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

Percentage Accurate: 99.8% → 99.8%

Time: 6.7s

Alternatives: 10

Speedup: 1.0×

• 27.5% 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

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

FPCore

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

C

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

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 = (5.0d0 * y) + ((t + (((z + y) + z) + y)) * x)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

Alternative 2: 63.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(2, y, t\right) \cdot x\\ \mathbf{if}\;x \leq -6 \cdot 10^{+22}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -1.15 \cdot 10^{-85}:\\ \;\;\;\;\left(z \cdot x\right) \cdot 2\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-51}:\\ \;\;\;\;5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (fma 2.0 y t) x)))
   (if (<= x -6e+22)
     t_1
     (if (<= x -1.15e-85) (* (* z x) 2.0) (if (<= x 5.8e-51) (* 5.0 y) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(2.0, y, t) * x;
	double tmp;
	if (x <= -6e+22) {
		tmp = t_1;
	} else if (x <= -1.15e-85) {
		tmp = (z * x) * 2.0;
	} else if (x <= 5.8e-51) {
		tmp = 5.0 * y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(fma(2.0, y, t) * x)
	tmp = 0.0
	if (x <= -6e+22)
		tmp = t_1;
	elseif (x <= -1.15e-85)
		tmp = Float64(Float64(z * x) * 2.0);
	elseif (x <= 5.8e-51)
		tmp = Float64(5.0 * y);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(2.0 * y + t), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -6e+22], t$95$1, If[LessEqual[x, -1.15e-85], N[(N[(z * x), $MachinePrecision] * 2.0), $MachinePrecision], If[LessEqual[x, 5.8e-51], N[(5.0 * y), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if x < -6e22 or 5.79999999999999945e-51 < 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 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 68.5

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

   5. Applied rewrites 68.5%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 67.0%

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

   if -6e22 < x < -1.15e-85

   1. Initial program 99.8%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 53.6

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

   5. Applied rewrites 53.6%

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

   if -1.15e-85 < x < 5.79999999999999945e-51

   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. *-commutative N/A

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

      2. lower-*.f64 64.1

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

   5. Applied rewrites 64.1%

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

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{+22}:\\ \;\;\;\;\mathsf{fma}\left(2, y, t\right) \cdot x\\ \mathbf{elif}\;x \leq -1.15 \cdot 10^{-85}:\\ \;\;\;\;\left(z \cdot x\right) \cdot 2\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-51}:\\ \;\;\;\;5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, y, t\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z + y, 2, t\right) \cdot x\\ \mathbf{if}\;x \leq -2.5:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 2.45:\\ \;\;\;\;\mathsf{fma}\left(y, 5, \mathsf{fma}\left(2, z, t\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (fma (+ z y) 2.0 t) x)))
   (if (<= x -2.5) t_1 (if (<= x 2.45) (fma y 5.0 (* (fma 2.0 z t) x)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma((z + y), 2.0, t) * x;
	double tmp;
	if (x <= -2.5) {
		tmp = t_1;
	} else if (x <= 2.45) {
		tmp = fma(y, 5.0, (fma(2.0, z, t) * x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(fma(Float64(z + y), 2.0, t) * x)
	tmp = 0.0
	if (x <= -2.5)
		tmp = t_1;
	elseif (x <= 2.45)
		tmp = fma(y, 5.0, Float64(fma(2.0, z, t) * x));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(z + y), $MachinePrecision] * 2.0 + t), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -2.5], t$95$1, If[LessEqual[x, 2.45], N[(y * 5.0 + N[(N[(2.0 * z + t), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.5 or 2.4500000000000002 < 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-out N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 99.3

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

   5. Applied rewrites 99.3%

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

   if -2.5 < x < 2.4500000000000002

   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. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 100.0

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 100.0

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

      8. lift-+.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. associate-+l+ N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. flip-+ N/A

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

      15. +-inverses N/A

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

      16. +-inverses N/A

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

      17. +-inverses N/A

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

      18. +-inverses N/A

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

      19. flip-+ N/A

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

      20. count-2 N/A

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

      21. lower-fma.f64 98.2

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

   4. Applied rewrites 98.2%

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

Alternative 4: 89.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z + y, 2, t\right) \cdot x\\ \mathbf{if}\;x \leq -0.35:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{-52}:\\ \;\;\;\;\mathsf{fma}\left(y, 5, \left(2 \cdot z\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (fma (+ z y) 2.0 t) x)))
   (if (<= x -0.35) t_1 (if (<= x 5.5e-52) (fma y 5.0 (* (* 2.0 z) x)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma((z + y), 2.0, t) * x;
	double tmp;
	if (x <= -0.35) {
		tmp = t_1;
	} else if (x <= 5.5e-52) {
		tmp = fma(y, 5.0, ((2.0 * z) * x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(fma(Float64(z + y), 2.0, t) * x)
	tmp = 0.0
	if (x <= -0.35)
		tmp = t_1;
	elseif (x <= 5.5e-52)
		tmp = fma(y, 5.0, Float64(Float64(2.0 * z) * x));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(z + y), $MachinePrecision] * 2.0 + t), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -0.35], t$95$1, If[LessEqual[x, 5.5e-52], N[(y * 5.0 + N[(N[(2.0 * z), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.34999999999999998 or 5.5e-52 < 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-out N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 98.1

