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

Percentage Accurate: 99.9% → 99.9%

Time: 10.1s

Alternatives: 13

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return (x + cos(y)) - (z * 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 = (x + cos(y)) - (z * sin(y))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return (x + cos(y)) - (z * 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 = (x + cos(y)) - (z * sin(y))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. cancel-sign-sub-inv 99.9%

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

   2. +-commutative 99.9%

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

   3. associate-+l+ 99.9%

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

   4. +-commutative 99.9%

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

   5. distribute-lft-neg-out 99.9%

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

   6. distribute-rgt-neg-in 99.9%

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

   7. sin-neg 99.9%

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

   8. fma-define 99.9%

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

   9. sin-neg 99.9%

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

3. Simplified 99.9%

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

5. Final simplification 99.9%

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

Alternative 2: 86.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{+24}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3.7 \cdot 10^{+97}:\\ \;\;\;\;\cos y - z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -6e+24) x (if (<= x 3.7e+97) (- (cos y) (* z (sin y))) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -6e+24) {
		tmp = x;
	} else if (x <= 3.7e+97) {
		tmp = cos(y) - (z * sin(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 <= (-6d+24)) then
        tmp = x
    else if (x <= 3.7d+97) then
        tmp = cos(y) - (z * sin(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 <= -6e+24) {
		tmp = x;
	} else if (x <= 3.7e+97) {
		tmp = Math.cos(y) - (z * Math.sin(y));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -6e+24:
		tmp = x
	elif x <= 3.7e+97:
		tmp = math.cos(y) - (z * math.sin(y))
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -6e+24)
		tmp = x;
	elseif (x <= 3.7e+97)
		tmp = Float64(cos(y) - Float64(z * sin(y)));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -6e+24)
		tmp = x;
	elseif (x <= 3.7e+97)
		tmp = cos(y) - (z * sin(y));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -6e+24], x, If[LessEqual[x, 3.7e+97], N[(N[Cos[y], $MachinePrecision] - N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if x < -5.9999999999999999e24 or 3.70000000000000001e97 < x

   1. Initial program 99.9%

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

      1. cancel-sign-sub-inv 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-+l+ 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-lft-neg-out 99.9%

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

      6. distribute-rgt-neg-in 99.9%

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

      7. sin-neg 99.9%

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

      8. fma-define 100.0%

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

      9. sin-neg 100.0%

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

   3. Simplified 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-undefine 99.9%

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

      3. distribute-rgt-neg-in 99.9%

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

      4. associate-+l+ 99.9%

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

      5. distribute-lft-neg-in 99.9%

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

      6. add-cube-cbrt 99.7%

         \[\leadsto \left(-z\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\sin y} \cdot \sqrt[3]{\sin y}\right) \cdot \sqrt[3]{\sin y}\right)} + \left(x + \cos y\right) \]

      7. associate-*r* 99.7%

         \[\leadsto \color{blue}{\left(\left(-z\right) \cdot \left(\sqrt[3]{\sin y} \cdot \sqrt[3]{\sin y}\right)\right) \cdot \sqrt[3]{\sin y}} + \left(x + \cos y\right) \]

      8. fma-define 99.7%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(-z\right) \cdot \left(\sqrt[3]{\sin y} \cdot \sqrt[3]{\sin y}\right), \sqrt[3]{\sin y}, x + \cos y\right)} \]

      9. pow2 99.7%

         \[\leadsto \mathsf{fma}\left(\left(-z\right) \cdot \color{blue}{{\left(\sqrt[3]{\sin y}\right)}^{2}}, \sqrt[3]{\sin y}, x + \cos y\right) \]

   6. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(-z\right) \cdot {\left(\sqrt[3]{\sin y}\right)}^{2}, \sqrt[3]{\sin y}, x + \cos y\right)} \]

   7. Taylor expanded in x around inf 85.0%

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

   if -5.9999999999999999e24 < x < 3.70000000000000001e97

   1. Initial program 99.9%

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

      1. cancel-sign-sub-inv 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-+l+ 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-lft-neg-out 99.9%

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

      6. distribute-rgt-neg-in 99.9%

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

      7. sin-neg 99.9%

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

      8. fma-define 99.9%

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

      9. sin-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 94.9%

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

      1. neg-mul-1 94.9%

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

      2. sub-neg 94.9%

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

   7. Simplified 94.9%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{+24}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3.7 \cdot 10^{+97}:\\ \;\;\;\;\cos y - z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return (cos(y) + x) - (z * 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 = (cos(y) + x) - (z * sin(y))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Final simplification 99.9%

