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

Percentage Accurate: 99.9% → 99.9%

Time: 8.8s

Alternatives: 14

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, 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]

Derivation

1. Initial program 99.9%

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

Alternative 2: 83.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \cos y\\ \mathbf{if}\;z \leq -3.55 \cdot 10^{+33}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq -1.05 \cdot 10^{-152}:\\ \;\;\;\;x + z\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{-13}:\\ \;\;\;\;x + \sin y\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{+114}:\\ \;\;\;\;x + z\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (cos y))))
   (if (<= z -3.55e+33)
     t_0
     (if (<= z -1.05e-152)
       (+ x z)
       (if (<= z 4.8e-13) (+ x (sin y)) (if (<= z 3.1e+114) (+ x z) t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = z * cos(y);
	double tmp;
	if (z <= -3.55e+33) {
		tmp = t_0;
	} else if (z <= -1.05e-152) {
		tmp = x + z;
	} else if (z <= 4.8e-13) {
		tmp = x + sin(y);
	} else if (z <= 3.1e+114) {
		tmp = x + z;
	} 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 <= (-3.55d+33)) then
        tmp = t_0
    else if (z <= (-1.05d-152)) then
        tmp = x + z
    else if (z <= 4.8d-13) then
        tmp = x + sin(y)
    else if (z <= 3.1d+114) then
        tmp = x + z
    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 <= -3.55e+33) {
		tmp = t_0;
	} else if (z <= -1.05e-152) {
		tmp = x + z;
	} else if (z <= 4.8e-13) {
		tmp = x + Math.sin(y);
	} else if (z <= 3.1e+114) {
		tmp = x + z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * math.cos(y)
	tmp = 0
	if z <= -3.55e+33:
		tmp = t_0
	elif z <= -1.05e-152:
		tmp = x + z
	elif z <= 4.8e-13:
		tmp = x + math.sin(y)
	elif z <= 3.1e+114:
		tmp = x + z
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * cos(y))
	tmp = 0.0
	if (z <= -3.55e+33)
		tmp = t_0;
	elseif (z <= -1.05e-152)
		tmp = Float64(x + z);
	elseif (z <= 4.8e-13)
		tmp = Float64(x + sin(y));
	elseif (z <= 3.1e+114)
		tmp = Float64(x + z);
	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 <= -3.55e+33)
		tmp = t_0;
	elseif (z <= -1.05e-152)
		tmp = x + z;
	elseif (z <= 4.8e-13)
		tmp = x + sin(y);
	elseif (z <= 3.1e+114)
		tmp = x + z;
	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, -3.55e+33], t$95$0, If[LessEqual[z, -1.05e-152], N[(x + z), $MachinePrecision], If[LessEqual[z, 4.8e-13], N[(x + N[Sin[y], $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3.1e+114], N[(x + z), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.54999999999999989e33 or 3.1e114 < z

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \color{blue}{\cos y}\right) \]

      2. cos-lowering-cos.f64 85.8%

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{cos.f64}\left(y\right)\right) \]

   5. Simplified 85.8%

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

   if -3.54999999999999989e33 < z < -1.04999999999999999e-152 or 4.7999999999999997e-13 < z < 3.1e114

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 89.1%

         \[\leadsto \mathsf{+.f64}\left(z, \color{blue}{x}\right) \]

   5. Simplified 89.1%

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

   if -1.04999999999999999e-152 < z < 4.7999999999999997e-13

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\sin y, \color{blue}{x}\right) \]

      3. sin-lowering-sin.f64 93.0%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{sin.f64}\left(y\right), x\right) \]

   5. Simplified 93.0%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.55 \cdot 10^{+33}:\\ \;\;\;\;z \cdot \cos y\\ \mathbf{elif}\;z \leq -1.05 \cdot 10^{-152}:\\ \;\;\;\;x + z\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{-13}:\\ \;\;\;\;x + \sin y\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{+114}:\\ \;\;\;\;x + z\\ \mathbf{else}:\\ \;\;\;\;z \cdot \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + z \cdot \cos y\\ \mathbf{if}\;z \leq -65000000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 0.0001:\\ \;\;\;\;\left(x + \sin y\right) + z\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (* z (cos y)))))
   (if (<= z -65000000000000.0)
     t_0
     (if (<= z 0.0001) (+ (+ x (sin y)) z) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x + (z * cos(y));
	double tmp;
	if (z <= -65000000000000.0) {
		tmp = t_0;
	} else if (z <= 0.0001) {
		tmp = (x + sin(y)) + z;
	} 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 = x + (z * cos(y))
    if (z <= (-65000000000000.0d0)) then
        tmp = t_0
    else if (z <= 0.0001d0) then
        tmp = (x + sin(y)) + z
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x + (z * Math.cos(y));
	double tmp;
	if (z <= -65000000000000.0) {
		tmp = t_0;
	} else if (z <= 0.0001) {
		tmp = (x + Math.sin(y)) + z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x + (z * math.cos(y))
	tmp = 0
	if z <= -65000000000000.0:
		tmp = t_0
	elif z <= 0.0001:
		tmp = (x + math.sin(y)) + z
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x + Float64(z * cos(y)))
	tmp = 0.0
	if (z <= -65000000000000.0)
		tmp = t_0;
	elseif (z <= 0.0001)
		tmp = Float64(Float64(x + sin(y)) + z);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x + (z * cos(y));
	tmp = 0.0;
	if (z <= -65000000000000.0)
		tmp = t_0;
	elseif (z <= 0.0001)
		tmp = (x + sin(y)) + z;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x + N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -65000000000000.0], t$95$0, If[LessEqual[z, 0.0001], N[(N[(x + N[Sin[y], $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -6.5e13 or 1.00000000000000005e-4 < 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

