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

Percentage Accurate: 99.8% → 99.8%

Time: 8.5s

Alternatives: 16

Speedup: 1.0×

• 26.4% 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: 99.8% 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: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return (x * ((y + (z + (y + z))) + 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 + (y + z))) + t)) + (y * 5.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := N[(N[(x * N[(N[(y + N[(z + N[(y + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]), $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. Final simplification 99.9%

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

Alternative 2: 87.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \mathsf{fma}\left(2, y + z, t\right)\\ \mathbf{if}\;x \leq -650:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -1.05 \cdot 10^{-278}:\\ \;\;\;\;\mathsf{fma}\left(5, y, x \cdot \mathsf{fma}\left(y, 2, t\right)\right)\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(z + z, x, y \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (fma 2.0 (+ y z) t))))
   (if (<= x -650.0)
     t_1
     (if (<= x -1.05e-278)
       (fma 5.0 y (* x (fma y 2.0 t)))
       (if (<= x 1.4e-8) (fma (+ z z) x (* y 5.0)) 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 <= -650.0) {
		tmp = t_1;
	} else if (x <= -1.05e-278) {
		tmp = fma(5.0, y, (x * fma(y, 2.0, t)));
	} else if (x <= 1.4e-8) {
		tmp = fma((z + z), x, (y * 5.0));
	} 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 <= -650.0)
		tmp = t_1;
	elseif (x <= -1.05e-278)
		tmp = fma(5.0, y, Float64(x * fma(y, 2.0, t)));
	elseif (x <= 1.4e-8)
		tmp = fma(Float64(z + z), x, Float64(y * 5.0));
	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, -650.0], t$95$1, If[LessEqual[x, -1.05e-278], N[(5.0 * y + N[(x * N[(y * 2.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.4e-8], N[(N[(z + z), $MachinePrecision] * x + N[(y * 5.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if x < -650 or 1.4e-8 < 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 99.6

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

   5. Applied rewrites 99.6%

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

   if -650 < x < -1.05000000000000007e-278

   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. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 80.1

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

   5. Applied rewrites 80.1%

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

   if -1.05000000000000007e-278 < x < 1.4e-8

   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

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. distribute-rgt-in N/A

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

      5. associate-+l+ N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. lift-+.f64 N/A

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

      12. flip-+ N/A

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

      13. +-inverses N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{0}}{\left(y + z\right) - \left(y + z\right)}, x, t \cdot x + y \cdot 5\right) \]

      14. +-inverses N/A

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

      15. +-inverses N/A

          \[\leadsto \mathsf{fma}\left(\frac{z \cdot z - z \cdot z}{\color{blue}{0}}, x, t \cdot x + y \cdot 5\right) \]

      16. +-inverses N/A

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

      17. flip-+ N/A

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

      18. lower-+.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-fma.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in t around 0

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

      1. lower-*.f64 88.3

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

   7. Applied rewrites 88.3%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -650:\\ \;\;\;\;x \cdot \mathsf{fma}\left(2, y + z, t\right)\\ \mathbf{elif}\;x \leq -1.05 \cdot 10^{-278}:\\ \;\;\;\;\mathsf{fma}\left(5, y, x \cdot \mathsf{fma}\left(y, 2, t\right)\right)\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(z + z, x, y \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(2, y + z, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{-28}:\\ \;\;\;\;x \cdot \left(\left(y + z\right) + \left(\left(y + z\right) + t\right)\right)\\ \mathbf{elif}\;x \leq -1.05 \cdot 10^{-278}:\\ \;\;\;\;\mathsf{fma}\left(x, t, y \cdot 5\right)\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(z + z, x, y \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(2, y + z, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -3.4e-28)
   (* x (+ (+ y z) (+ (+ y z) t)))
   (if (<= x -1.05e-278)
     (fma x t (* y 5.0))
     (if (<= x 1.4e-8) (fma (+ z z) x (* y 5.0)) (* x (fma 2.0 (+ y z) t))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -3.4e-28) {
		tmp = x * ((y + z) + ((y + z) + t));
	} else if (x <= -1.05e-278) {
		tmp = fma(x, t, (y * 5.0));
	} else if (x <= 1.4e-8) {
		tmp = fma((z + z), x, (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 (x <= -3.4e-28)
		tmp = Float64(x * Float64(Float64(y + z) + Float64(Float64(y + z) + t)));
	elseif (x <= -1.05e-278)
		tmp = fma(x, t, Float64(y * 5.0));
	elseif (x <= 1.4e-8)
		tmp = fma(Float64(z + z), x, 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[x, -3.4e-28], N[(x * N[(N[(y + z), $MachinePrecision] + N[(N[(y + z), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.05e-278], N[(x * t + N[(y * 5.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.4e-8], N[(N[(z + z), $MachinePrecision] * x + N[(y * 5.0), $MachinePrecision]), $MachinePrecision], N[(x * N[(2.0 * N[(y + z), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -3.4000000000000001e-28

