdot x + -5\right)\right)\right)} + t \cdot x \]

      15. metadata-eval N/A

          \[\leadsto y \cdot \left(\mathsf{neg}\left(\left(-2 \cdot x + \color{blue}{\left(\mathsf{neg}\left(5\right)\right)}\right)\right)\right) + t \cdot x \]

      16. sub-neg N/A

          \[\leadsto y \cdot \left(\mathsf{neg}\left(\color{blue}{\left(-2 \cdot x - 5\right)}\right)\right) + t \cdot x \]

      17. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(y, \mathsf{neg}\left(\left(-2 \cdot x - 5\right)\right), t \cdot x\right)} \]

   8. Simplified 80.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(x, 2, 5\right), x \cdot t\right)} \]

   9. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{5}, x \cdot t\right) \]
   10. Step-by-step derivation

   11. Simplified 80.6%

       \[\leadsto \mathsf{fma}\left(y, \color{blue}{5}, x \cdot t\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 79.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \mathsf{fma}\left(x, 2, 5\right)\\ \mathbf{if}\;y \leq -5.5 \cdot 10^{+53}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{+52}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(2, z, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* y (fma x 2.0 5.0))))
   (if (<= y -5.5e+53) t_1 (if (<= y 1.7e+52) (* x (fma 2.0 z t)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = y * fma(x, 2.0, 5.0);
	double tmp;
	if (y <= -5.5e+53) {
		tmp = t_1;
	} else if (y <= 1.7e+52) {
		tmp = x * fma(2.0, z, t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(y * fma(x, 2.0, 5.0))
	tmp = 0.0
	if (y <= -5.5e+53)
		tmp = t_1;
	elseif (y <= 1.7e+52)
		tmp = Float64(x * fma(2.0, z, t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y * N[(x * 2.0 + 5.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -5.5e+53], t$95$1, If[LessEqual[y, 1.7e+52], N[(x * N[(2.0 * z + t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -5.49999999999999975e53 or 1.7e52 < y

   1. Initial program 99.9%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(5 + 2 \cdot x\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto y \cdot \color{blue}{\left(2 \cdot x + 5\right)} \]

      2. metadata-eval N/A

         \[\leadsto y \cdot \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot x + 5\right) \]

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

         \[\leadsto y \cdot \left(\color{blue}{\left(\mathsf{neg}\left(-2 \cdot x\right)\right)} + 5\right) \]

      4. neg-sub0 N/A

         \[\leadsto y \cdot \left(\color{blue}{\left(0 - -2 \cdot x\right)} + 5\right) \]

      5. associate--r- N/A

         \[\leadsto y \cdot \color{blue}{\left(0 - \left(-2 \cdot x - 5\right)\right)} \]

      6. neg-sub0 N/A

         \[\leadsto y \cdot \color{blue}{\left(\mathsf{neg}\left(\left(-2 \cdot x - 5\right)\right)\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot \left(\mathsf{neg}\left(\left(-2 \cdot x - 5\right)\right)\right)} \]

      8. neg-sub0 N/A

         \[\leadsto y \cdot \color{blue}{\left(0 - \left(-2 \cdot x - 5\right)\right)} \]

      9. associate--r- N/A

         \[\leadsto y \cdot \color{blue}{\left(\left(0 - -2 \cdot x\right) + 5\right)} \]

      10. neg-sub0 N/A

          \[\leadsto y \cdot \left(\color{blue}{\left(\mathsf{neg}\left(-2 \cdot x\right)\right)} + 5\right) \]

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

          \[\leadsto y \cdot \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right) \cdot x} + 5\right) \]

      12. metadata-eval N/A

          \[\leadsto y \cdot \left(\color{blue}{2} \cdot x + 5\right) \]

      13. *-commutative N/A

          \[\leadsto y \cdot \left(\color{blue}{x \cdot 2} + 5\right) \]

      14. lower-fma.f64 80.1

          \[\leadsto y \cdot \color{blue}{\mathsf{fma}\left(x, 2, 5\right)} \]