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{x}, \mathsf{*.f64}\left(z, \mathsf{cos.f64}\left(y\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 99.9%

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

   if -6.5e13 < z < 1.00000000000000005e-4

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, \mathsf{sin.f64}\left(y\right)\right), \color{blue}{z}\right) \]
   4. Step-by-step derivation

   5. Simplified 99.9%

      \[\leadsto \left(x + \sin y\right) + \color{blue}{z} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 93.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + z \cdot \cos y\\ \mathbf{if}\;z \leq -1.05 \cdot 10^{-152}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{-10}:\\ \;\;\;\;x + \sin y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (* z (cos y)))))
   (if (<= z -1.05e-152) t_0 (if (<= z 1.35e-10) (+ x (sin y)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x + (z * cos(y));
	double tmp;
	if (z <= -1.05e-152) {
		tmp = t_0;
	} else if (z <= 1.35e-10) {
		tmp = x + sin(y);
	} 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 = x + (z * cos(y))
    if (z <= (-1.05d-152)) then
        tmp = t_0
    else if (z <= 1.35d-10) then
        tmp = x + sin(y)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x + (z * Math.cos(y));
	double tmp;
	if (z <= -1.05e-152) {
		tmp = t_0;
	} else if (z <= 1.35e-10) {
		tmp = x + Math.sin(y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x + (z * math.cos(y))
	tmp = 0
	if z <= -1.05e-152:
		tmp = t_0
	elif z <= 1.35e-10:
		tmp = x + math.sin(y)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x + Float64(z * cos(y)))
	tmp = 0.0
	if (z <= -1.05e-152)
		tmp = t_0;
	elseif (z <= 1.35e-10)
		tmp = Float64(x + sin(y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x + (z * cos(y));
	tmp = 0.0;
	if (z <= -1.05e-152)
		tmp = t_0;
	elseif (z <= 1.35e-10)
		tmp = x + sin(y);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x + N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.05e-152], t$95$0, If[LessEqual[z, 1.35e-10], N[(x + N[Sin[y], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.04999999999999999e-152 or 1.35e-10 < 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

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{x}, \mathsf{*.f64}\left(z, \mathsf{cos.f64}\left(y\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 98.2%

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

   if -1.04999999999999999e-152 < z < 1.35e-10

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\sin y, \color{blue}{x}\right) \]

      3. sin-lowering-sin.f64 93.0%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{sin.f64}\left(y\right), x\right) \]

   5. Simplified 93.0%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.05 \cdot 10^{-152}:\\ \;\;\;\;x + z \cdot \cos y\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{-10}:\\ \;\;\;\;x + \sin y\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \cos y\\ \mathbf{if}\;z \leq -2.95 \cdot 10^{+35}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 4.3 \cdot 10^{+112}:\\ \;\;\;\;x + z\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (cos y))))
   (if (<= z -2.95e+35) t_0 (if (<= z 4.3e+112) (+ x z) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = z * cos(y);
	double tmp;
	if (z <= -2.95e+35) {
		tmp = t_0;
	} else if (z <= 4.3e+112) {
		tmp = x + z;
	} 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 <= (-2.95d+35)) then
        tmp = t_0
    else if (z <= 4.3d+112) then
        tmp = x + z
    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 <= -2.95e+35) {
		tmp = t_0;
	} else if (z <= 4.3e+112) {
		tmp = x + z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * math.cos(y)
	tmp = 0
	if z <= -2.95e+35:
		tmp = t_0
	elif z <= 4.3e+112:
		tmp = x + z
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * cos(y))
	tmp = 0.0
	if (z <= -2.95e+35)
		tmp = t_0;
	elseif (z <= 4.3e+112)
		tmp = Float64(x + z);
	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 <= -2.95e+35)
		tmp = t_0;
	elseif (z <= 4.3e+112)
		tmp = x + z;
	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, -2.95e+35], t$95$0, If[LessEqual[z, 4.3e+112], N[(x + z), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.94999999999999993e35 or 4.29999999999999983e112 < z

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \color{blue}{\cos y}\right) \]

      2. cos-lowering-cos.f64 85.8%

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{cos.f64}\left(y\right)\right) \]

   5. Simplified 85.8%

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

   if -2.94999999999999993e35 < z < 4.29999999999999983e112

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 72.6%

         \[\leadsto \mathsf{+.f64}\left(z, \color{blue}{x}\right) \]