   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 95.8

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

   5. Applied rewrites 95.8%

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

   7. Applied rewrites 95.8%

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

   if -3.4000000000000001e-28 < x < -1.05000000000000007e-278

   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

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. distribute-rgt-in N/A

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

      5. associate-+l+ N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. lift-+.f64 N/A

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

      12. flip-+ N/A

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

      13. +-inverses N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{0}}{\left(y + z\right) - \left(y + z\right)}, x, t \cdot x + y \cdot 5\right) \]

      14. +-inverses N/A

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

      15. +-inverses N/A

          \[\leadsto \mathsf{fma}\left(\frac{z \cdot z - z \cdot z}{\color{blue}{0}}, x, t \cdot x + y \cdot 5\right) \]

      16. +-inverses N/A

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

      17. flip-+ N/A

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

      18. lower-+.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-fma.f64 100.0

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

   4. Applied rewrites 100.0%

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

   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. lower-fma.f64 N/A

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

      4. lower-*.f64 82.3

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

   7. Applied rewrites 82.3%

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

   if -1.05000000000000007e-278 < x < 1.4e-8

   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

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. distribute-rgt-in N/A

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

      5. associate-+l+ N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. lift-+.f64 N/A

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

      12. flip-+ N/A

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

      13. +-inverses N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{0}}{\left(y + z\right) - \left(y + z\right)}, x, t \cdot x + y \cdot 5\right) \]

      14. +-inverses N/A

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

      15. +-inverses N/A

          \[\leadsto \mathsf{fma}\left(\frac{z \cdot z - z \cdot z}{\color{blue}{0}}, x, t \cdot x + y \cdot 5\right) \]

      16. +-inverses N/A

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

      17. flip-+ N/A

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

      18. lower-+.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-fma.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in t around 0

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

      1. lower-*.f64 88.3

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

   7. Applied rewrites 88.3%

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

   if 1.4e-8 < 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. Applied rewrites 100.0%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{-28}:\\ \;\;\;\;x \cdot \left(\left(y + z\right) + \left(\left(y + z\right) + t\right)\right)\\ \mathbf{elif}\;x \leq -1.05 \cdot 10^{-278}:\\ \;\;\;\;\mathsf{fma}\left(x, t, y \cdot 5\right)\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(z + z, x, y \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(2, y + z, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \mathsf{fma}\left(2, y + z, t\right)\\ \mathbf{if}\;x \leq -3.4 \cdot 10^{-28}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -1.05 \cdot 10^{-278}:\\ \;\;\;\;\mathsf{fma}\left(x, t, y \cdot 5\right)\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(z + z, x, y \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (fma 2.0 (+ y z) t))))
   (if (<= x -3.4e-28)
     t_1
     (if (<= x -1.05e-278)
       (fma x t (* y 5.0))
       (if (<= x 1.4e-8) (fma (+ z z) x (* y 5.0)) 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 <= -3.4e-28) {
		tmp = t_1;
	} else if (x <= -1.05e-278) {
		tmp = fma(x, t, (y * 5.0));
	} else if (x <= 1.4e-8) {
		tmp = fma((z + z), x, (y * 5.0));
	} 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 <= -3.4e-28)
		tmp = t_1;
	elseif (x <= -1.05e-278)
		tmp = fma(x, t, Float64(y * 5.0));
	elseif (x <= 1.4e-8)
		tmp = fma(Float64(z + z), x, Float64(y * 5.0));
	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, -3.4e-28], t$95$1, If[LessEqual[x, -1.05e-278], N[(x * t + N[(y * 5.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.4e-8], N[(N[(z + z), $MachinePrecision] * x + N[(y * 5.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.4000000000000001e-28 or 1.4e-8 < 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 97.2

