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

Percentage Accurate: 99.9% → 100.0%

Time: 6.2s

Alternatives: 12

Speedup: 1.0×

• Unsound rule application detected in e-graph. Results may not be sound.

• 27.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_, t_] := N[(y * 5.0 + N[(x * N[(N[(y + z), $MachinePrecision] * 2.0 + t), $MachinePrecision]), $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. Step-by-step derivation

   1. +-commutative 99.9%

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

   2. fma-def 100.0%

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

   3. distribute-rgt-in 95.7%

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

   4. associate-+l+ 95.7%

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

   5. +-commutative 95.7%

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

   6. count-2 95.7%

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

   7. distribute-rgt-in 100.0%

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

   8. *-commutative 100.0%

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

   9. fma-def 100.0%

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

3. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 56.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 \cdot \left(x \cdot z\right)\\ t_2 := y \cdot \left(5 + x \cdot 2\right)\\ \mathbf{if}\;y \leq -7.5 \cdot 10^{+111}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -2.9 \cdot 10^{-88}:\\ \;\;\;\;x \cdot t\\ \mathbf{elif}\;y \leq -2.6 \cdot 10^{-205}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -3.9 \cdot 10^{-277}:\\ \;\;\;\;x \cdot t\\ \mathbf{elif}\;y \leq 4 \cdot 10^{-13}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* 2.0 (* x z))) (t_2 (* y (+ 5.0 (* x 2.0)))))
   (if (<= y -7.5e+111)
     t_2
     (if (<= y -2.9e-88)
       (* x t)
       (if (<= y -2.6e-205)
         t_1
         (if (<= y -3.9e-277) (* x t) (if (<= y 4e-13) t_1 t_2)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = 2.0 * (x * z);
	double t_2 = y * (5.0 + (x * 2.0));
	double tmp;
	if (y <= -7.5e+111) {
		tmp = t_2;
	} else if (y <= -2.9e-88) {
		tmp = x * t;
	} else if (y <= -2.6e-205) {
		tmp = t_1;
	} else if (y <= -3.9e-277) {
		tmp = x * t;
	} else if (y <= 4e-13) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_1 = 2.0d0 * (x * z)
    t_2 = y * (5.0d0 + (x * 2.0d0))
    if (y <= (-7.5d+111)) then
        tmp = t_2
    else if (y <= (-2.9d-88)) then
        tmp = x * t
    else if (y <= (-2.6d-205)) then
        tmp = t_1
    else if (y <= (-3.9d-277)) then
        tmp = x * t
    else if (y <= 4d-13) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = 2.0 * (x * z);
	double t_2 = y * (5.0 + (x * 2.0));
	double tmp;
	if (y <= -7.5e+111) {
		tmp = t_2;
	} else if (y <= -2.9e-88) {
		tmp = x * t;
	} else if (y <= -2.6e-205) {
		tmp = t_1;
	} else if (y <= -3.9e-277) {
		tmp = x * t;
	} else if (y <= 4e-13) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = 2.0 * (x * z)
	t_2 = y * (5.0 + (x * 2.0))
	tmp = 0
	if y <= -7.5e+111:
		tmp = t_2
	elif y <= -2.9e-88:
		tmp = x * t
	elif y <= -2.6e-205:
		tmp = t_1
	elif y <= -3.9e-277:
		tmp = x * t
	elif y <= 4e-13:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(2.0 * Float64(x * z))
	t_2 = Float64(y * Float64(5.0 + Float64(x * 2.0)))
	tmp = 0.0
	if (y <= -7.5e+111)
		tmp = t_2;
	elseif (y <= -2.9e-88)
		tmp = Float64(x * t);
	elseif (y <= -2.6e-205)
		tmp = t_1;
	elseif (y <= -3.9e-277)
		tmp = Float64(x * t);
	elseif (y <= 4e-13)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = 2.0 * (x * z);
	t_2 = y * (5.0 + (x * 2.0));
	tmp = 0.0;
	if (y <= -7.5e+111)
		tmp = t_2;
	elseif (y <= -2.9e-88)
		tmp = x * t;
	elseif (y <= -2.6e-205)
		tmp = t_1;
	elseif (y <= -3.9e-277)
		tmp = x * t;
	elseif (y <= 4e-13)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(2.0 * N[(x * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(y * N[(5.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -7.5e+111], t$95$2, If[LessEqual[y, -2.9e-88], N[(x * t), $MachinePrecision], If[LessEqual[y, -2.6e-205], t$95$1, If[LessEqual[y, -3.9e-277], N[(x * t), $MachinePrecision], If[LessEqual[y, 4e-13], t$95$1, t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.49999999999999948e111 or 4.0000000000000001e-13 < 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. Step-by-step derivation