   5. Simplified 72.6%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.95 \cdot 10^{+35}:\\ \;\;\;\;z \cdot \cos y\\ \mathbf{elif}\;z \leq 4.3 \cdot 10^{+112}:\\ \;\;\;\;x + z\\ \mathbf{else}:\\ \;\;\;\;z \cdot \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 70.3% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -80:\\ \;\;\;\;x + z\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-25}:\\ \;\;\;\;\left(x + y \cdot \left(1 + \left(y \cdot y\right) \cdot \left(-0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right) + z \cdot \left(1 + \left(y \cdot y\right) \cdot \left(-0.5 + \left(y \cdot y\right) \cdot \left(0.041666666666666664 + \left(y \cdot y\right) \cdot -0.001388888888888889\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -80.0)
   (+ x z)
   (if (<= y 5e-25)
     (+
      (+
       x
       (*
        y
        (+
         1.0
         (*
          (* y y)
          (+ -0.16666666666666666 (* (* y y) 0.008333333333333333))))))
      (*
       z
       (+
        1.0
        (*
         (* y y)
         (+
          -0.5
          (*
           (* y y)
           (+ 0.041666666666666664 (* (* y y) -0.001388888888888889))))))))
     (+ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -80.0) {
		tmp = x + z;
	} else if (y <= 5e-25) {
		tmp = (x + (y * (1.0 + ((y * y) * (-0.16666666666666666 + ((y * y) * 0.008333333333333333)))))) + (z * (1.0 + ((y * y) * (-0.5 + ((y * y) * (0.041666666666666664 + ((y * y) * -0.001388888888888889)))))));
	} else {
		tmp = x + z;
	}
	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 (y <= (-80.0d0)) then
        tmp = x + z
    else if (y <= 5d-25) then
        tmp = (x + (y * (1.0d0 + ((y * y) * ((-0.16666666666666666d0) + ((y * y) * 0.008333333333333333d0)))))) + (z * (1.0d0 + ((y * y) * ((-0.5d0) + ((y * y) * (0.041666666666666664d0 + ((y * y) * (-0.001388888888888889d0))))))))
    else
        tmp = x + z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -80.0) {
		tmp = x + z;
	} else if (y <= 5e-25) {
		tmp = (x + (y * (1.0 + ((y * y) * (-0.16666666666666666 + ((y * y) * 0.008333333333333333)))))) + (z * (1.0 + ((y * y) * (-0.5 + ((y * y) * (0.041666666666666664 + ((y * y) * -0.001388888888888889)))))));
	} else {
		tmp = x + z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -80.0:
		tmp = x + z
	elif y <= 5e-25:
		tmp = (x + (y * (1.0 + ((y * y) * (-0.16666666666666666 + ((y * y) * 0.008333333333333333)))))) + (z * (1.0 + ((y * y) * (-0.5 + ((y * y) * (0.041666666666666664 + ((y * y) * -0.001388888888888889)))))))
	else:
		tmp = x + z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -80.0)
		tmp = Float64(x + z);
	elseif (y <= 5e-25)
		tmp = Float64(Float64(x + Float64(y * Float64(1.0 + Float64(Float64(y * y) * Float64(-0.16666666666666666 + Float64(Float64(y * y) * 0.008333333333333333)))))) + Float64(z * Float64(1.0 + Float64(Float64(y * y) * Float64(-0.5 + Float64(Float64(y * y) * Float64(0.041666666666666664 + Float64(Float64(y * y) * -0.001388888888888889))))))));
	else
		tmp = Float64(x + z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -80.0)
		tmp = x + z;
	elseif (y <= 5e-25)
		tmp = (x + (y * (1.0 + ((y * y) * (-0.16666666666666666 + ((y * y) * 0.008333333333333333)))))) + (z * (1.0 + ((y * y) * (-0.5 + ((y * y) * (0.041666666666666664 + ((y * y) * -0.001388888888888889)))))));
	else
		tmp = x + z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -80.0], N[(x + z), $MachinePrecision], If[LessEqual[y, 5e-25], N[(N[(x + N[(y * N[(1.0 + N[(N[(y * y), $MachinePrecision] * N[(-0.16666666666666666 + N[(N[(y * y), $MachinePrecision] * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(z * N[(1.0 + N[(N[(y * y), $MachinePrecision] * N[(-0.5 + N[(N[(y * y), $MachinePrecision] * N[(0.041666666666666664 + N[(N[(y * y), $MachinePrecision] * -0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -80 or 4.99999999999999962e-25 < y

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 42.9%

         \[\leadsto \mathsf{+.f64}\left(z, \color{blue}{x}\right) \]

   5. Simplified 42.9%

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

   if -80 < y < 4.99999999999999962e-25

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(x + y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)\right)\right)}, \mathsf{*.f64}\left(z, \mathsf{cos.f64}\left(y\right)\right)\right) \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{z}, \mathsf{cos.f64}\left(y\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \left(1 + {y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)\right)\right)\right), \mathsf{*.f64}\left(z, \mathsf{cos.f64}\left(y\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \left({y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(z, \mathsf{cos.f64}\left(y\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({y}^{2}\right), \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(z, \mathsf{cos.f64}\left(y\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(y \cdot y\right), \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(z, \mathsf{cos.f64}\left(y\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(z, \mathsf{cos.f64}\left(y\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\frac{1}{120} \cdot {y}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(z, \mathsf{cos.f64}\left(y\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\frac{1}{120} \cdot {y}^{2} + \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(z, \mathsf{cos.f64}\left(y\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\frac{-1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(z, \mathsf{cos.f64}\left(y\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{6}, \left(\frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(z, \mathsf{cos.f64}\left(y\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{6}, \left({y}^{2} \cdot \frac{1}{120}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(z, \mathsf{cos.f64}\left(y\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left({y}^{2}\right), \frac{1}{120}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(z, \mathsf{cos.f64}\left(y\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left(y \cdot y\right), \frac{1}{120}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(z, \mathsf{cos.f64}\left(y\right)\right)\right) \]

      14. *-lowering-*.f64 100.0%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(z, \mathsf{cos.f64}\left(y\right)\right)\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\left(x + y \cdot \left(1 + \left(y \cdot y\right) \cdot \left(-0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)} + z \cdot \cos y \]