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

   5. Applied rewrites 97.2%

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

   if -3.4000000000000001e-28 < x < -1.05000000000000007e-278

   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

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. distribute-rgt-in N/A

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

      5. associate-+l+ N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. lift-+.f64 N/A

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

      12. flip-+ N/A

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

      13. +-inverses N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{0}}{\left(y + z\right) - \left(y + z\right)}, x, t \cdot x + y \cdot 5\right) \]

      14. +-inverses N/A

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

      15. +-inverses N/A

          \[\leadsto \mathsf{fma}\left(\frac{z \cdot z - z \cdot z}{\color{blue}{0}}, x, t \cdot x + y \cdot 5\right) \]

      16. +-inverses N/A

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

      17. flip-+ N/A

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

      18. lower-+.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-fma.f64 100.0

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

   4. Applied rewrites 100.0%

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

   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. lower-fma.f64 N/A

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

      4. lower-*.f64 82.3

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

   7. Applied rewrites 82.3%

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

   if -1.05000000000000007e-278 < x < 1.4e-8

   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

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. distribute-rgt-in N/A

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

      5. associate-+l+ N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. lift-+.f64 N/A

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

      12. flip-+ N/A

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

      13. +-inverses N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{0}}{\left(y + z\right) - \left(y + z\right)}, x, t \cdot x + y \cdot 5\right) \]

      14. +-inverses N/A

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

      15. +-inverses N/A

          \[\leadsto \mathsf{fma}\left(\frac{z \cdot z - z \cdot z}{\color{blue}{0}}, x, t \cdot x + y \cdot 5\right) \]

      16. +-inverses N/A

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

      17. flip-+ N/A

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

      18. lower-+.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-fma.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in t around 0

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

      1. lower-*.f64 88.3

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

   7. Applied rewrites 88.3%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{-28}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(2, y + z, t\right)\\ \mathbf{elif}\;x \leq -1.05 \cdot 10^{-278}:\\ \;\;\;\;\mathsf{fma}\left(x, t, y \cdot 5\right)\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(z + z, x, y \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(2, y + z, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \mathsf{fma}\left(x, 2, 5\right)\\ \mathbf{if}\;y \leq -6.2 \cdot 10^{+17}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{-112}:\\ \;\;\;\;x \cdot \left(z + \left(z + t\right)\right)\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{+179}:\\ \;\;\;\;\mathsf{fma}\left(5, y, x \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* y (fma x 2.0 5.0))))
   (if (<= y -6.2e+17)
     t_1
     (if (<= y 1.75e-112)
       (* x (+ z (+ z t)))
       (if (<= y 1.45e+179) (fma 5.0 y (* x 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 <= -6.2e+17) {
		tmp = t_1;
	} else if (y <= 1.75e-112) {
		tmp = x * (z + (z + t));
	} else if (y <= 1.45e+179) {
		tmp = fma(5.0, y, (x * 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 <= -6.2e+17)
		tmp = t_1;
	elseif (y <= 1.75e-112)
		tmp = Float64(x * Float64(z + Float64(z + t)));
	elseif (y <= 1.45e+179)
		tmp = fma(5.0, y, Float64(x * 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, -6.2e+17], t$95$1, If[LessEqual[y, 1.75e-112], N[(x * N[(z + N[(z + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.45e+179], N[(5.0 * y + N[(x * t), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.2e17 or 1.45000000000000009e179 < 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 85.7

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

   5. Applied rewrites 85.7%

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

   if -6.2e17 < y < 1.74999999999999997e-112

   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

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 99.9

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. flip-+ N/A

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

      11. +-inverses N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{0}}{\left(y + z\right) - \left(y + z\right)} + t, x, y \cdot 5\right) \]

      12. +-inverses N/A

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

      13. +-inverses N/A

          \[\leadsto \mathsf{fma}\left(\frac{z \cdot z - z \cdot z}{\color{blue}{0}} + t, x, y \cdot 5\right) \]