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

Alternative 4: 81.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(-\sin y\right)\\ \mathbf{if}\;z \leq -8.6 \cdot 10^{+204}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq -7.8 \cdot 10^{+64}:\\ \;\;\;\;\left(x + 1\right) - y \cdot z\\ \mathbf{elif}\;z \leq -1.6 \cdot 10^{+54} \lor \neg \left(z \leq 1.06 \cdot 10^{+70}\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\cos y + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (- (sin y)))))
   (if (<= z -8.6e+204)
     t_0
     (if (<= z -7.8e+64)
       (- (+ x 1.0) (* y z))
       (if (or (<= z -1.6e+54) (not (<= z 1.06e+70))) t_0 (+ (cos y) x))))))

C

double code(double x, double y, double z) {
	double t_0 = z * -sin(y);
	double tmp;
	if (z <= -8.6e+204) {
		tmp = t_0;
	} else if (z <= -7.8e+64) {
		tmp = (x + 1.0) - (y * z);
	} else if ((z <= -1.6e+54) || !(z <= 1.06e+70)) {
		tmp = t_0;
	} else {
		tmp = cos(y) + 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) :: t_0
    real(8) :: tmp
    t_0 = z * -sin(y)
    if (z <= (-8.6d+204)) then
        tmp = t_0
    else if (z <= (-7.8d+64)) then
        tmp = (x + 1.0d0) - (y * z)
    else if ((z <= (-1.6d+54)) .or. (.not. (z <= 1.06d+70))) then
        tmp = t_0
    else
        tmp = cos(y) + x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = z * -Math.sin(y);
	double tmp;
	if (z <= -8.6e+204) {
		tmp = t_0;
	} else if (z <= -7.8e+64) {
		tmp = (x + 1.0) - (y * z);
	} else if ((z <= -1.6e+54) || !(z <= 1.06e+70)) {
		tmp = t_0;
	} else {
		tmp = Math.cos(y) + x;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * -math.sin(y)
	tmp = 0
	if z <= -8.6e+204:
		tmp = t_0
	elif z <= -7.8e+64:
		tmp = (x + 1.0) - (y * z)
	elif (z <= -1.6e+54) or not (z <= 1.06e+70):
		tmp = t_0
	else:
		tmp = math.cos(y) + x
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(-sin(y)))
	tmp = 0.0
	if (z <= -8.6e+204)
		tmp = t_0;
	elseif (z <= -7.8e+64)
		tmp = Float64(Float64(x + 1.0) - Float64(y * z));
	elseif ((z <= -1.6e+54) || !(z <= 1.06e+70))
		tmp = t_0;
	else
		tmp = Float64(cos(y) + x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * -sin(y);
	tmp = 0.0;
	if (z <= -8.6e+204)
		tmp = t_0;
	elseif (z <= -7.8e+64)
		tmp = (x + 1.0) - (y * z);
	elseif ((z <= -1.6e+54) || ~((z <= 1.06e+70)))
		tmp = t_0;
	else
		tmp = cos(y) + x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * (-N[Sin[y], $MachinePrecision])), $MachinePrecision]}, If[LessEqual[z, -8.6e+204], t$95$0, If[LessEqual[z, -7.8e+64], N[(N[(x + 1.0), $MachinePrecision] - N[(y * z), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[z, -1.6e+54], N[Not[LessEqual[z, 1.06e+70]], $MachinePrecision]], t$95$0, N[(N[Cos[y], $MachinePrecision] + x), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -8.6e204 or -7.7999999999999996e64 < z < -1.6e54 or 1.06e70 < z

   1. Initial program 99.7%

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

      1. cancel-sign-sub-inv 99.7%

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

      2. +-commutative 99.7%

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

      3. associate-+l+ 99.7%

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

      4. +-commutative 99.7%

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

      5. distribute-lft-neg-out 99.7%

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

      6. distribute-rgt-neg-in 99.7%

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

      7. sin-neg 99.7%

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

      8. fma-define 99.8%

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

      9. sin-neg 99.8%

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

   3. Simplified 99.8%

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

      1. +-commutative 99.8%

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

      2. fma-undefine 99.7%

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

      3. distribute-rgt-neg-in 99.7%

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

      4. associate-+l+ 99.7%

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

      5. distribute-lft-neg-in 99.7%

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

      6. add-cube-cbrt 98.2%

         \[\leadsto \left(-z\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\sin y} \cdot \sqrt[3]{\sin y}\right) \cdot \sqrt[3]{\sin y}\right)} + \left(x + \cos y\right) \]

      7. associate-*r* 98.2%

         \[\leadsto \color{blue}{\left(\left(-z\right) \cdot \left(\sqrt[3]{\sin y} \cdot \sqrt[3]{\sin y}\right)\right) \cdot \sqrt[3]{\sin y}} + \left(x + \cos y\right) \]

      8. fma-define 98.3%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(-z\right) \cdot \left(\sqrt[3]{\sin y} \cdot \sqrt[3]{\sin y}\right), \sqrt[3]{\sin y}, x + \cos y\right)} \]

      9. pow2 98.3%

         \[\leadsto \mathsf{fma}\left(\left(-z\right) \cdot \color{blue}{{\left(\sqrt[3]{\sin y}\right)}^{2}}, \sqrt[3]{\sin y}, x + \cos y\right) \]