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

   5. Applied rewrites 98.1%

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

   if -0.34999999999999998 < x < 5.5e-52

   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. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 100.0

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 100.0

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

      8. lift-+.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. associate-+l+ N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. flip-+ N/A

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

      15. +-inverses N/A

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

      16. +-inverses N/A

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

      17. +-inverses N/A

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

      18. +-inverses N/A

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

      19. flip-+ N/A

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

      20. count-2 N/A

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

      21. lower-fma.f64 98.6

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

   4. Applied rewrites 98.6%

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

   5. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 84.9

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

   7. Applied rewrites 84.9%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.35:\\ \;\;\;\;\mathsf{fma}\left(z + y, 2, t\right) \cdot x\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{-52}:\\ \;\;\;\;\mathsf{fma}\left(y, 5, \left(2 \cdot z\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z + y, 2, t\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z + y, 2, t\right) \cdot x\\ \mathbf{if}\;x \leq -1.15 \cdot 10^{-85}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.86 \cdot 10^{-162}:\\ \;\;\;\;\mathsf{fma}\left(y, 5, t \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (fma (+ z y) 2.0 t) x)))
   (if (<= x -1.15e-85) t_1 (if (<= x 1.86e-162) (fma y 5.0 (* t x)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma((z + y), 2.0, t) * x;
	double tmp;
	if (x <= -1.15e-85) {
		tmp = t_1;
	} else if (x <= 1.86e-162) {
		tmp = fma(y, 5.0, (t * x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(fma(Float64(z + y), 2.0, t) * x)
	tmp = 0.0
	if (x <= -1.15e-85)
		tmp = t_1;
	elseif (x <= 1.86e-162)
		tmp = fma(y, 5.0, Float64(t * x));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(z + y), $MachinePrecision] * 2.0 + t), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -1.15e-85], t$95$1, If[LessEqual[x, 1.86e-162], N[(y * 5.0 + N[(t * x), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.15e-85 or 1.85999999999999997e-162 < 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-out N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 91.2

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

   5. Applied rewrites 91.2%

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

   if -1.15e-85 < x < 1.85999999999999997e-162

   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. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 100.0

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 100.0

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

      8. lift-+.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. associate-+l+ N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. flip-+ N/A

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

      15. +-inverses N/A

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

      16. +-inverses N/A

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

      17. +-inverses N/A

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

      18. +-inverses N/A

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

      19. flip-+ N/A

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

      20. count-2 N/A

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

      21. lower-fma.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in t around inf

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

      1. lower-*.f64 84.4

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

   7. Applied rewrites 84.4%

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

Alternative 6: 79.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(2, x, 5\right) \cdot y\\ \mathbf{if}\;y \leq -5 \cdot 10^{+14}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 10^{+40}:\\ \;\;\;\;\mathsf{fma}\left(z, 2, t\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (fma 2.0 x 5.0) y)))
   (if (<= y -5e+14) t_1 (if (<= y 1e+40) (* (fma z 2.0 t) x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(2.0, x, 5.0) * y;
	double tmp;
	if (y <= -5e+14) {
		tmp = t_1;
	} else if (y <= 1e+40) {
		tmp = fma(z, 2.0, t) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(fma(2.0, x, 5.0) * y)
	tmp = 0.0
	if (y <= -5e+14)
		tmp = t_1;
	elseif (y <= 1e+40)
		tmp = Float64(fma(z, 2.0, t) * x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(2.0 * x + 5.0), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[y, -5e+14], t$95$1, If[LessEqual[y, 1e+40], N[(N[(z * 2.0 + t), $MachinePrecision] * x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -5e14 or 1.00000000000000003e40 < 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. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. neg-sub0 N/A

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

      10. associate--r- N/A

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

      11. neg-sub0 N/A

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

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

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

      13. metadata-eval N/A

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

      14. lower-fma.f64 80.3

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

   5. Applied rewrites 80.3%

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

   if -5e14 < y < 1.00000000000000003e40

   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 y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 81.8

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

   5. Applied rewrites 81.8%

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

Alternative 7: 64.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(2, y, t\right) \cdot x\\ \mathbf{if}\;x \leq -370000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-51}:\\ \;\;\;\;\mathsf{fma}\left(2, x, 5\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (fma 2.0 y t) x)))
   (if (<= x -370000.0) t_1 (if (<= x 5.8e-51) (* (fma 2.0 x 5.0) y) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(2.0, y, t) * x;
	double tmp;
	if (x <= -370000.0) {
		tmp = t_1;
	} else if (x <= 5.8e-51) {
		tmp = fma(2.0, x, 5.0) * y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(fma(2.0, y, t) * x)
	tmp = 0.0
	if (x <= -370000.0)
		tmp = t_1;
	elseif (x <= 5.8e-51)
		tmp = Float64(fma(2.0, x, 5.0) * y);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(2.0 * y + t), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -370000.0], t$95$1, If[LessEqual[x, 5.8e-51], N[(N[(2.0 * x + 5.0), $MachinePrecision] * y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.7e5 or 5.79999999999999945e-51 < 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 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 67.6