      14. +-inverses N/A

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

      15. flip-+ N/A

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

      16. lower-+.f64 96.0

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

   4. Applied rewrites 96.0%

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

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x \cdot \left(t + 2 \cdot z\right)} \]
   6. 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 83.2

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

   7. Applied rewrites 83.2%

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

   9. Applied rewrites 83.2%

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

   if 1.74999999999999997e-112 < y < 1.45000000000000009e179

   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

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 99.9

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. flip-+ N/A

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

      11. +-inverses N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{0}}{\left(y + z\right) - \left(y + z\right)} + t, x, y \cdot 5\right) \]

      12. +-inverses N/A

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

      13. +-inverses N/A

          \[\leadsto \mathsf{fma}\left(\frac{z \cdot z - z \cdot z}{\color{blue}{0}} + t, x, y \cdot 5\right) \]

      14. +-inverses N/A

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

      15. flip-+ N/A

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

      16. lower-+.f64 90.4

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

   4. Applied rewrites 90.4%

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

   5. Taylor expanded in z around 0

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

      1. lower-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 69.6

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

   7. Applied rewrites 69.6%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.2 \cdot 10^{+17}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(x, 2, 5\right)\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{-112}:\\ \;\;\;\;x \cdot \left(z + \left(z + t\right)\right)\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{+179}:\\ \;\;\;\;\mathsf{fma}\left(5, y, x \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(x, 2, 5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \mathsf{fma}\left(x, 2, 5\right)\\ \mathbf{if}\;y \leq -6.2 \cdot 10^{+17}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{-112}:\\ \;\;\;\;x \cdot \left(z + \left(z + t\right)\right)\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{+179}:\\ \;\;\;\;\mathsf{fma}\left(x, t, y \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* y (fma x 2.0 5.0))))
   (if (<= y -6.2e+17)
     t_1
     (if (<= y 1.75e-112)
       (* x (+ z (+ z t)))
       (if (<= y 1.45e+179) (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 <= -6.2e+17) {
		tmp = t_1;
	} else if (y <= 1.75e-112) {
		tmp = x * (z + (z + t));
	} else if (y <= 1.45e+179) {
		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 <= -6.2e+17)
		tmp = t_1;
	elseif (y <= 1.75e-112)
		tmp = Float64(x * Float64(z + Float64(z + t)));
	elseif (y <= 1.45e+179)
		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, -6.2e+17], t$95$1, If[LessEqual[y, 1.75e-112], N[(x * N[(z + N[(z + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.45e+179], N[(x * t + N[(y * 5.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.2e17 or 1.45000000000000009e179 < 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 85.7

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

   5. Applied rewrites 85.7%

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

   if -6.2e17 < y < 1.74999999999999997e-112

   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

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 99.9

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. flip-+ N/A

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

      11. +-inverses N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{0}}{\left(y + z\right) - \left(y + z\right)} + t, x, y \cdot 5\right) \]

      12. +-inverses N/A

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

      13. +-inverses N/A

          \[\leadsto \mathsf{fma}\left(\frac{z \cdot z - z \cdot z}{\color{blue}{0}} + t, x, y \cdot 5\right) \]

      14. +-inverses N/A

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

      15. flip-+ N/A

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

      16. lower-+.f64 96.0

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

   4. Applied rewrites 96.0%

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

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x \cdot \left(t + 2 \cdot z\right)} \]
   6. 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 83.2

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

   7. Applied rewrites 83.2%

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

   9. Applied rewrites 83.2%

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

   if 1.74999999999999997e-112 < y < 1.45000000000000009e179

   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

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. distribute-rgt-in N/A

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

      5. associate-+l+ N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. lift-+.f64 N/A

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

      12. flip-+ N/A

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

      13. +-inverses N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{0}}{\left(y + z\right) - \left(y + z\right)}, x, t \cdot x + y \cdot 5\right) \]

      14. +-inverses N/A

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

      15. +-inverses N/A

          \[\leadsto \mathsf{fma}\left(\frac{z \cdot z - z \cdot z}{\color{blue}{0}}, x, t \cdot x + y \cdot 5\right) \]