   6. Taylor expanded in y around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(z, \color{blue}{\left(1 + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) - \frac{1}{2}\right)\right)}\right)\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \color{blue}{\left({y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) - \frac{1}{2}\right)\right)}\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) - \frac{1}{2}\right)}\right)\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(y \cdot y\right), \left(\color{blue}{{y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right)} - \frac{1}{2}\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\color{blue}{{y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right)} - \frac{1}{2}\right)\right)\right)\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left({y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left({y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) + \frac{-1}{2}\right)\right)\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\frac{-1}{2} + \color{blue}{{y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right)}\right)\right)\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right)\right)}\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left(y \cdot y\right), \left(\color{blue}{\frac{1}{24}} + \frac{-1}{720} \cdot {y}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\color{blue}{\frac{1}{24}} + \frac{-1}{720} \cdot {y}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{24}, \color{blue}{\left(\frac{-1}{720} \cdot {y}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{24}, \left({y}^{2} \cdot \color{blue}{\frac{-1}{720}}\right)\right)\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\frac{-1}{720}}\right)\right)\right)\right)\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(y \cdot y\right), \frac{-1}{720}\right)\right)\right)\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f64 99.6%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{720}\right)\right)\right)\right)\right)\right)\right)\right) \]

   8. Simplified 99.6%

      \[\leadsto \left(x + y \cdot \left(1 + \left(y \cdot y\right) \cdot \left(-0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right) + z \cdot \color{blue}{\left(1 + \left(y \cdot y\right) \cdot \left(-0.5 + \left(y \cdot y\right) \cdot \left(0.041666666666666664 + \left(y \cdot y\right) \cdot -0.001388888888888889\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -80:\\ \;\;\;\;x + z\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-25}:\\ \;\;\;\;\left(x + y \cdot \left(1 + \left(y \cdot y\right) \cdot \left(-0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right) + z \cdot \left(1 + \left(y \cdot y\right) \cdot \left(-0.5 + \left(y \cdot y\right) \cdot \left(0.041666666666666664 + \left(y \cdot y\right) \cdot -0.001388888888888889\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + z\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 70.3% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.5:\\ \;\;\;\;x + z\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{-25}:\\ \;\;\;\;\left(x + y \cdot \left(1 + \left(y \cdot y\right) \cdot \left(-0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right) + \left(z + \left(y \cdot y\right) \cdot \left(z \cdot \left(-0.5 + \left(y \cdot y\right) \cdot 0.041666666666666664\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -7.5)
   (+ x z)
   (if (<= y 6.2e-25)
     (+
      (+
       x
       (*
        y
        (+
         1.0
         (*
          (* y y)
          (+ -0.16666666666666666 (* (* y y) 0.008333333333333333))))))
      (+ z (* (* y y) (* z (+ -0.5 (* (* y y) 0.041666666666666664))))))
     (+ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -7.5) {
		tmp = x + z;
	} else if (y <= 6.2e-25) {
		tmp = (x + (y * (1.0 + ((y * y) * (-0.16666666666666666 + ((y * y) * 0.008333333333333333)))))) + (z + ((y * y) * (z * (-0.5 + ((y * y) * 0.041666666666666664)))));
	} else {
		tmp = x + z;
	}
	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 (y <= (-7.5d0)) then
        tmp = x + z
    else if (y <= 6.2d-25) then
        tmp = (x + (y * (1.0d0 + ((y * y) * ((-0.16666666666666666d0) + ((y * y) * 0.008333333333333333d0)))))) + (z + ((y * y) * (z * ((-0.5d0) + ((y * y) * 0.041666666666666664d0)))))
    else
        tmp = x + z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -7.5) {
		tmp = x + z;
	} else if (y <= 6.2e-25) {
		tmp = (x + (y * (1.0 + ((y * y) * (-0.16666666666666666 + ((y * y) * 0.008333333333333333)))))) + (z + ((y * y) * (z * (-0.5 + ((y * y) * 0.041666666666666664)))));
	} else {
		tmp = x + z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -7.5:
		tmp = x + z
	elif y <= 6.2e-25:
		tmp = (x + (y * (1.0 + ((y * y) * (-0.16666666666666666 + ((y * y) * 0.008333333333333333)))))) + (z + ((y * y) * (z * (-0.5 + ((y * y) * 0.041666666666666664)))))
	else:
		tmp = x + z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -7.5)
		tmp = Float64(x + z);
	elseif (y <= 6.2e-25)
		tmp = Float64(Float64(x + Float64(y * Float64(1.0 + Float64(Float64(y * y) * Float64(-0.16666666666666666 + Float64(Float64(y * y) * 0.008333333333333333)))))) + Float64(z + Float64(Float64(y * y) * Float64(z * Float64(-0.5 + Float64(Float64(y * y) * 0.041666666666666664))))));
	else
		tmp = Float64(x + z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -7.5)
		tmp = x + z;
	elseif (y <= 6.2e-25)
		tmp = (x + (y * (1.0 + ((y * y) * (-0.16666666666666666 + ((y * y) * 0.008333333333333333)))))) + (z + ((y * y) * (z * (-0.5 + ((y * y) * 0.041666666666666664)))));
	else
		tmp = x + z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -7.5], N[(x + z), $MachinePrecision], If[LessEqual[y, 6.2e-25], N[(N[(x + N[(y * N[(1.0 + N[(N[(y * y), $MachinePrecision] * N[(-0.16666666666666666 + N[(N[(y * y), $MachinePrecision] * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(z + N[(N[(y * y), $MachinePrecision] * N[(z * N[(-0.5 + N[(N[(y * y), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -7.5 or 6.19999999999999989e-25 < y