      1. fma-def 99.9%

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

      2. associate-+l+ 99.9%

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

      3. +-commutative 99.9%

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

      4. count-2 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around inf 80.4%

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

   if -7.49999999999999948e111 < y < -2.9000000000000001e-88 or -2.5999999999999998e-205 < y < -3.89999999999999987e-277

   1. Initial program 99.9%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]

   2. Taylor expanded in t around inf 55.8%

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

   if -2.9000000000000001e-88 < y < -2.5999999999999998e-205 or -3.89999999999999987e-277 < y < 4.0000000000000001e-13

   1. Initial program 100.0%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]

   2. Taylor expanded in z around inf 59.6%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.5 \cdot 10^{+111}:\\ \;\;\;\;y \cdot \left(5 + x \cdot 2\right)\\ \mathbf{elif}\;y \leq -2.9 \cdot 10^{-88}:\\ \;\;\;\;x \cdot t\\ \mathbf{elif}\;y \leq -2.6 \cdot 10^{-205}:\\ \;\;\;\;2 \cdot \left(x \cdot z\right)\\ \mathbf{elif}\;y \leq -3.9 \cdot 10^{-277}:\\ \;\;\;\;x \cdot t\\ \mathbf{elif}\;y \leq 4 \cdot 10^{-13}:\\ \;\;\;\;2 \cdot \left(x \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(5 + x \cdot 2\right)\\ \end{array} \]

Alternative 3: 89.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot 5 + 2 \cdot \left(x \cdot z\right)\\ t_2 := x \cdot \left(t + \left(y + z\right) \cdot 2\right)\\ \mathbf{if}\;x \leq -2.5 \cdot 10^{-34}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{-149}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{-67}:\\ \;\;\;\;y \cdot 5 + x \cdot t\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-40}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (* y 5.0) (* 2.0 (* x z)))) (t_2 (* x (+ t (* (+ y z) 2.0)))))
   (if (<= x -2.5e-34)
     t_2
     (if (<= x 8.5e-149)
       t_1
       (if (<= x 3.2e-67)
         (+ (* y 5.0) (* x t))
         (if (<= x 2.8e-40) t_1 t_2))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (y * 5.0) + (2.0 * (x * z));
	double t_2 = x * (t + ((y + z) * 2.0));
	double tmp;
	if (x <= -2.5e-34) {
		tmp = t_2;
	} else if (x <= 8.5e-149) {
		tmp = t_1;
	} else if (x <= 3.2e-67) {
		tmp = (y * 5.0) + (x * t);
	} else if (x <= 2.8e-40) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_1 = (y * 5.0d0) + (2.0d0 * (x * z))
    t_2 = x * (t + ((y + z) * 2.0d0))
    if (x <= (-2.5d-34)) then
        tmp = t_2
    else if (x <= 8.5d-149) then
        tmp = t_1
    else if (x <= 3.2d-67) then
        tmp = (y * 5.0d0) + (x * t)
    else if (x <= 2.8d-40) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (y * 5.0) + (2.0 * (x * z));
	double t_2 = x * (t + ((y + z) * 2.0));
	double tmp;
	if (x <= -2.5e-34) {
		tmp = t_2;
	} else if (x <= 8.5e-149) {
		tmp = t_1;
	} else if (x <= 3.2e-67) {
		tmp = (y * 5.0) + (x * t);
	} else if (x <= 2.8e-40) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (y * 5.0) + (2.0 * (x * z))
	t_2 = x * (t + ((y + z) * 2.0))
	tmp = 0
	if x <= -2.5e-34:
		tmp = t_2
	elif x <= 8.5e-149:
		tmp = t_1
	elif x <= 3.2e-67:
		tmp = (y * 5.0) + (x * t)
	elif x <= 2.8e-40:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(y * 5.0) + Float64(2.0 * Float64(x * z)))
	t_2 = Float64(x * Float64(t + Float64(Float64(y + z) * 2.0)))
	tmp = 0.0
	if (x <= -2.5e-34)
		tmp = t_2;
	elseif (x <= 8.5e-149)
		tmp = t_1;
	elseif (x <= 3.2e-67)
		tmp = Float64(Float64(y * 5.0) + Float64(x * t));
	elseif (x <= 2.8e-40)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (y * 5.0) + (2.0 * (x * z));
	t_2 = x * (t + ((y + z) * 2.0));
	tmp = 0.0;
	if (x <= -2.5e-34)
		tmp = t_2;
	elseif (x <= 8.5e-149)
		tmp = t_1;
	elseif (x <= 3.2e-67)
		tmp = (y * 5.0) + (x * t);
	elseif (x <= 2.8e-40)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y * 5.0), $MachinePrecision] + N[(2.0 * N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x * N[(t + N[(N[(y + z), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.5e-34], t$95$2, If[LessEqual[x, 8.5e-149], t$95$1, If[LessEqual[x, 3.2e-67], N[(N[(y * 5.0), $MachinePrecision] + N[(x * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.8e-40], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.5000000000000001e-34 or 2.8e-40 < 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. Step-by-step derivation

