Graphics.Rasterific.Svg.PathConverter:segmentToBezier from rasterific-svg-0.2.3.1, C

Percentage Accurate: 99.9% → 99.9%

Time: 7.4s

Alternatives: 11

Speedup: 1.0×

Specification

Math

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

FPCore

(FPCore (x y z) :precision binary64 (+ (+ x (sin y)) (* z (cos y))))

C

double code(double x, double y, double z) {
	return (x + sin(y)) + (z * cos(y));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x + sin(y)) + (z * cos(y))
end function

Java

public static double code(double x, double y, double z) {
	return (x + Math.sin(y)) + (z * Math.cos(y));
}

Python

def code(x, y, z):
	return (x + math.sin(y)) + (z * math.cos(y))

Julia

function code(x, y, z)
	return Float64(Float64(x + sin(y)) + Float64(z * cos(y)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x + sin(y)) + (z * cos(y));
end

Wolfram

code[x_, y_, z_] := N[(N[(x + N[Sin[y], $MachinePrecision]), $MachinePrecision] + N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z) :precision binary64 (+ (+ x (sin y)) (* z (cos y))))

C

double code(double x, double y, double z) {
	return (x + sin(y)) + (z * cos(y));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x + sin(y)) + (z * cos(y))
end function

Java

public static double code(double x, double y, double z) {
	return (x + Math.sin(y)) + (z * Math.cos(y));
}

Python

def code(x, y, z):
	return (x + math.sin(y)) + (z * math.cos(y))

Julia

function code(x, y, z)
	return Float64(Float64(x + sin(y)) + Float64(z * cos(y)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x + sin(y)) + (z * cos(y));
end

Wolfram

code[x_, y_, z_] := N[(N[(x + N[Sin[y], $MachinePrecision]), $MachinePrecision] + N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(z, \cos y, x + \sin y\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fma z (cos y) (+ x (sin y))))

C

double code(double x, double y, double z) {
	return fma(z, cos(y), (x + sin(y)));
}

Julia

function code(x, y, z)
	return fma(z, cos(y), Float64(x + sin(y)))
end

Wolfram

code[x_, y_, z_] := N[(z * N[Cos[y], $MachinePrecision] + N[(x + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. +-commutative 100.0%

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

   2. fma-define 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(z, \cos y, x + \sin y\right)} \]
4. Add Preprocessing
5. Add Preprocessing

Alternative 2: 95.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \cos y\\ \mathbf{if}\;x \leq -1.8 \cdot 10^{-43}:\\ \;\;\;\;\mathsf{fma}\left(z, \cos y, x\right)\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-31}:\\ \;\;\;\;\sin y + t\_0\\ \mathbf{else}:\\ \;\;\;\;x + t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (cos y))))
   (if (<= x -1.8e-43)
     (fma z (cos y) x)
     (if (<= x 7.5e-31) (+ (sin y) t_0) (+ x t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = z * cos(y);
	double tmp;
	if (x <= -1.8e-43) {
		tmp = fma(z, cos(y), x);
	} else if (x <= 7.5e-31) {
		tmp = sin(y) + t_0;
	} else {
		tmp = x + t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(z * cos(y))
	tmp = 0.0
	if (x <= -1.8e-43)
		tmp = fma(z, cos(y), x);
	elseif (x <= 7.5e-31)
		tmp = Float64(sin(y) + t_0);
	else
		tmp = Float64(x + t_0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.8e-43], N[(z * N[Cos[y], $MachinePrecision] + x), $MachinePrecision], If[LessEqual[x, 7.5e-31], N[(N[Sin[y], $MachinePrecision] + t$95$0), $MachinePrecision], N[(x + t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.7999999999999999e-43

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-define 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \cos y, x + \sin y\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 99.5%