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

   5. Applied rewrites 67.6%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 66.1%

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

   if -3.7e5 < x < 5.79999999999999945e-51

   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. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. neg-sub0 N/A

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

      10. associate--r- N/A

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

      11. neg-sub0 N/A

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

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

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

      13. metadata-eval N/A

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

      14. lower-fma.f64 60.7

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

   5. Applied rewrites 60.7%

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

Alternative 8: 45.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(z \cdot x\right) \cdot 2\\ \mathbf{if}\;z \leq -3 \cdot 10^{-30}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 4.4 \cdot 10^{+54}:\\ \;\;\;\;t \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (* z x) 2.0)))
   (if (<= z -3e-30) t_1 (if (<= z 4.4e+54) (* t x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * x) * 2.0;
	double tmp;
	if (z <= -3e-30) {
		tmp = t_1;
	} else if (z <= 4.4e+54) {
		tmp = t * x;
	} 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 = (z * x) * 2.0d0
    if (z <= (-3d-30)) then
        tmp = t_1
    else if (z <= 4.4d+54) then
        tmp = t * x
    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 = (z * x) * 2.0;
	double tmp;
	if (z <= -3e-30) {
		tmp = t_1;
	} else if (z <= 4.4e+54) {
		tmp = t * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (z * x) * 2.0
	tmp = 0
	if z <= -3e-30:
		tmp = t_1
	elif z <= 4.4e+54:
		tmp = t * x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z * x) * 2.0)
	tmp = 0.0
	if (z <= -3e-30)
		tmp = t_1;
	elseif (z <= 4.4e+54)
		tmp = Float64(t * x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (z * x) * 2.0;
	tmp = 0.0;
	if (z <= -3e-30)
		tmp = t_1;
	elseif (z <= 4.4e+54)
		tmp = t * x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z * x), $MachinePrecision] * 2.0), $MachinePrecision]}, If[LessEqual[z, -3e-30], t$95$1, If[LessEqual[z, 4.4e+54], N[(t * x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.9999999999999999e-30 or 4.3999999999999998e54 < z

   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 inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 61.2

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

   5. Applied rewrites 61.2%

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

   if -2.9999999999999999e-30 < z < 4.3999999999999998e54

   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. lower-*.f64 41.0

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

   5. Applied rewrites 41.0%

      \[\leadsto \color{blue}{t \cdot x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 47.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.35:\\ \;\;\;\;t \cdot x\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-51}:\\ \;\;\;\;5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;t \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -0.35) (* t x) (if (<= x 5.8e-51) (* 5.0 y) (* t x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -0.35) {
		tmp = t * x;
	} else if (x <= 5.8e-51) {
		tmp = 5.0 * y;
	} else {
		tmp = t * x;
	}
	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 <= (-0.35d0)) then
        tmp = t * x
    else if (x <= 5.8d-51) then
        tmp = 5.0d0 * y
    else
        tmp = t * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -0.35) {
		tmp = t * x;
	} else if (x <= 5.8e-51) {
		tmp = 5.0 * y;
	} else {
		tmp = t * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -0.35:
		tmp = t * x
	elif x <= 5.8e-51:
		tmp = 5.0 * y
	else:
		tmp = t * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -0.35)
		tmp = Float64(t * x);
	elseif (x <= 5.8e-51)
		tmp = Float64(5.0 * y);
	else
		tmp = Float64(t * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -0.35)
		tmp = t * x;
	elseif (x <= 5.8e-51)
		tmp = 5.0 * y;
	else
		tmp = t * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -0.35], N[(t * x), $MachinePrecision], If[LessEqual[x, 5.8e-51], N[(5.0 * y), $MachinePrecision], N[(t * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.34999999999999998 or 5.79999999999999945e-51 < 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 t around inf

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

      1. lower-*.f64 36.5

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

   5. Applied rewrites 36.5%

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

   if -0.34999999999999998 < x < 5.79999999999999945e-51

   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. *-commutative N/A

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

      2. lower-*.f64 59.1

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

   5. Applied rewrites 59.1%

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

4. Final simplification 46.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.35:\\ \;\;\;\;t \cdot x\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-51}:\\ \;\;\;\;5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;t \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 30.3% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ t \cdot x \end{array} \]

FPCore

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

C

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

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 = t * x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := N[(t * x), $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 t around inf

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

   1. lower-*.f64 27.6

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

5. Applied rewrites 27.6%

   \[\leadsto \color{blue}{t \cdot x} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024277 
(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)))