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 42.9%

         \[\leadsto \mathsf{+.f64}\left(z, \color{blue}{x}\right) \]

   5. Simplified 42.9%

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

   if -7.5 < y < 6.19999999999999989e-25

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(x + y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)\right)\right)}, \mathsf{*.f64}\left(z, \mathsf{cos.f64}\left(y\right)\right)\right) \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{z}, \mathsf{cos.f64}\left(y\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \left(1 + {y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)\right)\right)\right), \mathsf{*.f64}\left(z, \mathsf{cos.f64}\left(y\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \left({y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(z, \mathsf{cos.f64}\left(y\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({y}^{2}\right), \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(z, \mathsf{cos.f64}\left(y\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(y \cdot y\right), \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(z, \mathsf{cos.f64}\left(y\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(z, \mathsf{cos.f64}\left(y\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\frac{1}{120} \cdot {y}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(z, \mathsf{cos.f64}\left(y\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\frac{1}{120} \cdot {y}^{2} + \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(z, \mathsf{cos.f64}\left(y\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\frac{-1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(z, \mathsf{cos.f64}\left(y\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{6}, \left(\frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(z, \mathsf{cos.f64}\left(y\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{6}, \left({y}^{2} \cdot \frac{1}{120}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(z, \mathsf{cos.f64}\left(y\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left({y}^{2}\right), \frac{1}{120}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(z, \mathsf{cos.f64}\left(y\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left(y \cdot y\right), \frac{1}{120}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(z, \mathsf{cos.f64}\left(y\right)\right)\right) \]

      14. *-lowering-*.f64 100.0%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(z, \mathsf{cos.f64}\left(y\right)\right)\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\left(x + y \cdot \left(1 + \left(y \cdot y\right) \cdot \left(-0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)} + z \cdot \cos y \]

   6. Taylor expanded in y around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right)\right), \color{blue}{\left(z + {y}^{2} \cdot \left(\frac{-1}{2} \cdot z + \frac{1}{24} \cdot \left({y}^{2} \cdot z\right)\right)\right)}\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(z, \color{blue}{\left({y}^{2} \cdot \left(\frac{-1}{2} \cdot z + \frac{1}{24} \cdot \left({y}^{2} \cdot z\right)\right)\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\left(\frac{-1}{2} \cdot z + \frac{1}{24} \cdot \left({y}^{2} \cdot z\right)\right)}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(\left(y \cdot y\right), \left(\color{blue}{\frac{-1}{2} \cdot z} + \frac{1}{24} \cdot \left({y}^{2} \cdot z\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\color{blue}{\frac{-1}{2} \cdot z} + \frac{1}{24} \cdot \left({y}^{2} \cdot z\right)\right)\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\frac{-1}{2} \cdot z + \left(\frac{1}{24} \cdot {y}^{2}\right) \cdot \color{blue}{z}\right)\right)\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(z \cdot \color{blue}{\left(\frac{-1}{2} + \frac{1}{24} \cdot {y}^{2}\right)}\right)\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(z \cdot \left(\frac{1}{24} \cdot {y}^{2} + \color{blue}{\frac{-1}{2}}\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(z \cdot \left(\frac{1}{24} \cdot {y}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right)\right)\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(z \cdot \left(\frac{1}{24} \cdot {y}^{2} - \color{blue}{\frac{1}{2}}\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(z, \color{blue}{\left(\frac{1}{24} \cdot {y}^{2} - \frac{1}{2}\right)}\right)\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(z, \left(\frac{1}{24} \cdot {y}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(z, \left(\frac{1}{24} \cdot {y}^{2} + \frac{-1}{2}\right)\right)\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(z, \left(\frac{-1}{2} + \color{blue}{\frac{1}{24} \cdot {y}^{2}}\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left(\frac{1}{24} \cdot {y}^{2}\right)}\right)\right)\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{-1}{2}, \left({y}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left(y \cdot y\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f64 99.5%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 99.5%