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

   if -1.7999999999999999e-43 < x < 7.49999999999999975e-31

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 96.4%

      \[\leadsto \color{blue}{\sin y} + z \cdot \cos y \]

   if 7.49999999999999975e-31 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 98.7%

      \[\leadsto \color{blue}{x} + z \cdot \cos y \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 95.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.45 \cdot 10^{-32}:\\ \;\;\;\;\mathsf{fma}\left(z, \cos y, x\right)\\ \mathbf{elif}\;z \leq 7.4 \cdot 10^{-34}:\\ \;\;\;\;x + \sin y\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \cos y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -2.45e-32)
   (fma z (cos y) x)
   (if (<= z 7.4e-34) (+ x (sin y)) (+ x (* z (cos y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.45e-32) {
		tmp = fma(z, cos(y), x);
	} else if (z <= 7.4e-34) {
		tmp = x + sin(y);
	} else {
		tmp = x + (z * cos(y));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -2.45e-32)
		tmp = fma(z, cos(y), x);
	elseif (z <= 7.4e-34)
		tmp = Float64(x + sin(y));
	else
		tmp = Float64(x + Float64(z * cos(y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -2.45e-32], N[(z * N[Cos[y], $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 7.4e-34], N[(x + N[Sin[y], $MachinePrecision]), $MachinePrecision], N[(x + N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.4499999999999999e-32

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. fma-define 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \cos y, x + \sin y\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 97.4%

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

   if -2.4499999999999999e-32 < z < 7.39999999999999976e-34

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-define 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \cos y, x + \sin y\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 94.3%

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

      1. +-commutative 94.3%

         \[\leadsto \color{blue}{\sin y + x} \]

   7. Simplified 94.3%

      \[\leadsto \color{blue}{\sin y + x} \]

   if 7.39999999999999976e-34 < z

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 99.9%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.45 \cdot 10^{-32}:\\ \;\;\;\;\mathsf{fma}\left(z, \cos y, x\right)\\ \mathbf{elif}\;z \leq 7.4 \cdot 10^{-34}:\\ \;\;\;\;x + \sin y\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.9% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z) :precision binary64 (+ (* z (cos y)) (+ x (sin y))))

C

double code(double x, double y, double z) {
	return (z * cos(y)) + (x + sin(y));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (z * cos(y)) + (x + sin(y))
end function

Java

public static double code(double x, double y, double z) {
	return (z * Math.cos(y)) + (x + Math.sin(y));
}

Python

def code(x, y, z):
	return (z * math.cos(y)) + (x + math.sin(y))

Julia

function code(x, y, z)
	return Float64(Float64(z * cos(y)) + Float64(x + sin(y)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (z * cos(y)) + (x + sin(y));
end

Wolfram

code[x_, y_, z_] := N[(N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision] + N[(x + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Final simplification 100.0%

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

Alternative 5: 84.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \cos y\\ \mathbf{if}\;z \leq -7.2 \cdot 10^{+220}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq -1.2 \cdot 10^{-42}:\\ \;\;\;\;z + x\\ \mathbf{elif}\;z \leq 9.2 \cdot 10^{-32}:\\ \;\;\;\;x + \sin y\\ \mathbf{elif}\;z \leq 3 \cdot 10^{+151}:\\ \;\;\;\;z + x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (cos y))))
   (if (<= z -7.2e+220)
     t_0
     (if (<= z -1.2e-42)
       (+ z x)
       (if (<= z 9.2e-32) (+ x (sin y)) (if (<= z 3e+151) (+ z x) t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = z * cos(y);
	double tmp;
	if (z <= -7.2e+220) {
		tmp = t_0;
	} else if (z <= -1.2e-42) {
		tmp = z + x;
	} else if (z <= 9.2e-32) {
		tmp = x + sin(y);
	} else if (z <= 3e+151) {
		tmp = z + x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = z * cos(y)
    if (z <= (-7.2d+220)) then
        tmp = t_0
    else if (z <= (-1.2d-42)) then
        tmp = z + x
    else if (z <= 9.2d-32) then
        tmp = x + sin(y)
    else if (z <= 3d+151) then
        tmp = z + x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = z * Math.cos(y);
	double tmp;
	if (z <= -7.2e+220) {
		tmp = t_0;
	} else if (z <= -1.2e-42) {
		tmp = z + x;
	} else if (z <= 9.2e-32) {
		tmp = x + Math.sin(y);
	} else if (z <= 3e+151) {
		tmp = z + x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * math.cos(y)
	tmp = 0
	if z <= -7.2e+220:
		tmp = t_0
	elif z <= -1.2e-42:
		tmp = z + x
	elif z <= 9.2e-32:
		tmp = x + math.sin(y)
	elif z <= 3e+151:
		tmp = z + x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * cos(y))
	tmp = 0.0
	if (z <= -7.2e+220)
		tmp = t_0;
	elseif (z <= -1.2e-42)
		tmp = Float64(z + x);
	elseif (z <= 9.2e-32)
		tmp = Float64(x + sin(y));
	elseif (z <= 3e+151)
		tmp = Float64(z + x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * cos(y);
	tmp = 0.0;
	if (z <= -7.2e+220)
		tmp = t_0;
	elseif (z <= -1.2e-42)
		tmp = z + x;
	elseif (z <= 9.2e-32)
		tmp = x + sin(y);
	elseif (z <= 3e+151)
		tmp = z + x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -7.2e+220], t$95$0, If[LessEqual[z, -1.2e-42], N[(z + x), $MachinePrecision], If[LessEqual[z, 9.2e-32], N[(x + N[Sin[y], $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3e+151], N[(z + x), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -7.20000000000000038e220 or 2.9999999999999999e151 < z