   6. Applied egg-rr 98.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(-z\right) \cdot {\left(\sqrt[3]{\sin y}\right)}^{2}, \sqrt[3]{\sin y}, x + \cos y\right)} \]

   7. Taylor expanded in z around inf 82.2%

      \[\leadsto \color{blue}{-1 \cdot \left(z \cdot \sin y\right)} \]
   8. Step-by-step derivation

      1. mul-1-neg 82.2%

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

      2. *-commutative 82.2%

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

      3. distribute-rgt-neg-in 82.2%

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

   9. Simplified 82.2%

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

   if -8.6e204 < z < -7.7999999999999996e64

   1. Initial program 99.9%

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

      1. cancel-sign-sub-inv 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-+l+ 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-lft-neg-out 99.9%

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

      6. distribute-rgt-neg-in 99.9%

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

      7. sin-neg 99.9%

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

      8. fma-define 99.9%

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

      9. sin-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in y around 0 65.2%

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

      1. associate-+r+ 65.2%

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

      2. mul-1-neg 65.2%

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

      3. unsub-neg 65.2%

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

      4. +-commutative 65.2%

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

      5. *-commutative 65.2%

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

   7. Simplified 65.2%

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

   if -1.6e54 < z < 1.06e70

   1. Initial program 100.0%

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

      1. cancel-sign-sub-inv 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-+l+ 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-lft-neg-out 100.0%

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

      6. distribute-rgt-neg-in 100.0%

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

      7. sin-neg 100.0%

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

      8. fma-define 100.0%

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

      9. sin-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in z around 0 95.2%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8.6 \cdot 10^{+204}:\\ \;\;\;\;z \cdot \left(-\sin y\right)\\ \mathbf{elif}\;z \leq -7.8 \cdot 10^{+64}:\\ \;\;\;\;\left(x + 1\right) - y \cdot z\\ \mathbf{elif}\;z \leq -1.6 \cdot 10^{+54} \lor \neg \left(z \leq 1.06 \cdot 10^{+70}\right):\\ \;\;\;\;z \cdot \left(-\sin y\right)\\ \mathbf{else}:\\ \;\;\;\;\cos y + x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 69.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{+58}:\\ \;\;\;\;x + 1\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{+30}:\\ \;\;\;\;\left(x + 1\right) - y \cdot z\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{+175}:\\ \;\;\;\;\cos y\\ \mathbf{else}:\\ \;\;\;\;x + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.2e+58)
   (+ x 1.0)
   (if (<= y 1.7e+30)
     (- (+ x 1.0) (* y z))
     (if (<= y 1.4e+175) (cos y) (+ x 1.0)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.2e+58) {
		tmp = x + 1.0;
	} else if (y <= 1.7e+30) {
		tmp = (x + 1.0) - (y * z);
	} else if (y <= 1.4e+175) {
		tmp = cos(y);
	} else {
		tmp = x + 1.0;
	}
	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) :: tmp
    if (y <= (-1.2d+58)) then
        tmp = x + 1.0d0
    else if (y <= 1.7d+30) then
        tmp = (x + 1.0d0) - (y * z)
    else if (y <= 1.4d+175) then
        tmp = cos(y)
    else
        tmp = x + 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.2e+58) {
		tmp = x + 1.0;
	} else if (y <= 1.7e+30) {
		tmp = (x + 1.0) - (y * z);
	} else if (y <= 1.4e+175) {
		tmp = Math.cos(y);
	} else {
		tmp = x + 1.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.2e+58:
		tmp = x + 1.0
	elif y <= 1.7e+30:
		tmp = (x + 1.0) - (y * z)
	elif y <= 1.4e+175:
		tmp = math.cos(y)
	else:
		tmp = x + 1.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.2e+58)
		tmp = Float64(x + 1.0);
	elseif (y <= 1.7e+30)
		tmp = Float64(Float64(x + 1.0) - Float64(y * z));
	elseif (y <= 1.4e+175)
		tmp = cos(y);
	else
		tmp = Float64(x + 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1.2e+58)
		tmp = x + 1.0;
	elseif (y <= 1.7e+30)
		tmp = (x + 1.0) - (y * z);
	elseif (y <= 1.4e+175)
		tmp = cos(y);
	else
		tmp = x + 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.2e+58], N[(x + 1.0), $MachinePrecision], If[LessEqual[y, 1.7e+30], N[(N[(x + 1.0), $MachinePrecision] - N[(y * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.4e+175], N[Cos[y], $MachinePrecision], N[(x + 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.2e58 or 1.4000000000000001e175 < y

   1. Initial program 99.8%

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

      1. cancel-sign-sub-inv 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+l+ 99.8%