      1. fma-def 100.0%

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

      2. associate-+l+ 100.0%

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

      3. +-commutative 100.0%

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

      4. count-2 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 97.9%

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

   if -2.5000000000000001e-34 < x < 8.5000000000000006e-149 or 3.20000000000000021e-67 < x < 2.8e-40

   1. Initial program 99.9%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]

   2. Taylor expanded in y around 0 99.9%

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

   4. Simplified 99.9%

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

   5. Taylor expanded in t around 0 92.7%

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

   if 8.5000000000000006e-149 < x < 3.20000000000000021e-67

   1. Initial program 100.0%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]

   2. Taylor expanded in y around 0 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in z around 0 92.5%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{-34}:\\ \;\;\;\;x \cdot \left(t + \left(y + z\right) \cdot 2\right)\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{-149}:\\ \;\;\;\;y \cdot 5 + 2 \cdot \left(x \cdot z\right)\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{-67}:\\ \;\;\;\;y \cdot 5 + x \cdot t\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-40}:\\ \;\;\;\;y \cdot 5 + 2 \cdot \left(x \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(t + \left(y + z\right) \cdot 2\right)\\ \end{array} \]

Alternative 4: 46.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 \cdot \left(x \cdot z\right)\\ \mathbf{if}\;x \leq -1 \cdot 10^{+224}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -3.8 \cdot 10^{+183}:\\ \;\;\;\;x \cdot t\\ \mathbf{elif}\;x \leq -2.3 \cdot 10^{+137}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.7 \cdot 10^{+66}:\\ \;\;\;\;x \cdot t\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-68}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-41}:\\ \;\;\;\;y \cdot 5\\ \mathbf{else}:\\ \;\;\;\;x \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* 2.0 (* x z))))
   (if (<= x -1e+224)
     t_1
     (if (<= x -3.8e+183)
       (* x t)
       (if (<= x -2.3e+137)
         t_1
         (if (<= x -1.7e+66)
           (* x t)
           (if (<= x -2e-68) t_1 (if (<= x 5.8e-41) (* y 5.0) (* x t)))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = 2.0 * (x * z);
	double tmp;
	if (x <= -1e+224) {
		tmp = t_1;
	} else if (x <= -3.8e+183) {
		tmp = x * t;
	} else if (x <= -2.3e+137) {
		tmp = t_1;
	} else if (x <= -1.7e+66) {
		tmp = x * t;
	} else if (x <= -2e-68) {
		tmp = t_1;
	} else if (x <= 5.8e-41) {