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

      2. fma-define 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \cos y, x + \sin y\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 91.7%

      \[\leadsto \color{blue}{z \cdot \cos y} \]

   if -7.20000000000000038e220 < z < -1.20000000000000001e-42 or 9.2000000000000002e-32 < z < 2.9999999999999999e151

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-define 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \cos y, x + \sin y\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 81.4%

      \[\leadsto \color{blue}{x + z} \]

   if -1.20000000000000001e-42 < z < 9.2000000000000002e-32

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-define 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \cos y, x + \sin y\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 95.1%

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

      1. +-commutative 95.1%

         \[\leadsto \color{blue}{\sin y + x} \]

   7. Simplified 95.1%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.2 \cdot 10^{+220}:\\ \;\;\;\;z \cdot \cos y\\ \mathbf{elif}\;z \leq -1.2 \cdot 10^{-42}:\\ \;\;\;\;z + x\\ \mathbf{elif}\;z \leq 9.2 \cdot 10^{-32}:\\ \;\;\;\;x + \sin y\\ \mathbf{elif}\;z \leq 3 \cdot 10^{+151}:\\ \;\;\;\;z + x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.18 \cdot 10^{-54}:\\ \;\;\;\;z + x\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-52}:\\ \;\;\;\;z \cdot \cos y\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-31}:\\ \;\;\;\;\sin y\\ \mathbf{else}:\\ \;\;\;\;z + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.18e-54)
   (+ z x)
   (if (<= x 2e-52) (* z (cos y)) (if (<= x 2.5e-31) (sin y) (+ z x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.18e-54) {
		tmp = z + x;
	} else if (x <= 2e-52) {
		tmp = z * cos(y);
	} else if (x <= 2.5e-31) {
		tmp = sin(y);
	} else {
		tmp = z + x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-1.18d-54)) then
        tmp = z + x
    else if (x <= 2d-52) then
        tmp = z * cos(y)
    else if (x <= 2.5d-31) then
        tmp = sin(y)
    else
        tmp = z + x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.18e-54) {
		tmp = z + x;
	} else if (x <= 2e-52) {
		tmp = z * Math.cos(y);
	} else if (x <= 2.5e-31) {
		tmp = Math.sin(y);
	} else {
		tmp = z + x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.18e-54:
		tmp = z + x
	elif x <= 2e-52:
		tmp = z * math.cos(y)
	elif x <= 2.5e-31:
		tmp = math.sin(y)
	else:
		tmp = z + x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.18e-54)
		tmp = Float64(z + x);
	elseif (x <= 2e-52)
		tmp = Float64(z * cos(y));
	elseif (x <= 2.5e-31)
		tmp = sin(y);
	else
		tmp = Float64(z + x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.18e-54)
		tmp = z + x;
	elseif (x <= 2e-52)
		tmp = z * cos(y);
	elseif (x <= 2.5e-31)
		tmp = sin(y);
	else
		tmp = z + x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.18e-54], N[(z + x), $MachinePrecision], If[LessEqual[x, 2e-52], N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.5e-31], N[Sin[y], $MachinePrecision], N[(z + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.17999999999999996e-54 or 2.5e-31 < x