      16. +-inverses N/A

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

      17. flip-+ N/A

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

      18. lower-+.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-fma.f64 90.4

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

   4. Applied rewrites 90.4%

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

   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. lower-fma.f64 N/A

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

      4. lower-*.f64 69.6

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

   7. Applied rewrites 69.6%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.2 \cdot 10^{+17}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(x, 2, 5\right)\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{-112}:\\ \;\;\;\;x \cdot \left(z + \left(z + t\right)\right)\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{+179}:\\ \;\;\;\;\mathsf{fma}\left(x, t, y \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(x, 2, 5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \mathsf{fma}\left(x, 2, 5\right)\\ \mathbf{if}\;y \leq -6.2 \cdot 10^{+17}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{-112}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(2, z, t\right)\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{+179}:\\ \;\;\;\;\mathsf{fma}\left(x, t, y \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* y (fma x 2.0 5.0))))
   (if (<= y -6.2e+17)
     t_1
     (if (<= y 1.75e-112)
       (* x (fma 2.0 z t))
       (if (<= y 1.45e+179) (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 <= -6.2e+17) {
		tmp = t_1;
	} else if (y <= 1.75e-112) {
		tmp = x * fma(2.0, z, t);
	} else if (y <= 1.45e+179) {
		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 <= -6.2e+17)
		tmp = t_1;
	elseif (y <= 1.75e-112)
		tmp = Float64(x * fma(2.0, z, t));
	elseif (y <= 1.45e+179)
		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, -6.2e+17], t$95$1, If[LessEqual[y, 1.75e-112], N[(x * N[(2.0 * z + t), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.45e+179], N[(x * t + N[(y * 5.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.2e17 or 1.45000000000000009e179 < 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 85.7

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

   5. Applied rewrites 85.7%

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

   if -6.2e17 < y < 1.74999999999999997e-112

   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 83.2

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

   5. Applied rewrites 83.2%

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

   if 1.74999999999999997e-112 < y < 1.45000000000000009e179

   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

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. distribute-rgt-in N/A

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

      5. associate-+l+ N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. lift-+.f64 N/A

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

      12. flip-+ N/A

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

      13. +-inverses N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{0}}{\left(y + z\right) - \left(y + z\right)}, x, t \cdot x + y \cdot 5\right) \]

      14. +-inverses N/A

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

      15. +-inverses N/A

          \[\leadsto \mathsf{fma}\left(\frac{z \cdot z - z \cdot z}{\color{blue}{0}}, x, t \cdot x + y \cdot 5\right) \]

      16. +-inverses N/A

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

      17. flip-+ N/A

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

      18. lower-+.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-fma.f64 90.4

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

   4. Applied rewrites 90.4%

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

   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. lower-fma.f64 N/A

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

      4. lower-*.f64 69.6

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

   7. Applied rewrites 69.6%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.2 \cdot 10^{+17}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(x, 2, 5\right)\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{-112}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(2, z, t\right)\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{+179}:\\ \;\;\;\;\mathsf{fma}\left(x, t, y \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(x, 2, 5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \mathsf{fma}\left(2, y + z, t\right)\\ \mathbf{if}\;x \leq -5700:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(z + \left(z + t\right), x, y \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (fma 2.0 (+ y z) t))))
   (if (<= x -5700.0)
     t_1
     (if (<= x 1.8e-6) (fma (+ z (+ z t)) x (* y 5.0)) 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 <= -5700.0) {
		tmp = t_1;
	} else if (x <= 1.8e-6) {
		tmp = fma((z + (z + t)), x, (y * 5.0));
	} 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 <= -5700.0)
		tmp = t_1;
	elseif (x <= 1.8e-6)
		tmp = fma(Float64(z + Float64(z + t)), x, Float64(y * 5.0));
	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, -5700.0], t$95$1, If[LessEqual[x, 1.8e-6], N[(N[(z + N[(z + t), $MachinePrecision]), $MachinePrecision] * x + N[(y * 5.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -5700 or 1.79999999999999992e-6 < 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 99.6

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

   5. Applied rewrites 99.6%

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

   if -5700 < x < 1.79999999999999992e-6

   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

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 99.9

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. flip-+ N/A

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

      11. +-inverses N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{0}}{\left(y + z\right) - \left(y + z\right)} + t, x, y \cdot 5\right) \]