		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) :: t_1
    real(8) :: tmp
    t_1 = 2.0d0 * (x * z)
    if (x <= (-1d+224)) then
        tmp = t_1
    else if (x <= (-3.8d+183)) then
        tmp = x * t
    else if (x <= (-2.3d+137)) then
        tmp = t_1
    else if (x <= (-1.7d+66)) then
        tmp = x * t
    else if (x <= (-2d-68)) then
        tmp = t_1
    else if (x <= 5.8d-41) 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 t_1 = 2.0 * (x * z);
	double tmp;
	if (x <= -1e+224) {
		tmp = t_1;
	} else if (x <= -3.8e+183) {
		tmp = x * t;
	} else if (x <= -2.3e+137) {
		tmp = t_1;
	} else if (x <= -1.7e+66) {
		tmp = x * t;
	} else if (x <= -2e-68) {
		tmp = t_1;
	} else if (x <= 5.8e-41) {
		tmp = y * 5.0;
	} else {
		tmp = x * t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = 2.0 * (x * z)
	tmp = 0
	if x <= -1e+224:
		tmp = t_1
	elif x <= -3.8e+183:
		tmp = x * t
	elif x <= -2.3e+137:
		tmp = t_1
	elif x <= -1.7e+66:
		tmp = x * t
	elif x <= -2e-68:
		tmp = t_1
	elif x <= 5.8e-41:
		tmp = y * 5.0
	else:
		tmp = x * t
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(2.0 * Float64(x * z))
	tmp = 0.0
	if (x <= -1e+224)
		tmp = t_1;
	elseif (x <= -3.8e+183)
		tmp = Float64(x * t);
	elseif (x <= -2.3e+137)
		tmp = t_1;
	elseif (x <= -1.7e+66)
		tmp = Float64(x * t);
	elseif (x <= -2e-68)
		tmp = t_1;
	elseif (x <= 5.8e-41)
		tmp = Float64(y * 5.0);
	else
		tmp = Float64(x * t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = 2.0 * (x * z);
	tmp = 0.0;
	if (x <= -1e+224)
		tmp = t_1;
	elseif (x <= -3.8e+183)
		tmp = x * t;
	elseif (x <= -2.3e+137)
		tmp = t_1;
	elseif (x <= -1.7e+66)
		tmp = x * t;
	elseif (x <= -2e-68)
		tmp = t_1;
	elseif (x <= 5.8e-41)
		tmp = y * 5.0;
	else
		tmp = x * t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(2.0 * N[(x * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1e+224], t$95$1, If[LessEqual[x, -3.8e+183], N[(x * t), $MachinePrecision], If[LessEqual[x, -2.3e+137], t$95$1, If[LessEqual[x, -1.7e+66], N[(x * t), $MachinePrecision], If[LessEqual[x, -2e-68], t$95$1, If[LessEqual[x, 5.8e-41], N[(y * 5.0), $MachinePrecision], N[(x * t), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -9.9999999999999997e223 or -3.80000000000000001e183 < x < -2.29999999999999999e137 or -1.70000000000000015e66 < x < -2.00000000000000013e-68

   1. Initial program 100.0%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]

   2. Taylor expanded in z around inf 60.5%

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

   if -9.9999999999999997e223 < x < -3.80000000000000001e183 or -2.29999999999999999e137 < x < -1.70000000000000015e66 or 5.79999999999999955e-41 < 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. Taylor expanded in t around inf 51.1%