   5. Simplified 80.1%

      \[\leadsto \color{blue}{y \cdot \mathsf{fma}\left(x, 2, 5\right)} \]

   if -5.49999999999999975e53 < y < 1.7e52

   1. Initial program 99.9%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 80.1

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

   5. Simplified 80.1%

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

Alternative 12: 48.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.55 \cdot 10^{-33}:\\ \;\;\;\;x \cdot t\\ \mathbf{elif}\;x \leq 4.45 \cdot 10^{-7}:\\ \;\;\;\;y \cdot 5\\ \mathbf{else}:\\ \;\;\;\;x \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -2.55e-33) (* x t) (if (<= x 4.45e-7) (* y 5.0) (* x t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -2.55e-33) {
		tmp = x * t;
	} else if (x <= 4.45e-7) {
		tmp = y * 5.0;
	} else {
		tmp = x * t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (x <= (-2.55d-33)) then
        tmp = x * t
    else if (x <= 4.45d-7) then
        tmp = y * 5.0d0
    else
        tmp = x * t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -2.55e-33) {
		tmp = x * t;
	} else if (x <= 4.45e-7) {
		tmp = y * 5.0;
	} else {
		tmp = x * t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -2.55e-33:
		tmp = x * t
	elif x <= 4.45e-7:
		tmp = y * 5.0
	else:
		tmp = x * t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -2.55e-33)
		tmp = Float64(x * t);
	elseif (x <= 4.45e-7)
		tmp = Float64(y * 5.0);
	else
		tmp = Float64(x * t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -2.55e-33)
		tmp = x * t;
	elseif (x <= 4.45e-7)
		tmp = y * 5.0;
	else
		tmp = x * t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -2.55e-33], N[(x * t), $MachinePrecision], If[LessEqual[x, 4.45e-7], N[(y * 5.0), $MachinePrecision], N[(x * t), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.55000000000000004e-33 or 4.45e-7 < x

   1. Initial program 99.9%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot x} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{x \cdot t} \]

      2. lower-*.f64 39.1

         \[\leadsto \color{blue}{x \cdot t} \]

   5. Simplified 39.1%

      \[\leadsto \color{blue}{x \cdot t} \]

   if -2.55000000000000004e-33 < x < 4.45e-7

   1. Initial program 99.9%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{5 \cdot y} \]
   4. Step-by-step derivation

      1. lower-*.f64 57.6

         \[\leadsto \color{blue}{5 \cdot y} \]

   5. Simplified 57.6%

      \[\leadsto \color{blue}{5 \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.55 \cdot 10^{-33}:\\ \;\;\;\;x \cdot t\\ \mathbf{elif}\;x \leq 4.45 \cdot 10^{-7}:\\ \;\;\;\;y \cdot 5\\ \mathbf{else}:\\ \;\;\;\;x \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 30.3% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ y \cdot 5 \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (* y 5.0))

C

double code(double x, double y, double z, double t) {
	return y * 5.0;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = y * 5.0d0
end function

Java

public static double code(double x, double y, double z, double t) {
	return y * 5.0;
}

Python

def code(x, y, z, t):
	return y * 5.0

Julia

function code(x, y, z, t)
	return Float64(y * 5.0)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = y * 5.0;
end

Wolfram

code[x_, y_, z_, t_] := N[(y * 5.0), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{5 \cdot y} \]
4. Step-by-step derivation

   1. lower-*.f64 32.0

      \[\leadsto \color{blue}{5 \cdot y} \]

5. Simplified 32.0%

   \[\leadsto \color{blue}{5 \cdot y} \]

6. Final simplification 32.0%

   \[\leadsto y \cdot 5 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024212 
(FPCore (x y z t)
  :name "Graphics.Rendering.Plot.Render.Plot.Legend:renderLegendOutside from plot-0.2.3.4, B"
  :precision binary64
  (+ (* x (+ (+ (+ (+ y z) z) y) t)) (* y 5.0)))