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

      4. +-commutative 99.8%

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

      5. distribute-lft-neg-out 99.8%

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

      6. distribute-rgt-neg-in 99.8%

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

      7. sin-neg 99.8%

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

      8. fma-define 99.8%

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

      9. sin-neg 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around 0 41.5%

      \[\leadsto \color{blue}{1 + x} \]
   6. Step-by-step derivation

      1. +-commutative 41.5%

         \[\leadsto \color{blue}{x + 1} \]

   7. Simplified 41.5%

      \[\leadsto \color{blue}{x + 1} \]

   if -1.2e58 < y < 1.7000000000000001e30

   1. Initial program 100.0%

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

      1. cancel-sign-sub-inv 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-+l+ 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-lft-neg-out 100.0%

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

      6. distribute-rgt-neg-in 100.0%

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

      7. sin-neg 100.0%

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

      8. fma-define 100.0%

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

      9. sin-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 92.1%

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

      1. associate-+r+ 92.1%

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

      2. mul-1-neg 92.1%

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

      3. unsub-neg 92.1%

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

      4. +-commutative 92.1%

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

      5. *-commutative 92.1%

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

   7. Simplified 92.1%

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

   if 1.7000000000000001e30 < y < 1.4000000000000001e175

   1. Initial program 99.9%

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

      1. cancel-sign-sub-inv 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-+l+ 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-lft-neg-out 99.9%

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

      6. distribute-rgt-neg-in 99.9%

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

      7. sin-neg 99.9%

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

      8. fma-define 99.9%

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

      9. sin-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 84.6%

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

      1. neg-mul-1 84.6%

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

      2. sub-neg 84.6%

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

   7. Simplified 84.6%

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

   8. Taylor expanded in z around 0 40.9%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{+58}:\\ \;\;\;\;x + 1\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{+30}:\\ \;\;\;\;\left(x + 1\right) - y \cdot z\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{+175}:\\ \;\;\;\;\cos y\\ \mathbf{else}:\\ \;\;\;\;x + 1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.15 \cdot 10^{+19} \lor \neg \left(y \leq 11000000\right):\\ \;\;\;\;\cos y + x\\ \mathbf{else}:\\ \;\;\;\;1 + \left(x + y \cdot \left(y \cdot \left(0.16666666666666666 \cdot \left(y \cdot z\right) - 0.5\right) - z\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -3.15e+19) (not (<= y 11000000.0)))
   (+ (cos y) x)
   (+ 1.0 (+ x (* y (- (* y (- (* 0.16666666666666666 (* y z)) 0.5)) z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -3.15e+19) || !(y <= 11000000.0)) {
		tmp = cos(y) + x;
	} else {
		tmp = 1.0 + (x + (y * ((y * ((0.16666666666666666 * (y * z)) - 0.5)) - 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 <= (-3.15d+19)) .or. (.not. (y <= 11000000.0d0))) then
        tmp = cos(y) + x
    else
        tmp = 1.0d0 + (x + (y * ((y * ((0.16666666666666666d0 * (y * z)) - 0.5d0)) - z)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -3.15e+19) || !(y <= 11000000.0)) {
		tmp = Math.cos(y) + x;
	} else {
		tmp = 1.0 + (x + (y * ((y * ((0.16666666666666666 * (y * z)) - 0.5)) - z)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -3.15e+19) or not (y <= 11000000.0):
		tmp = math.cos(y) + x
	else:
		tmp = 1.0 + (x + (y * ((y * ((0.16666666666666666 * (y * z)) - 0.5)) - z)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -3.15e+19) || !(y <= 11000000.0))
		tmp = Float64(cos(y) + x);
	else
		tmp = Float64(1.0 + Float64(x + Float64(y * Float64(Float64(y * Float64(Float64(0.16666666666666666 * Float64(y * z)) - 0.5)) - z))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -3.15e+19) || ~((y <= 11000000.0)))
		tmp = cos(y) + x;
	else
		tmp = 1.0 + (x + (y * ((y * ((0.16666666666666666 * (y * z)) - 0.5)) - z)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -3.15e+19], N[Not[LessEqual[y, 11000000.0]], $MachinePrecision]], N[(N[Cos[y], $MachinePrecision] + x), $MachinePrecision], N[(1.0 + N[(x + N[(y * N[(N[(y * N[(N[(0.16666666666666666 * N[(y * z), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.15e19 or 1.1e7 < y

   1. Initial program 99.8%

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

      1. cancel-sign-sub-inv 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+l+ 99.8%

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

      4. +-commutative 99.8%

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

      5. distribute-lft-neg-out 99.8%

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

      6. distribute-rgt-neg-in 99.8%

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

      7. sin-neg 99.8%

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

      8. fma-define 99.8%

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

      9. sin-neg 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in z around 0 53.5%