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

   if -2.00000000000000013e-68 < x < 5.79999999999999955e-41

   1. Initial program 99.9%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]

   2. Taylor expanded in x around 0 68.2%

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

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+224}:\\ \;\;\;\;2 \cdot \left(x \cdot z\right)\\ \mathbf{elif}\;x \leq -3.8 \cdot 10^{+183}:\\ \;\;\;\;x \cdot t\\ \mathbf{elif}\;x \leq -2.3 \cdot 10^{+137}:\\ \;\;\;\;2 \cdot \left(x \cdot z\right)\\ \mathbf{elif}\;x \leq -1.7 \cdot 10^{+66}:\\ \;\;\;\;x \cdot t\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-68}:\\ \;\;\;\;2 \cdot \left(x \cdot z\right)\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-41}:\\ \;\;\;\;y \cdot 5\\ \mathbf{else}:\\ \;\;\;\;x \cdot t\\ \end{array} \]

Alternative 5: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+35} \lor \neg \left(x \leq 2.5\right):\\ \;\;\;\;x \cdot \left(t + \left(y + z\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(t + \left(z + z\right)\right) + y \cdot 5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -5e+35) (not (<= x 2.5)))
   (* x (+ t (* (+ y z) 2.0)))
   (+ (* x (+ t (+ z z))) (* y 5.0))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -5e+35) || !(x <= 2.5)) {
		tmp = x * (t + ((y + z) * 2.0));
	} else {
		tmp = (x * (t + (z + z))) + (y * 5.0);
	}
	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 <= (-5d+35)) .or. (.not. (x <= 2.5d0))) then
        tmp = x * (t + ((y + z) * 2.0d0))
    else
        tmp = (x * (t + (z + z))) + (y * 5.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -5e+35) || !(x <= 2.5)) {
		tmp = x * (t + ((y + z) * 2.0));
	} else {
		tmp = (x * (t + (z + z))) + (y * 5.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -5e+35) or not (x <= 2.5):
		tmp = x * (t + ((y + z) * 2.0))
	else:
		tmp = (x * (t + (z + z))) + (y * 5.0)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -5e+35) || !(x <= 2.5))
		tmp = Float64(x * Float64(t + Float64(Float64(y + z) * 2.0)));
	else
		tmp = Float64(Float64(x * Float64(t + Float64(z + z))) + Float64(y * 5.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -5e+35) || ~((x <= 2.5)))
		tmp = x * (t + ((y + z) * 2.0));
	else
		tmp = (x * (t + (z + z))) + (y * 5.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -5e+35], N[Not[LessEqual[x, 2.5]], $MachinePrecision]], N[(x * N[(t + N[(N[(y + z), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(t + N[(z + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * 5.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.00000000000000021e35 or 2.5 < x

   1. Initial program 100.0%

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

      1. fma-def 100.0%

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

      2. associate-+l+ 100.0%

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

      3. +-commutative 100.0%

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

      4. count-2 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 99.1%

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

   if -5.00000000000000021e35 < x < 2.5

   1. Initial program 99.9%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]

   2. Taylor expanded in y around 0 99.7%

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

   4. Simplified 99.7%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+35} \lor \neg \left(x \leq 2.5\right):\\ \;\;\;\;x \cdot \left(t + \left(y + z\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(t + \left(z + z\right)\right) + y \cdot 5\\ \end{array} \]

Alternative 6: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   \[\leadsto x \cdot \left(t + \left(y + \left(z + \left(y + z\right)\right)\right)\right) + y \cdot 5 \]