      12. +-inverses N/A

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

      13. +-inverses N/A

          \[\leadsto \mathsf{fma}\left(\frac{z \cdot z - z \cdot z}{\color{blue}{0}} + t, x, y \cdot 5\right) \]

      14. +-inverses N/A

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

      15. flip-+ N/A

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

      16. lower-+.f64 98.8

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

   4. Applied rewrites 98.8%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 98.8

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

   6. Applied rewrites 98.8%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5700:\\ \;\;\;\;x \cdot \mathsf{fma}\left(2, y + z, t\right)\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(z + \left(z + t\right), x, y \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(2, y + z, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \mathsf{fma}\left(2, y + z, t\right)\\ \mathbf{if}\;x \leq -5700:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(y, 5, x \cdot \left(t + \left(z + z\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (fma 2.0 (+ y z) t))))
   (if (<= x -5700.0)
     t_1
     (if (<= x 1.8e-6) (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 <= -5700.0) {
		tmp = t_1;
	} else if (x <= 1.8e-6) {
		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 <= -5700.0)
		tmp = t_1;
	elseif (x <= 1.8e-6)
		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, -5700.0], t$95$1, If[LessEqual[x, 1.8e-6], 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 < -5700 or 1.79999999999999992e-6 < 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 99.6

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

   5. Applied rewrites 99.6%

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

   if -5700 < x < 1.79999999999999992e-6

   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

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. distribute-rgt-in N/A

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

      5. associate-+l+ N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. lift-+.f64 N/A

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

      12. flip-+ N/A

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

      13. +-inverses N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{0}}{\left(y + z\right) - \left(y + z\right)}, x, t \cdot x + y \cdot 5\right) \]

      14. +-inverses N/A

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

      15. +-inverses N/A

          \[\leadsto \mathsf{fma}\left(\frac{z \cdot z - z \cdot z}{\color{blue}{0}}, x, t \cdot x + y \cdot 5\right) \]

      16. +-inverses N/A

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

      17. flip-+ N/A

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

      18. lower-+.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-fma.f64 98.8

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

   4. Applied rewrites 98.8%

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

      1. lift-fma.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate-+r+ N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. count-2 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-out N/A

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

      12. lower-*.f64 N/A

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

      13. lower-+.f64 98.7

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

      14. lift-*.f64 N/A

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

      15. count-2 N/A

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

      16. lift-+.f64 98.7

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

   6. Applied rewrites 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 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5700:\\ \;\;\;\;x \cdot \mathsf{fma}\left(2, y + z, t\right)\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(y, 5, x \cdot \left(t + \left(z + z\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(2, y + z, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 42.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \left(z + z\right)\\ \mathbf{if}\;x \leq -8.5 \cdot 10^{+36}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -6.6 \cdot 10^{-224}:\\ \;\;\;\;x \cdot t\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-9}:\\ \;\;\;\;y \cdot 5\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (+ z z))))
   (if (<= x -8.5e+36)
     t_1
     (if (<= x -6.6e-224) (* x t) (if (<= x 2.8e-9) (* y 5.0) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * (z + z);
	double tmp;
	if (x <= -8.5e+36) {
		tmp = t_1;
	} else if (x <= -6.6e-224) {
		tmp = x * t;
	} else if (x <= 2.8e-9) {
		tmp = y * 5.0;
	} 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 * (z + z)
    if (x <= (-8.5d+36)) then
        tmp = t_1
    else if (x <= (-6.6d-224)) then
        tmp = x * t
    else if (x <= 2.8d-9) then
        tmp = y * 5.0d0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x * (z + z);
	double tmp;
	if (x <= -8.5e+36) {
		tmp = t_1;
	} else if (x <= -6.6e-224) {
		tmp = x * t;
	} else if (x <= 2.8e-9) {
		tmp = y * 5.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * (z + z)
	tmp = 0
	if x <= -8.5e+36:
		tmp = t_1
	elif x <= -6.6e-224:
		tmp = x * t
	elif x <= 2.8e-9:
		tmp = y * 5.0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * Float64(z + z))
	tmp = 0.0
	if (x <= -8.5e+36)
		tmp = t_1;
	elseif (x <= -6.6e-224)
		tmp = Float64(x * t);
	elseif (x <= 2.8e-9)
		tmp = Float64(y * 5.0);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * (z + z);
	tmp = 0.0;
	if (x <= -8.5e+36)
		tmp = t_1;
	elseif (x <= -6.6e-224)
		tmp = x * t;
	elseif (x <= 2.8e-9)
		tmp = y * 5.0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[(z + z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -8.5e+36], t$95$1, If[LessEqual[x, -6.6e-224], N[(x * t), $MachinePrecision], If[LessEqual[x, 2.8e-9], N[(y * 5.0), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.50000000000000014e36 or 2.79999999999999984e-9 < 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 z around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 49.5