      \[\leadsto \left(x + y \cdot \left(1 + \left(y \cdot y\right) \cdot \left(-0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right) + \color{blue}{\left(z + \left(y \cdot y\right) \cdot \left(z \cdot \left(-0.5 + \left(y \cdot y\right) \cdot 0.041666666666666664\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.5:\\ \;\;\;\;x + z\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{-25}:\\ \;\;\;\;\left(x + y \cdot \left(1 + \left(y \cdot y\right) \cdot \left(-0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right) + \left(z + \left(y \cdot y\right) \cdot \left(z \cdot \left(-0.5 + \left(y \cdot y\right) \cdot 0.041666666666666664\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + z\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 70.2% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -370:\\ \;\;\;\;x + z\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{-25}:\\ \;\;\;\;\left(x + z\right) + y \cdot \left(1 + y \cdot \left(z \cdot -0.5 + y \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -370.0)
   (+ x z)
   (if (<= y 6.2e-25)
     (+ (+ x z) (* y (+ 1.0 (* y (+ (* z -0.5) (* y -0.16666666666666666))))))
     (+ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -370.0) {
		tmp = x + z;
	} else if (y <= 6.2e-25) {
		tmp = (x + z) + (y * (1.0 + (y * ((z * -0.5) + (y * -0.16666666666666666)))));
	} else {
		tmp = x + z;
	}
	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 (y <= (-370.0d0)) then
        tmp = x + z
    else if (y <= 6.2d-25) then
        tmp = (x + z) + (y * (1.0d0 + (y * ((z * (-0.5d0)) + (y * (-0.16666666666666666d0))))))
    else
        tmp = x + z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -370.0) {
		tmp = x + z;
	} else if (y <= 6.2e-25) {
		tmp = (x + z) + (y * (1.0 + (y * ((z * -0.5) + (y * -0.16666666666666666)))));
	} else {
		tmp = x + z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -370.0:
		tmp = x + z
	elif y <= 6.2e-25:
		tmp = (x + z) + (y * (1.0 + (y * ((z * -0.5) + (y * -0.16666666666666666)))))
	else:
		tmp = x + z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -370.0)
		tmp = Float64(x + z);
	elseif (y <= 6.2e-25)
		tmp = Float64(Float64(x + z) + Float64(y * Float64(1.0 + Float64(y * Float64(Float64(z * -0.5) + Float64(y * -0.16666666666666666))))));
	else
		tmp = Float64(x + z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -370.0)
		tmp = x + z;
	elseif (y <= 6.2e-25)
		tmp = (x + z) + (y * (1.0 + (y * ((z * -0.5) + (y * -0.16666666666666666)))));
	else
		tmp = x + z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -370.0], N[(x + z), $MachinePrecision], If[LessEqual[y, 6.2e-25], N[(N[(x + z), $MachinePrecision] + N[(y * N[(1.0 + N[(y * N[(N[(z * -0.5), $MachinePrecision] + N[(y * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -370 or 6.19999999999999989e-25 < y

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 42.9%

         \[\leadsto \mathsf{+.f64}\left(z, \color{blue}{x}\right) \]

   5. Simplified 42.9%

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

   if -370 < y < 6.19999999999999989e-25

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + \left(z + y \cdot \left(1 + y \cdot \left(\frac{-1}{2} \cdot z + \frac{-1}{6} \cdot y\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \left(x + z\right) + \color{blue}{y \cdot \left(1 + y \cdot \left(\frac{-1}{2} \cdot z + \frac{-1}{6} \cdot y\right)\right)} \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(x + z\right), \color{blue}{\left(y \cdot \left(1 + y \cdot \left(\frac{-1}{2} \cdot z + \frac{-1}{6} \cdot y\right)\right)\right)}\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(z + x\right), \left(\color{blue}{y} \cdot \left(1 + y \cdot \left(\frac{-1}{2} \cdot z + \frac{-1}{6} \cdot y\right)\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(z, x\right), \left(\color{blue}{y} \cdot \left(1 + y \cdot \left(\frac{-1}{2} \cdot z + \frac{-1}{6} \cdot y\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(z, x\right), \mathsf{*.f64}\left(y, \color{blue}{\left(1 + y \cdot \left(\frac{-1}{2} \cdot z + \frac{-1}{6} \cdot y\right)\right)}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(z, x\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \color{blue}{\left(y \cdot \left(\frac{-1}{2} \cdot z + \frac{-1}{6} \cdot y\right)\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(z, x\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{-1}{2} \cdot z + \frac{-1}{6} \cdot y\right)}\right)\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(z, x\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{-1}{2} \cdot z\right), \color{blue}{\left(\frac{-1}{6} \cdot y\right)}\right)\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(z, x\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\left(z \cdot \frac{-1}{2}\right), \left(\color{blue}{\frac{-1}{6}} \cdot y\right)\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(z, x\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \frac{-1}{2}\right), \left(\color{blue}{\frac{-1}{6}} \cdot y\right)\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(z, x\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \frac{-1}{2}\right), \left(y \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 99.5%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(z, x\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \frac{-1}{2}\right), \mathsf{*.f64}\left(y, \color{blue}{\frac{-1}{6}}\right)\right)\right)\right)\right)\right) \]

   5. Simplified 99.5%

      \[\leadsto \color{blue}{\left(z + x\right) + y \cdot \left(1 + y \cdot \left(z \cdot -0.5 + y \cdot -0.16666666666666666\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -370:\\ \;\;\;\;x + z\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{-25}:\\ \;\;\;\;\left(x + z\right) + y \cdot \left(1 + y \cdot \left(z \cdot -0.5 + y \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + z\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 70.2% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.6:\\ \;\;\;\;x + z\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{-25}:\\ \;\;\;\;\left(x + y\right) + z \cdot \left(1 + y \cdot \left(y \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -4.6)
   (+ x z)
   (if (<= y 6.2e-25) (+ (+ x y) (* z (+ 1.0 (* y (* y -0.5))))) (+ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -4.6) {
		tmp = x + z;
	} else if (y <= 6.2e-25) {
		tmp = (x + y) + (z * (1.0 + (y * (y * -0.5))));
	} else {
		tmp = x + z;
	}
	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 (y <= (-4.6d0)) then
        tmp = x + z
    else if (y <= 6.2d-25) then
        tmp = (x + y) + (z * (1.0d0 + (y * (y * (-0.5d0)))))
    else
        tmp = x + z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -4.6) {
		tmp = x + z;
	} else if (y <= 6.2e-25) {
		tmp = (x + y) + (z * (1.0 + (y * (y * -0.5))));
	} else {
		tmp = x + z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -4.6:
		tmp = x + z
	elif y <= 6.2e-25:
		tmp = (x + y) + (z * (1.0 + (y * (y * -0.5))))
	else:
		tmp = x + z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -4.6)
		tmp = Float64(x + z);
	elseif (y <= 6.2e-25)
		tmp = Float64(Float64(x + y) + Float64(z * Float64(1.0 + Float64(y * Float64(y * -0.5)))));
	else
		tmp = Float64(x + z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -4.6)
		tmp = x + z;
	elseif (y <= 6.2e-25)
		tmp = (x + y) + (z * (1.0 + (y * (y * -0.5))));
	else
		tmp = x + z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -4.6], N[(x + z), $MachinePrecision], If[LessEqual[y, 6.2e-25], N[(N[(x + y), $MachinePrecision] + N[(z * N[(1.0 + N[(y * N[(y * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.5999999999999996 or 6.19999999999999989e-25 < y