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

   if -3.15e19 < y < 1.1e7

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 97.0%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.15 \cdot 10^{+19} \lor \neg \left(y \leq 11000000\right):\\ \;\;\;\;\cos y + x\\ \mathbf{else}:\\ \;\;\;\;1 + \left(x + y \cdot \left(y \cdot \left(0.16666666666666666 \cdot \left(y \cdot z\right) - 0.5\right) - z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 70.9% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{+19} \lor \neg \left(y \leq 3000000000\right):\\ \;\;\;\;x + 1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(x + y \cdot \left(y \cdot \left(0.16666666666666666 \cdot \left(y \cdot z\right) - 0.5\right) - z\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -5.5e+19) (not (<= y 3000000000.0)))
   (+ x 1.0)
   (+ 1.0 (+ x (* y (- (* y (- (* 0.16666666666666666 (* y z)) 0.5)) z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -5.5e+19) || !(y <= 3000000000.0)) {
		tmp = x + 1.0;
	} else {
		tmp = 1.0 + (x + (y * ((y * ((0.16666666666666666 * (y * z)) - 0.5)) - 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 <= (-5.5d+19)) .or. (.not. (y <= 3000000000.0d0))) then
        tmp = x + 1.0d0
    else
        tmp = 1.0d0 + (x + (y * ((y * ((0.16666666666666666d0 * (y * z)) - 0.5d0)) - z)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -5.5e+19) || !(y <= 3000000000.0)) {
		tmp = x + 1.0;
	} else {
		tmp = 1.0 + (x + (y * ((y * ((0.16666666666666666 * (y * z)) - 0.5)) - z)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -5.5e+19) or not (y <= 3000000000.0):
		tmp = x + 1.0
	else:
		tmp = 1.0 + (x + (y * ((y * ((0.16666666666666666 * (y * z)) - 0.5)) - z)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -5.5e+19) || !(y <= 3000000000.0))
		tmp = Float64(x + 1.0);
	else
		tmp = Float64(1.0 + Float64(x + Float64(y * Float64(Float64(y * Float64(Float64(0.16666666666666666 * Float64(y * z)) - 0.5)) - z))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -5.5e+19) || ~((y <= 3000000000.0)))
		tmp = x + 1.0;
	else
		tmp = 1.0 + (x + (y * ((y * ((0.16666666666666666 * (y * z)) - 0.5)) - z)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -5.5e+19], N[Not[LessEqual[y, 3000000000.0]], $MachinePrecision]], N[(x + 1.0), $MachinePrecision], N[(1.0 + N[(x + N[(y * N[(N[(y * N[(N[(0.16666666666666666 * N[(y * z), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -5.5e19 or 3e9 < y

   1. Initial program 99.8%

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

      1. cancel-sign-sub-inv 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+l+ 99.8%

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

      4. +-commutative 99.8%

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

      5. distribute-lft-neg-out 99.8%

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

      6. distribute-rgt-neg-in 99.8%

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

      7. sin-neg 99.8%

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

      8. fma-define 99.8%

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

      9. sin-neg 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around 0 35.6%

      \[\leadsto \color{blue}{1 + x} \]
   6. Step-by-step derivation

      1. +-commutative 35.6%

         \[\leadsto \color{blue}{x + 1} \]

   7. Simplified 35.6%

      \[\leadsto \color{blue}{x + 1} \]

   if -5.5e19 < y < 3e9

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 97.0%

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

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{+19} \lor \neg \left(y \leq 3000000000\right):\\ \;\;\;\;x + 1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(x + y \cdot \left(y \cdot \left(0.16666666666666666 \cdot \left(y \cdot z\right) - 0.5\right) - z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 70.8% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{+54} \lor \neg \left(y \leq 22\right):\\ \;\;\;\;x + 1\\ \mathbf{else}:\\ \;\;\;\;\left(x + 1\right) - y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -3e+54) (not (<= y 22.0))) (+ x 1.0) (- (+ x 1.0) (* y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -3e+54) || !(y <= 22.0)) {
		tmp = x + 1.0;
	} else {
		tmp = (x + 1.0) - (y * 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 <= (-3d+54)) .or. (.not. (y <= 22.0d0))) then
        tmp = x + 1.0d0
    else
        tmp = (x + 1.0d0) - (y * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -3e+54) || !(y <= 22.0)) {
		tmp = x + 1.0;
	} else {
		tmp = (x + 1.0) - (y * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -3e+54) or not (y <= 22.0):
		tmp = x + 1.0
	else:
		tmp = (x + 1.0) - (y * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -3e+54) || !(y <= 22.0))
		tmp = Float64(x + 1.0);
	else
		tmp = Float64(Float64(x + 1.0) - Float64(y * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -3e+54) || ~((y <= 22.0)))
		tmp = x + 1.0;
	else
		tmp = (x + 1.0) - (y * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -3e+54], N[Not[LessEqual[y, 22.0]], $MachinePrecision]], N[(x + 1.0), $MachinePrecision], N[(N[(x + 1.0), $MachinePrecision] - N[(y * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.9999999999999999e54 or 22 < y

   1. Initial program 99.8%

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

      1. cancel-sign-sub-inv 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+l+ 99.8%