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

   5. Applied rewrites 49.5%

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

   7. Applied rewrites 49.5%

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

   if -8.50000000000000014e36 < x < -6.6000000000000003e-224

   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 41.1

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

   5. Applied rewrites 41.1%

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

   if -6.6000000000000003e-224 < x < 2.79999999999999984e-9

   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 70.5

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

   5. Applied rewrites 70.5%

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

4. Final simplification 55.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{+36}:\\ \;\;\;\;x \cdot \left(z + z\right)\\ \mathbf{elif}\;x \leq -6.6 \cdot 10^{-224}:\\ \;\;\;\;x \cdot t\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-9}:\\ \;\;\;\;y \cdot 5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(z + z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 88.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \mathsf{fma}\left(2, y + z, t\right)\\ \mathbf{if}\;x \leq -3.4 \cdot 10^{-28}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 4.3 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(5, y, x \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (fma 2.0 (+ y z) t))))
   (if (<= x -3.4e-28) t_1 (if (<= x 4.3e-10) (fma 5.0 y (* 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 <= -3.4e-28) {
		tmp = t_1;
	} else if (x <= 4.3e-10) {
		tmp = fma(5.0, y, (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 <= -3.4e-28)
		tmp = t_1;
	elseif (x <= 4.3e-10)
		tmp = fma(5.0, y, 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, -3.4e-28], t$95$1, If[LessEqual[x, 4.3e-10], N[(5.0 * y + N[(x * t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.4000000000000001e-28 or 4.30000000000000014e-10 < 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 97.2

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

   5. Applied rewrites 97.2%

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

   if -3.4000000000000001e-28 < x < 4.30000000000000014e-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. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 99.9

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. flip-+ N/A

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

      11. +-inverses N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{0}}{\left(y + z\right) - \left(y + z\right)} + t, x, y \cdot 5\right) \]

      12. +-inverses N/A

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

      13. +-inverses N/A

          \[\leadsto \mathsf{fma}\left(\frac{z \cdot z - z \cdot z}{\color{blue}{0}} + t, x, y \cdot 5\right) \]

      14. +-inverses N/A

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

      15. flip-+ N/A

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

      16. lower-+.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in z around 0

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

      1. lower-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 79.7

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

   7. Applied rewrites 79.7%

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

Alternative 12: 78.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \mathsf{fma}\left(x, 2, 5\right)\\ \mathbf{if}\;y \leq -6.2 \cdot 10^{+17}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 4.7 \cdot 10^{+41}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(2, z, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* y (fma x 2.0 5.0))))
   (if (<= y -6.2e+17) t_1 (if (<= y 4.7e+41) (* 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 <= -6.2e+17) {
		tmp = t_1;
	} else if (y <= 4.7e+41) {
		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 <= -6.2e+17)
		tmp = t_1;
	elseif (y <= 4.7e+41)
		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, -6.2e+17], t$95$1, If[LessEqual[y, 4.7e+41], N[(x * N[(2.0 * z + t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -6.2e17 or 4.70000000000000001e41 < 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 79.1

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

   5. Applied rewrites 79.1%

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

   if -6.2e17 < y < 4.70000000000000001e41

   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 79.4

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

   5. Applied rewrites 79.4%

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

Alternative 13: 64.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \mathsf{fma}\left(2, z, t\right)\\ \mathbf{if}\;x \leq -6.6 \cdot 10^{-224}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 5.4 \cdot 10^{-11}:\\ \;\;\;\;y \cdot 5\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (fma 2.0 z t))))
   (if (<= x -6.6e-224) t_1 (if (<= x 5.4e-11) (* y 5.0) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * fma(2.0, z, t);
	double tmp;
	if (x <= -6.6e-224) {
		tmp = t_1;
	} else if (x <= 5.4e-11) {
		tmp = y * 5.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(x * fma(2.0, z, t))
	tmp = 0.0
	if (x <= -6.6e-224)
		tmp = t_1;
	elseif (x <= 5.4e-11)
		tmp = Float64(y * 5.0);
	else
		tmp = t_1;
	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, -6.6e-224], t$95$1, If[LessEqual[x, 5.4e-11], N[(y * 5.0), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -6.6000000000000003e-224 or 5.40000000000000009e-11 < 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 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 71.0