Alternative 7: 61.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \left(t + y \cdot 2\right)\\ \mathbf{if}\;x \leq -4.5 \cdot 10^{+65}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -2.6 \cdot 10^{-68}:\\ \;\;\;\;2 \cdot \left(x \cdot z\right)\\ \mathbf{elif}\;x \leq 1300:\\ \;\;\;\;y \cdot \left(5 + x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (+ t (* y 2.0)))))
   (if (<= x -4.5e+65)
     t_1
     (if (<= x -2.6e-68)
       (* 2.0 (* x z))
       (if (<= x 1300.0) (* y (+ 5.0 (* x 2.0))) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * (t + (y * 2.0));
	double tmp;
	if (x <= -4.5e+65) {
		tmp = t_1;
	} else if (x <= -2.6e-68) {
		tmp = 2.0 * (x * z);
	} else if (x <= 1300.0) {
		tmp = y * (5.0 + (x * 2.0));
	} 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 * (t + (y * 2.0d0))
    if (x <= (-4.5d+65)) then
        tmp = t_1
    else if (x <= (-2.6d-68)) then
        tmp = 2.0d0 * (x * z)
    else if (x <= 1300.0d0) then
        tmp = y * (5.0d0 + (x * 2.0d0))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x * (t + (y * 2.0));
	double tmp;
	if (x <= -4.5e+65) {
		tmp = t_1;
	} else if (x <= -2.6e-68) {
		tmp = 2.0 * (x * z);
	} else if (x <= 1300.0) {
		tmp = y * (5.0 + (x * 2.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * (t + (y * 2.0))
	tmp = 0
	if x <= -4.5e+65:
		tmp = t_1
	elif x <= -2.6e-68:
		tmp = 2.0 * (x * z)
	elif x <= 1300.0:
		tmp = y * (5.0 + (x * 2.0))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * Float64(t + Float64(y * 2.0)))
	tmp = 0.0
	if (x <= -4.5e+65)
		tmp = t_1;
	elseif (x <= -2.6e-68)
		tmp = Float64(2.0 * Float64(x * z));
	elseif (x <= 1300.0)
		tmp = Float64(y * Float64(5.0 + Float64(x * 2.0)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * (t + (y * 2.0));
	tmp = 0.0;
	if (x <= -4.5e+65)
		tmp = t_1;
	elseif (x <= -2.6e-68)
		tmp = 2.0 * (x * z);
	elseif (x <= 1300.0)
		tmp = y * (5.0 + (x * 2.0));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[(t + N[(y * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4.5e+65], t$95$1, If[LessEqual[x, -2.6e-68], N[(2.0 * N[(x * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1300.0], N[(y * N[(5.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.5e65 or 1300 < 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. Step-by-step derivation

      1. fma-def 100.0%

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

      2. associate-+l+ 100.0%

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

      3. +-commutative 100.0%

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

      4. count-2 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in z around 0 71.3%

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

   5. Taylor expanded in x around inf 70.9%

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

   if -4.5e65 < x < -2.5999999999999998e-68

   1. Initial program 99.9%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]

   2. Taylor expanded in z around inf 66.5%

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

   if -2.5999999999999998e-68 < x < 1300

   1. Initial program 99.9%

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

      1. fma-def 99.9%

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

      2. associate-+l+ 99.9%

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

      3. +-commutative 99.9%

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

      4. count-2 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around inf 66.5%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{+65}:\\ \;\;\;\;x \cdot \left(t + y \cdot 2\right)\\ \mathbf{elif}\;x \leq -2.6 \cdot 10^{-68}:\\ \;\;\;\;2 \cdot \left(x \cdot z\right)\\ \mathbf{elif}\;x \leq 1300:\\ \;\;\;\;y \cdot \left(5 + x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(t + y \cdot 2\right)\\ \end{array} \]

Alternative 8: 88.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{-70} \lor \neg \left(x \leq 1.52 \cdot 10^{-52}\right):\\ \;\;\;\;x \cdot \left(t + \left(y + z\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 5 + x \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -5.2e-70) (not (<= x 1.52e-52)))
   (* x (+ t (* (+ y z) 2.0)))
   (+ (* y 5.0) (* x t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -5.2e-70) || !(x <= 1.52e-52)) {
		tmp = x * (t + ((y + z) * 2.0));
	} else {
		tmp = (y * 5.0) + (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 <= (-5.2d-70)) .or. (.not. (x <= 1.52d-52))) then
        tmp = x * (t + ((y + z) * 2.0d0))
    else
        tmp = (y * 5.0d0) + (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 <= -5.2e-70) || !(x <= 1.52e-52)) {
		tmp = x * (t + ((y + z) * 2.0));
	} else {
		tmp = (y * 5.0) + (x * t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -5.2e-70) or not (x <= 1.52e-52):
		tmp = x * (t + ((y + z) * 2.0))
	else:
		tmp = (y * 5.0) + (x * t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -5.2e-70) || !(x <= 1.52e-52))
		tmp = Float64(x * Float64(t + Float64(Float64(y + z) * 2.0)));
	else
		tmp = Float64(Float64(y * 5.0) + Float64(x * t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -5.2e-70) || ~((x <= 1.52e-52)))
		tmp = x * (t + ((y + z) * 2.0));
	else
		tmp = (y * 5.0) + (x * t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -5.2e-70], N[Not[LessEqual[x, 1.52e-52]], $MachinePrecision]], N[(x * N[(t + N[(N[(y + z), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * 5.0), $MachinePrecision] + N[(x * t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.20000000000000004e-70 or 1.5199999999999999e-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. Step-by-step derivation