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

      4. +-commutative 99.8%

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

      5. distribute-lft-neg-out 99.8%

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

      6. distribute-rgt-neg-in 99.8%

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

      7. sin-neg 99.8%

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

      8. fma-define 99.8%

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

      9. sin-neg 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around 0 37.0%

      \[\leadsto \color{blue}{1 + x} \]
   6. Step-by-step derivation

      1. +-commutative 37.0%

         \[\leadsto \color{blue}{x + 1} \]

   7. Simplified 37.0%

      \[\leadsto \color{blue}{x + 1} \]

   if -2.9999999999999999e54 < y < 22

   1. Initial program 100.0%

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

      1. cancel-sign-sub-inv 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-+l+ 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-lft-neg-out 100.0%

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

      6. distribute-rgt-neg-in 100.0%

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

      7. sin-neg 100.0%

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

      8. fma-define 100.0%

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

      9. sin-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 92.7%

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

      1. associate-+r+ 92.7%

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

      2. mul-1-neg 92.7%

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

      3. unsub-neg 92.7%

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

      4. +-commutative 92.7%

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

      5. *-commutative 92.7%

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

   7. Simplified 92.7%

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

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{+54} \lor \neg \left(y \leq 22\right):\\ \;\;\;\;x + 1\\ \mathbf{else}:\\ \;\;\;\;\left(x + 1\right) - y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 65.3% accurate, 13.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{+23}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.55 \cdot 10^{+38}:\\ \;\;\;\;1 - y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -2.9e+23) x (if (<= x 2.55e+38) (- 1.0 (* y z)) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.9e+23) {
		tmp = x;
	} else if (x <= 2.55e+38) {
		tmp = 1.0 - (y * 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 <= (-2.9d+23)) then
        tmp = x
    else if (x <= 2.55d+38) then
        tmp = 1.0d0 - (y * 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 <= -2.9e+23) {
		tmp = x;
	} else if (x <= 2.55e+38) {
		tmp = 1.0 - (y * z);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -2.9e+23:
		tmp = x
	elif x <= 2.55e+38:
		tmp = 1.0 - (y * z)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -2.9e+23)
		tmp = x;
	elseif (x <= 2.55e+38)
		tmp = Float64(1.0 - Float64(y * z));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -2.9e+23)
		tmp = x;
	elseif (x <= 2.55e+38)
		tmp = 1.0 - (y * z);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -2.9e+23], x, If[LessEqual[x, 2.55e+38], N[(1.0 - N[(y * z), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if x < -2.90000000000000013e23 or 2.5500000000000001e38 < x

   1. Initial program 99.9%

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

      1. cancel-sign-sub-inv 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-+l+ 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-lft-neg-out 99.9%

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

      6. distribute-rgt-neg-in 99.9%

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

      7. sin-neg 99.9%

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

      8. fma-define 99.9%

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

      9. sin-neg 99.9%

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

   3. Simplified 99.9%

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

      1. +-commutative 99.9%

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

      2. fma-undefine 99.9%

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

      3. distribute-rgt-neg-in 99.9%

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

      4. associate-+l+ 99.9%

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

      5. distribute-lft-neg-in 99.9%

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

      6. add-cube-cbrt 99.7%

         \[\leadsto \left(-z\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\sin y} \cdot \sqrt[3]{\sin y}\right) \cdot \sqrt[3]{\sin y}\right)} + \left(x + \cos y\right) \]

      7. associate-*r* 99.7%

         \[\leadsto \color{blue}{\left(\left(-z\right) \cdot \left(\sqrt[3]{\sin y} \cdot \sqrt[3]{\sin y}\right)\right) \cdot \sqrt[3]{\sin y}} + \left(x + \cos y\right) \]

      8. fma-define 99.7%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(-z\right) \cdot \left(\sqrt[3]{\sin y} \cdot \sqrt[3]{\sin y}\right), \sqrt[3]{\sin y}, x + \cos y\right)} \]

      9. pow2 99.7%

         \[\leadsto \mathsf{fma}\left(\left(-z\right) \cdot \color{blue}{{\left(\sqrt[3]{\sin y}\right)}^{2}}, \sqrt[3]{\sin y}, x + \cos y\right) \]

   6. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(-z\right) \cdot {\left(\sqrt[3]{\sin y}\right)}^{2}, \sqrt[3]{\sin y}, x + \cos y\right)} \]

   7. Taylor expanded in x around inf 80.7%

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

   if -2.90000000000000013e23 < x < 2.5500000000000001e38

   1. Initial program 99.9%

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

      1. cancel-sign-sub-inv 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-+l+ 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-lft-neg-out 99.9%