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

   5. Applied rewrites 71.0%

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

   if -6.6000000000000003e-224 < x < 5.40000000000000009e-11

   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 70.5

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

   5. Applied rewrites 70.5%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.6 \cdot 10^{-224}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(2, z, t\right)\\ \mathbf{elif}\;x \leq 5.4 \cdot 10^{-11}:\\ \;\;\;\;y \cdot 5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(2, z, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 58.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \mathsf{fma}\left(2, y, t\right)\\ \mathbf{if}\;x \leq -6.6 \cdot 10^{-224}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-8}:\\ \;\;\;\;y \cdot 5\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (fma 2.0 y t))))
   (if (<= x -6.6e-224) t_1 (if (<= x 1.1e-8) (* y 5.0) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * fma(2.0, y, t);
	double tmp;
	if (x <= -6.6e-224) {
		tmp = t_1;
	} else if (x <= 1.1e-8) {
		tmp = y * 5.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(x * fma(2.0, y, t))
	tmp = 0.0
	if (x <= -6.6e-224)
		tmp = t_1;
	elseif (x <= 1.1e-8)
		tmp = Float64(y * 5.0);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[(2.0 * y + t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -6.6e-224], t$95$1, If[LessEqual[x, 1.1e-8], N[(y * 5.0), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -6.6000000000000003e-224 or 1.0999999999999999e-8 < 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 89.8

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

   5. Applied rewrites 89.8%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 56.8%

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

   if -6.6000000000000003e-224 < x < 1.0999999999999999e-8

   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 70.5

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

   5. Applied rewrites 70.5%

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

4. Final simplification 62.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.6 \cdot 10^{-224}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(2, y, t\right)\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-8}:\\ \;\;\;\;y \cdot 5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(2, y, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 43.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.6 \cdot 10^{-224}:\\ \;\;\;\;x \cdot t\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-8}:\\ \;\;\;\;y \cdot 5\\ \mathbf{else}:\\ \;\;\;\;x \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -6.6e-224) (* x t) (if (<= x 1.4e-8) (* y 5.0) (* x t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -6.6e-224) {
		tmp = x * t;
	} else if (x <= 1.4e-8) {
		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 <= (-6.6d-224)) then
        tmp = x * t
    else if (x <= 1.4d-8) 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 <= -6.6e-224) {
		tmp = x * t;
	} else if (x <= 1.4e-8) {
		tmp = y * 5.0;
	} else {
		tmp = x * t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -6.6e-224:
		tmp = x * t
	elif x <= 1.4e-8:
		tmp = y * 5.0
	else:
		tmp = x * t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -6.6e-224)
		tmp = Float64(x * t);
	elseif (x <= 1.4e-8)
		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 <= -6.6e-224)
		tmp = x * t;
	elseif (x <= 1.4e-8)
		tmp = y * 5.0;
	else
		tmp = x * t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -6.6e-224], N[(x * t), $MachinePrecision], If[LessEqual[x, 1.4e-8], N[(y * 5.0), $MachinePrecision], N[(x * t), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -6.6000000000000003e-224 or 1.4e-8 < 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 36.9

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

   5. Applied rewrites 36.9%

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

   if -6.6000000000000003e-224 < x < 1.4e-8

   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 70.5

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

   5. Applied rewrites 70.5%

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

4. Final simplification 49.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.6 \cdot 10^{-224}:\\ \;\;\;\;x \cdot t\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-8}:\\ \;\;\;\;y \cdot 5\\ \mathbf{else}:\\ \;\;\;\;x \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 29.9% 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 33.6

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

5. Applied rewrites 33.6%

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

6. Final simplification 33.6%

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

Reproduce

herbie shell --seed 2024232 
(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)))