      1. fma-def 100.0%

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

      2. associate-+l+ 100.0%

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

      3. +-commutative 100.0%

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

      4. count-2 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 94.5%

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

   if -5.20000000000000004e-70 < x < 1.5199999999999999e-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. Taylor expanded in y around 0 99.9%

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

   4. Simplified 99.9%

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

   5. Taylor expanded in z around 0 81.5%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{-70} \lor \neg \left(x \leq 1.52 \cdot 10^{-52}\right):\\ \;\;\;\;x \cdot \left(t + \left(y + z\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 5 + x \cdot t\\ \end{array} \]

Alternative 9: 77.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{+115} \lor \neg \left(y \leq 3.1 \cdot 10^{-11}\right):\\ \;\;\;\;y \cdot \left(5 + x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(t + z \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -4.8e+115) (not (<= y 3.1e-11)))
   (* y (+ 5.0 (* x 2.0)))
   (* x (+ t (* z 2.0)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -4.8e+115) || !(y <= 3.1e-11)) {
		tmp = y * (5.0 + (x * 2.0));
	} else {
		tmp = x * (t + (z * 2.0));
	}
	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 ((y <= (-4.8d+115)) .or. (.not. (y <= 3.1d-11))) then
        tmp = y * (5.0d0 + (x * 2.0d0))
    else
        tmp = x * (t + (z * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -4.8e+115) || !(y <= 3.1e-11)) {
		tmp = y * (5.0 + (x * 2.0));
	} else {
		tmp = x * (t + (z * 2.0));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (y <= -4.8e+115) or not (y <= 3.1e-11):
		tmp = y * (5.0 + (x * 2.0))
	else:
		tmp = x * (t + (z * 2.0))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -4.8e+115) || !(y <= 3.1e-11))
		tmp = Float64(y * Float64(5.0 + Float64(x * 2.0)));
	else
		tmp = Float64(x * Float64(t + Float64(z * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((y <= -4.8e+115) || ~((y <= 3.1e-11)))
		tmp = y * (5.0 + (x * 2.0));
	else
		tmp = x * (t + (z * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -4.8e+115], N[Not[LessEqual[y, 3.1e-11]], $MachinePrecision]], N[(y * N[(5.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(t + N[(z * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.8000000000000001e115 or 3.10000000000000028e-11 < 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. Step-by-step derivation

      1. fma-def 99.9%

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

      2. associate-+l+ 99.9%

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

      3. +-commutative 99.9%

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

      4. count-2 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around inf 80.4%

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

   if -4.8000000000000001e115 < y < 3.10000000000000028e-11

   1. Initial program 100.0%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]

   2. Taylor expanded in y around 0 81.6%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{+115} \lor \neg \left(y \leq 3.1 \cdot 10^{-11}\right):\\ \;\;\;\;y \cdot \left(5 + x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(t + z \cdot 2\right)\\ \end{array} \]