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

      6. distribute-rgt-neg-in 99.9%

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

      7. sin-neg 99.9%

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

      8. fma-define 99.9%

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

      9. sin-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 97.2%

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

      1. neg-mul-1 97.2%

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

      2. sub-neg 97.2%

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

   7. Simplified 97.2%

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

   8. Taylor expanded in y around 0 56.0%

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

      1. mul-1-neg 56.0%

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

      2. unsub-neg 56.0%

         \[\leadsto \color{blue}{1 - y \cdot z} \]

   10. Simplified 56.0%

       \[\leadsto \color{blue}{1 - y \cdot z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{+23}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.55 \cdot 10^{+38}:\\ \;\;\;\;1 - y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 62.0% accurate, 18.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.00115:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{-10}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -0.00115) x (if (<= x 4.6e-10) 1.0 x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -0.00115) {
		tmp = x;
	} else if (x <= 4.6e-10) {
		tmp = 1.0;
	} 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 <= (-0.00115d0)) then
        tmp = x
    else if (x <= 4.6d-10) then
        tmp = 1.0d0
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -0.00115) {
		tmp = x;
	} else if (x <= 4.6e-10) {
		tmp = 1.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -0.00115:
		tmp = x
	elif x <= 4.6e-10:
		tmp = 1.0
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -0.00115)
		tmp = x;
	elseif (x <= 4.6e-10)
		tmp = 1.0;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -0.00115)
		tmp = x;
	elseif (x <= 4.6e-10)
		tmp = 1.0;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -0.00115], x, If[LessEqual[x, 4.6e-10], 1.0, x]]

Derivation

1. Split input into 2 regimes

2. if x < -0.00115 or 4.60000000000000014e-10 < x

   1. Initial program 99.9%

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

      1. cancel-sign-sub-inv 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-+l+ 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-lft-neg-out 99.9%

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

      6. distribute-rgt-neg-in 99.9%

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

      7. sin-neg 99.9%

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

      8. fma-define 99.9%

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

      9. sin-neg 99.9%

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

   3. Simplified 99.9%

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

      1. +-commutative 99.9%

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

      2. fma-undefine 99.9%

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

      3. distribute-rgt-neg-in 99.9%

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

      4. associate-+l+ 99.9%

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

      5. distribute-lft-neg-in 99.9%

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

      6. add-cube-cbrt 99.5%

         \[\leadsto \left(-z\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\sin y} \cdot \sqrt[3]{\sin y}\right) \cdot \sqrt[3]{\sin y}\right)} + \left(x + \cos y\right) \]

      7. associate-*r* 99.5%

         \[\leadsto \color{blue}{\left(\left(-z\right) \cdot \left(\sqrt[3]{\sin y} \cdot \sqrt[3]{\sin y}\right)\right) \cdot \sqrt[3]{\sin y}} + \left(x + \cos y\right) \]

      8. fma-define 99.5%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(-z\right) \cdot \left(\sqrt[3]{\sin y} \cdot \sqrt[3]{\sin y}\right), \sqrt[3]{\sin y}, x + \cos y\right)} \]

      9. pow2 99.5%

         \[\leadsto \mathsf{fma}\left(\left(-z\right) \cdot \color{blue}{{\left(\sqrt[3]{\sin y}\right)}^{2}}, \sqrt[3]{\sin y}, x + \cos y\right) \]

   6. Applied egg-rr 99.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(-z\right) \cdot {\left(\sqrt[3]{\sin y}\right)}^{2}, \sqrt[3]{\sin y}, x + \cos y\right)} \]

   7. Taylor expanded in x around inf 73.0%

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

   if -0.00115 < x < 4.60000000000000014e-10

   1. Initial program 99.9%

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

      1. cancel-sign-sub-inv 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-+l+ 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-lft-neg-out 99.9%

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

      6. distribute-rgt-neg-in 99.9%

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

      7. sin-neg 99.9%

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

      8. fma-define 99.9%

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

      9. sin-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 99.3%

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

      1. neg-mul-1 99.3%

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

      2. sub-neg 99.3%

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

   7. Simplified 99.3%

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

   8. Taylor expanded in y around 0 45.4%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00115:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{-10}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 63.2% accurate, 23.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1.65 \cdot 10^{+207}:\\ \;\;\;\;x + 1\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 1.65e+207) (+ x 1.0) (* z (- y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.65e+207) {
		tmp = x + 1.0;
	} else {
		tmp = z * -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 <= 1.65d+207) then
        tmp = x + 1.0d0
    else
        tmp = z * -y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.65e+207) {
		tmp = x + 1.0;
	} else {
		tmp = z * -y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 1.65e+207:
		tmp = x + 1.0
	else:
		tmp = z * -y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 1.65e+207)
		tmp = Float64(x + 1.0);
	else
		tmp = Float64(z * Float64(-y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 1.65e+207)
		tmp = x + 1.0;
	else
		tmp = z * -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 1.65e+207], N[(x + 1.0), $MachinePrecision], N[(z * (-y)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.65e207

   1. Initial program 99.9%

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

      1. cancel-sign-sub-inv 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-+l+ 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-lft-neg-out 99.9%