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 42.9%

         \[\leadsto \mathsf{+.f64}\left(z, \color{blue}{x}\right) \]

   5. Simplified 42.9%

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

   if -4.5999999999999996 < y < 6.19999999999999989e-25

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + \left(z + y \cdot \left(1 + \frac{-1}{2} \cdot \left(y \cdot z\right)\right)\right)} \]
   4. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto x + \left(z + \left(1 \cdot y + \color{blue}{\left(\frac{-1}{2} \cdot \left(y \cdot z\right)\right) \cdot y}\right)\right) \]

      2. *-lft-identity N/A

         \[\leadsto x + \left(z + \left(y + \color{blue}{\left(\frac{-1}{2} \cdot \left(y \cdot z\right)\right)} \cdot y\right)\right) \]

      3. associate-+r+ N/A

         \[\leadsto x + \left(\left(z + y\right) + \color{blue}{\left(\frac{-1}{2} \cdot \left(y \cdot z\right)\right) \cdot y}\right) \]

      4. +-commutative N/A

         \[\leadsto x + \left(\left(y + z\right) + \color{blue}{\left(\frac{-1}{2} \cdot \left(y \cdot z\right)\right)} \cdot y\right) \]

      5. *-commutative N/A

         \[\leadsto x + \left(\left(y + z\right) + \left(\frac{-1}{2} \cdot \left(z \cdot y\right)\right) \cdot y\right) \]

      6. associate-*r* N/A

         \[\leadsto x + \left(\left(y + z\right) + \left(\left(\frac{-1}{2} \cdot z\right) \cdot y\right) \cdot y\right) \]

      7. associate-*r* N/A

         \[\leadsto x + \left(\left(y + z\right) + \left(\frac{-1}{2} \cdot z\right) \cdot \color{blue}{\left(y \cdot y\right)}\right) \]

      8. unpow2 N/A

         \[\leadsto x + \left(\left(y + z\right) + \left(\frac{-1}{2} \cdot z\right) \cdot {y}^{\color{blue}{2}}\right) \]

      9. associate-+r+ N/A

         \[\leadsto x + \left(y + \color{blue}{\left(z + \left(\frac{-1}{2} \cdot z\right) \cdot {y}^{2}\right)}\right) \]

      10. associate-+r+ N/A

          \[\leadsto \left(x + y\right) + \color{blue}{\left(z + \left(\frac{-1}{2} \cdot z\right) \cdot {y}^{2}\right)} \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\left(x + y\right), \color{blue}{\left(z + \left(\frac{-1}{2} \cdot z\right) \cdot {y}^{2}\right)}\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\left(y + x\right), \left(\color{blue}{z} + \left(\frac{-1}{2} \cdot z\right) \cdot {y}^{2}\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(y, x\right), \left(\color{blue}{z} + \left(\frac{-1}{2} \cdot z\right) \cdot {y}^{2}\right)\right) \]

      14. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(y, x\right), \left(z + \frac{-1}{2} \cdot \color{blue}{\left(z \cdot {y}^{2}\right)}\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(y, x\right), \left(z + \frac{-1}{2} \cdot \left({y}^{2} \cdot \color{blue}{z}\right)\right)\right) \]

      16. *-lft-identity N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(y, x\right), \left(1 \cdot z + \color{blue}{\frac{-1}{2}} \cdot \left({y}^{2} \cdot z\right)\right)\right) \]

      17. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(y, x\right), \left(1 \cdot z + \left(\frac{-1}{2} \cdot {y}^{2}\right) \cdot \color{blue}{z}\right)\right) \]

      18. distribute-rgt-in N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(y, x\right), \left(z \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {y}^{2}\right)}\right)\right) \]

      19. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(y, x\right), \mathsf{*.f64}\left(z, \color{blue}{\left(1 + \frac{-1}{2} \cdot {y}^{2}\right)}\right)\right) \]

      20. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(y, x\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot {y}^{2}\right)}\right)\right)\right) \]

      21. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(y, x\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \left(\frac{-1}{2} \cdot \left(y \cdot \color{blue}{y}\right)\right)\right)\right)\right) \]