Alternative 10: 73.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{-67} \lor \neg \left(x \leq 6.7 \cdot 10^{-49}\right):\\ \;\;\;\;x \cdot \left(t + z \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 5 + x \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -1.25e-67) (not (<= x 6.7e-49)))
   (* x (+ t (* z 2.0)))
   (+ (* y 5.0) (* x t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.25e-67) || !(x <= 6.7e-49)) {
		tmp = x * (t + (z * 2.0));
	} else {
		tmp = (y * 5.0) + (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 <= (-1.25d-67)) .or. (.not. (x <= 6.7d-49))) then
        tmp = x * (t + (z * 2.0d0))
    else
        tmp = (y * 5.0d0) + (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 <= -1.25e-67) || !(x <= 6.7e-49)) {
		tmp = x * (t + (z * 2.0));
	} else {
		tmp = (y * 5.0) + (x * t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -1.25e-67) or not (x <= 6.7e-49):
		tmp = x * (t + (z * 2.0))
	else:
		tmp = (y * 5.0) + (x * t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -1.25e-67) || !(x <= 6.7e-49))
		tmp = Float64(x * Float64(t + Float64(z * 2.0)));
	else
		tmp = Float64(Float64(y * 5.0) + Float64(x * t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -1.25e-67) || ~((x <= 6.7e-49)))
		tmp = x * (t + (z * 2.0));
	else
		tmp = (y * 5.0) + (x * t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -1.25e-67], N[Not[LessEqual[x, 6.7e-49]], $MachinePrecision]], N[(x * N[(t + N[(z * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * 5.0), $MachinePrecision] + N[(x * t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.25e-67 or 6.7e-49 < 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. Taylor expanded in y around 0 78.3%

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

   if -1.25e-67 < x < 6.7e-49

   1. Initial program 99.9%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]

   2. Taylor expanded in y around 0 99.9%

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

   4. Simplified 99.9%

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

   5. Taylor expanded in z around 0 81.5%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{-67} \lor \neg \left(x \leq 6.7 \cdot 10^{-49}\right):\\ \;\;\;\;x \cdot \left(t + z \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 5 + x \cdot t\\ \end{array} \]

Alternative 11: 47.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{-35}:\\ \;\;\;\;x \cdot t\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-41}:\\ \;\;\;\;y \cdot 5\\ \mathbf{else}:\\ \;\;\;\;x \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -2.8e-35) (* x t) (if (<= x 2.2e-41) (* y 5.0) (* x t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -2.8e-35) {
		tmp = x * t;
	} else if (x <= 2.2e-41) {
		tmp = y * 5.0;
	} else {
		tmp = x * t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (x <= (-2.8d-35)) then
        tmp = x * t
    else if (x <= 2.2d-41) then
        tmp = y * 5.0d0
    else
        tmp = x * t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -2.8e-35) {
		tmp = x * t;
	} else if (x <= 2.2e-41) {
		tmp = y * 5.0;
	} else {
		tmp = x * t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -2.8e-35:
		tmp = x * t
	elif x <= 2.2e-41:
		tmp = y * 5.0
	else:
		tmp = x * t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -2.8e-35)
		tmp = Float64(x * t);
	elseif (x <= 2.2e-41)
		tmp = Float64(y * 5.0);
	else
		tmp = Float64(x * t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -2.8e-35)
		tmp = x * t;
	elseif (x <= 2.2e-41)
		tmp = y * 5.0;
	else
		tmp = x * t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -2.8e-35], N[(x * t), $MachinePrecision], If[LessEqual[x, 2.2e-41], N[(y * 5.0), $MachinePrecision], N[(x * t), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.8e-35 or 2.2e-41 < 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. Taylor expanded in t around inf 42.5%

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

   if -2.8e-35 < x < 2.2e-41

   1. Initial program 99.9%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]

   2. Taylor expanded in x around 0 63.7%

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

4. Final simplification 51.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{-35}:\\ \;\;\;\;x \cdot t\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-41}:\\ \;\;\;\;y \cdot 5\\ \mathbf{else}:\\ \;\;\;\;x \cdot t\\ \end{array} \]

Alternative 12: 29.2% accurate, 5.0× 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. Taylor expanded in x around 0 29.4%

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

3. Final simplification 29.4%

   \[\leadsto y \cdot 5 \]

Reproduce

herbie shell --seed 2023171 
(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)))