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-define 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \cos y, x + \sin y\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 90.7%

      \[\leadsto \color{blue}{x + z} \]

   if -1.17999999999999996e-54 < x < 2e-52

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. fma-define 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \cos y, x + \sin y\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 63.8%

      \[\leadsto \color{blue}{z \cdot \cos y} \]

   if 2e-52 < x < 2.5e-31

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{\sin y} + z \cdot \cos y \]

   4. Taylor expanded in z around 0 100.0%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.18 \cdot 10^{-54}:\\ \;\;\;\;z + x\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-52}:\\ \;\;\;\;z \cdot \cos y\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-31}:\\ \;\;\;\;\sin y\\ \mathbf{else}:\\ \;\;\;\;z + x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 95.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.8 \cdot 10^{-32} \lor \neg \left(z \leq 1.7 \cdot 10^{-32}\right):\\ \;\;\;\;x + z \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x + \sin y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -2.8e-32) (not (<= z 1.7e-32)))
   (+ x (* z (cos y)))
   (+ x (sin y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2.8e-32) || !(z <= 1.7e-32)) {
		tmp = x + (z * cos(y));
	} else {
		tmp = x + sin(y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-2.8d-32)) .or. (.not. (z <= 1.7d-32))) then
        tmp = x + (z * cos(y))
    else
        tmp = x + sin(y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2.8e-32) || !(z <= 1.7e-32)) {
		tmp = x + (z * Math.cos(y));
	} else {
		tmp = x + Math.sin(y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -2.8e-32) or not (z <= 1.7e-32):
		tmp = x + (z * math.cos(y))
	else:
		tmp = x + math.sin(y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -2.8e-32) || !(z <= 1.7e-32))
		tmp = Float64(x + Float64(z * cos(y)));
	else
		tmp = Float64(x + sin(y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -2.8e-32) || ~((z <= 1.7e-32)))
		tmp = x + (z * cos(y));
	else
		tmp = x + sin(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -2.8e-32], N[Not[LessEqual[z, 1.7e-32]], $MachinePrecision]], N[(x + N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[Sin[y], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.7999999999999999e-32 or 1.69999999999999989e-32 < z

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 98.5%

      \[\leadsto \color{blue}{x} + z \cdot \cos y \]

   if -2.7999999999999999e-32 < z < 1.69999999999999989e-32

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-define 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \cos y, x + \sin y\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 94.3%

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

      1. +-commutative 94.3%

         \[\leadsto \color{blue}{\sin y + x} \]

   7. Simplified 94.3%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.8 \cdot 10^{-32} \lor \neg \left(z \leq 1.7 \cdot 10^{-32}\right):\\ \;\;\;\;x + z \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x + \sin y\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 64.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.7 \cdot 10^{-277} \lor \neg \left(x \leq 4.4 \cdot 10^{-31}\right):\\ \;\;\;\;z + x\\ \mathbf{else}:\\ \;\;\;\;\sin y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -3.7e-277) (not (<= x 4.4e-31))) (+ z x) (sin y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -3.7e-277) || !(x <= 4.4e-31)) {
		tmp = z + x;
	} else {
		tmp = sin(y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-3.7d-277)) .or. (.not. (x <= 4.4d-31))) then
        tmp = z + x
    else
        tmp = sin(y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -3.7e-277) || !(x <= 4.4e-31)) {
		tmp = z + x;
	} else {
		tmp = Math.sin(y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -3.7e-277) or not (x <= 4.4e-31):
		tmp = z + x
	else:
		tmp = math.sin(y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -3.7e-277) || !(x <= 4.4e-31))
		tmp = Float64(z + x);
	else
		tmp = sin(y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -3.7e-277) || ~((x <= 4.4e-31)))
		tmp = z + x;
	else
		tmp = sin(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -3.7e-277], N[Not[LessEqual[x, 4.4e-31]], $MachinePrecision]], N[(z + x), $MachinePrecision], N[Sin[y], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.69999999999999985e-277 or 4.40000000000000019e-31 < x