   5. Simplified 99.5%

      \[\leadsto \color{blue}{\left(y + x\right) + z \cdot \left(1 + y \cdot \left(y \cdot -0.5\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.6:\\ \;\;\;\;x + z\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{-25}:\\ \;\;\;\;\left(x + y\right) + z \cdot \left(1 + y \cdot \left(y \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + z\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 70.4% accurate, 13.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.22 \cdot 10^{+71}:\\ \;\;\;\;x + z\\ \mathbf{elif}\;y \leq 1.26 \cdot 10^{+21}:\\ \;\;\;\;z + \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;x + z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.22e+71) (+ x z) (if (<= y 1.26e+21) (+ z (+ x y)) (+ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.22e+71) {
		tmp = x + z;
	} else if (y <= 1.26e+21) {
		tmp = z + (x + y);
	} else {
		tmp = x + z;
	}
	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 (y <= (-1.22d+71)) then
        tmp = x + z
    else if (y <= 1.26d+21) then
        tmp = z + (x + y)
    else
        tmp = x + z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.22e+71) {
		tmp = x + z;
	} else if (y <= 1.26e+21) {
		tmp = z + (x + y);
	} else {
		tmp = x + z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.22e+71:
		tmp = x + z
	elif y <= 1.26e+21:
		tmp = z + (x + y)
	else:
		tmp = x + z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.22e+71)
		tmp = Float64(x + z);
	elseif (y <= 1.26e+21)
		tmp = Float64(z + Float64(x + y));
	else
		tmp = Float64(x + z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1.22e+71)
		tmp = x + z;
	elseif (y <= 1.26e+21)
		tmp = z + (x + y);
	else
		tmp = x + z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.22e+71], N[(x + z), $MachinePrecision], If[LessEqual[y, 1.26e+21], N[(z + N[(x + y), $MachinePrecision]), $MachinePrecision], N[(x + z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.22000000000000001e71 or 1.26e21 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 42.3%

         \[\leadsto \mathsf{+.f64}\left(z, \color{blue}{x}\right) \]

   5. Simplified 42.3%

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

   if -1.22000000000000001e71 < y < 1.26e21

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(z, \color{blue}{\left(x + y\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(z, \left(y + \color{blue}{x}\right)\right) \]

      5. +-lowering-+.f64 92.9%

         \[\leadsto \mathsf{+.f64}\left(z, \mathsf{+.f64}\left(y, \color{blue}{x}\right)\right) \]

   5. Simplified 92.9%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.22 \cdot 10^{+71}:\\ \;\;\;\;x + z\\ \mathbf{elif}\;y \leq 1.26 \cdot 10^{+21}:\\ \;\;\;\;z + \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;x + z\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 54.9% accurate, 18.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{-39}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{+18}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -8e-39) x (if (<= x 2.3e+18) z x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -8e-39) {
		tmp = x;
	} else if (x <= 2.3e+18) {
		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 <= (-8d-39)) then
        tmp = x
    else if (x <= 2.3d+18) 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 <= -8e-39) {
		tmp = x;
	} else if (x <= 2.3e+18) {
		tmp = z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -8e-39:
		tmp = x
	elif x <= 2.3e+18:
		tmp = z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -8e-39)
		tmp = x;
	elseif (x <= 2.3e+18)
		tmp = z;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -8e-39)
		tmp = x;
	elseif (x <= 2.3e+18)
		tmp = z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -8e-39], x, If[LessEqual[x, 2.3e+18], z, x]]

Derivation

1. Split input into 2 regimes

2. if x < -7.99999999999999943e-39 or 2.3e18 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 65.3%

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

   if -7.99999999999999943e-39 < x < 2.3e18

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \color{blue}{\cos y}\right) \]

      2. cos-lowering-cos.f64 65.3%

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{cos.f64}\left(y\right)\right) \]

   5. Simplified 65.3%

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

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{z} \]
   7. Step-by-step derivation

   8. Simplified 40.3%

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

Alternative 12: 44.6% accurate, 18.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{-65}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-159}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -2.4e-65) x (if (<= x 1.4e-159) y x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.4e-65) {
		tmp = x;
	} else if (x <= 1.4e-159) {
		tmp = y;
	} 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 <= (-2.4d-65)) then
        tmp = x
    else if (x <= 1.4d-159) then
        tmp = y
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.4e-65) {
		tmp = x;
	} else if (x <= 1.4e-159) {
		tmp = y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -2.4e-65:
		tmp = x
	elif x <= 1.4e-159:
		tmp = y
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -2.4e-65)
		tmp = x;
	elseif (x <= 1.4e-159)
		tmp = y;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -2.4e-65)
		tmp = x;
	elseif (x <= 1.4e-159)
		tmp = y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -2.4e-65], x, If[LessEqual[x, 1.4e-159], y, x]]

Derivation

1. Split input into 2 regimes

2. if x < -2.4000000000000002e-65 or 1.4000000000000001e-159 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 54.1%

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

   if -2.4000000000000002e-65 < x < 1.4000000000000001e-159

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\sin y, \color{blue}{x}\right) \]

      3. sin-lowering-sin.f64 37.2%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{sin.f64}\left(y\right), x\right) \]

   5. Simplified 37.2%

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

   6. Taylor expanded in y around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{y}, x\right) \]
   7. Step-by-step derivation

   8. Simplified 17.4%

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

   9. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y} \]
   10. Step-by-step derivation

   11. Simplified 16.1%

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

Alternative 13: 66.3% accurate, 69.0× speedup

Math

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

FPCore

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

C

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

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 + z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. +-lowering-+.f64 66.6%

      \[\leadsto \mathsf{+.f64}\left(z, \color{blue}{x}\right) \]

5. Simplified 66.6%

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

6. Final simplification 66.6%

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

Alternative 14: 42.7% 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 99.9%

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

3. Taylor expanded in x around inf

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

5. Simplified 39.1%

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

Reproduce

herbie shell --seed 2024160 
(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))))