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-define 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \cos y, x + \sin y\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 83.2%

      \[\leadsto \color{blue}{x + z} \]

   if -3.69999999999999985e-277 < x < 4.40000000000000019e-31

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 98.4%

      \[\leadsto \color{blue}{\sin y} + z \cdot \cos y \]

   4. Taylor expanded in z around 0 51.4%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.7 \cdot 10^{-277} \lor \neg \left(x \leq 4.4 \cdot 10^{-31}\right):\\ \;\;\;\;z + x\\ \mathbf{else}:\\ \;\;\;\;\sin y\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 55.2% accurate, 18.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+17}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{+31}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -2e+17) x (if (<= x 5.2e+31) z x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -2e+17) {
		tmp = x;
	} else if (x <= 5.2e+31) {
		tmp = z;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-2d+17)) then
        tmp = x
    else if (x <= 5.2d+31) then
        tmp = z
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -2e+17) {
		tmp = x;
	} else if (x <= 5.2e+31) {
		tmp = z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -2e+17:
		tmp = x
	elif x <= 5.2e+31:
		tmp = z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -2e+17)
		tmp = x;
	elseif (x <= 5.2e+31)
		tmp = z;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -2e+17)
		tmp = x;
	elseif (x <= 5.2e+31)
		tmp = z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -2e+17], x, If[LessEqual[x, 5.2e+31], z, x]]

Derivation

1. Split input into 2 regimes

2. if x < -2e17 or 5.2e31 < x

   1. Initial program 100.0%

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

      1. log1p-expm1-u 99.9%

         \[\leadsto \left(x + \sin y\right) + z \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos y\right)\right)} \]

   4. Applied egg-rr 99.9%

      \[\leadsto \left(x + \sin y\right) + z \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos y\right)\right)} \]

   5. Taylor expanded in x around inf 83.6%

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

   if -2e17 < x < 5.2e31

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. fma-define 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \cos y, x + \sin y\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 60.8%

      \[\leadsto \color{blue}{z \cdot \cos y} \]

   6. Taylor expanded in y around 0 38.6%

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

Alternative 10: 66.9% accurate, 69.0× speedup

Math

\[\begin{array}{l} \\ z + x \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ z x))

C

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

Fortran

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

Java

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

Python

def code(x, y, z):
	return z + x

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(z + x), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. +-commutative 100.0%

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

   2. fma-define 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(z, \cos y, x + \sin y\right)} \]
4. Add Preprocessing

5. Taylor expanded in y around 0 70.0%

   \[\leadsto \color{blue}{x + z} \]

6. Final simplification 70.0%

   \[\leadsto z + x \]
7. Add Preprocessing

Alternative 11: 42.6% accurate, 207.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x
end function

Java

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

Python

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

Julia

function code(x, y, z)
	return x
end

MATLAB

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

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 100.0%

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

   1. log1p-expm1-u 99.9%

      \[\leadsto \left(x + \sin y\right) + z \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos y\right)\right)} \]

4. Applied egg-rr 99.9%

   \[\leadsto \left(x + \sin y\right) + z \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos y\right)\right)} \]

5. Taylor expanded in x around inf 48.4%

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

Reproduce

herbie shell --seed 2024144 
(FPCore (x y z)
  :name "Graphics.Rasterific.Svg.PathConverter:segmentToBezier from rasterific-svg-0.2.3.1, C"
  :precision binary64
  (+ (+ x (sin y)) (* z (cos y))))