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

      6. distribute-rgt-neg-in 99.9%

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

      7. sin-neg 99.9%

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

      8. fma-define 99.9%

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

      9. sin-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in y around 0 65.6%

      \[\leadsto \color{blue}{1 + x} \]
   6. Step-by-step derivation

      1. +-commutative 65.6%

         \[\leadsto \color{blue}{x + 1} \]

   7. Simplified 65.6%

      \[\leadsto \color{blue}{x + 1} \]

   if 1.65e207 < z

   1. Initial program 99.8%

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

      1. cancel-sign-sub-inv 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+l+ 99.8%

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

      4. +-commutative 99.8%

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

      5. distribute-lft-neg-out 99.8%

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

      6. distribute-rgt-neg-in 99.8%

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

      7. sin-neg 99.8%

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

      8. fma-define 99.9%

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

      9. sin-neg 99.9%

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

   3. Simplified 99.9%

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

      1. +-commutative 99.9%

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

      2. fma-undefine 99.8%

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

      3. distribute-rgt-neg-in 99.8%

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

      4. associate-+l+ 99.8%

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

      5. distribute-lft-neg-in 99.8%

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

      6. add-cube-cbrt 98.0%

         \[\leadsto \left(-z\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\sin y} \cdot \sqrt[3]{\sin y}\right) \cdot \sqrt[3]{\sin y}\right)} + \left(x + \cos y\right) \]

      7. associate-*r* 98.1%

         \[\leadsto \color{blue}{\left(\left(-z\right) \cdot \left(\sqrt[3]{\sin y} \cdot \sqrt[3]{\sin y}\right)\right) \cdot \sqrt[3]{\sin y}} + \left(x + \cos y\right) \]

      8. fma-define 98.1%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(-z\right) \cdot \left(\sqrt[3]{\sin y} \cdot \sqrt[3]{\sin y}\right), \sqrt[3]{\sin y}, x + \cos y\right)} \]

      9. pow2 98.1%

         \[\leadsto \mathsf{fma}\left(\left(-z\right) \cdot \color{blue}{{\left(\sqrt[3]{\sin y}\right)}^{2}}, \sqrt[3]{\sin y}, x + \cos y\right) \]

   6. Applied egg-rr 98.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(-z\right) \cdot {\left(\sqrt[3]{\sin y}\right)}^{2}, \sqrt[3]{\sin y}, x + \cos y\right)} \]

   7. Taylor expanded in z around inf 85.3%

      \[\leadsto \color{blue}{-1 \cdot \left(z \cdot \sin y\right)} \]
   8. Step-by-step derivation

      1. mul-1-neg 85.3%

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

      2. *-commutative 85.3%

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

      3. distribute-rgt-neg-in 85.3%

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

   9. Simplified 85.3%

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

   10. Taylor expanded in y around 0 34.7%

       \[\leadsto \color{blue}{-1 \cdot \left(y \cdot z\right)} \]
   11. Step-by-step derivation

       1. mul-1-neg 34.7%

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

       2. distribute-rgt-neg-in 34.7%

          \[\leadsto \color{blue}{y \cdot \left(-z\right)} \]

   12. Simplified 34.7%

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

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.65 \cdot 10^{+207}:\\ \;\;\;\;x + 1\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 62.9% accurate, 69.0× speedup

Math

\[\begin{array}{l} \\ x + 1 \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ x 1.0))

C

double code(double x, double y, double z) {
	return x + 1.0;
}

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 + 1.0d0
end function

Java

public static double code(double x, double y, double z) {
	return x + 1.0;
}

Python

def code(x, y, z):
	return x + 1.0

Julia

function code(x, y, z)
	return Float64(x + 1.0)
end

MATLAB

function tmp = code(x, y, z)
	tmp = x + 1.0;
end

Wolfram

code[x_, y_, z_] := N[(x + 1.0), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

   1. cancel-sign-sub-inv 99.9%

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

   2. +-commutative 99.9%

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

   3. associate-+l+ 99.9%

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

   4. +-commutative 99.9%

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

   5. distribute-lft-neg-out 99.9%

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

   6. distribute-rgt-neg-in 99.9%

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

   7. sin-neg 99.9%

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

   8. fma-define 99.9%

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

   9. sin-neg 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in y around 0 60.2%

   \[\leadsto \color{blue}{1 + x} \]
6. Step-by-step derivation

   1. +-commutative 60.2%

      \[\leadsto \color{blue}{x + 1} \]

7. Simplified 60.2%

   \[\leadsto \color{blue}{x + 1} \]

8. Final simplification 60.2%

   \[\leadsto x + 1 \]
9. Add Preprocessing

Alternative 13: 21.8% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 1.0)

C

double code(double x, double y, double z) {
	return 1.0;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 1.0d0
end function

Java

public static double code(double x, double y, double z) {
	return 1.0;
}

Python

def code(x, y, z):
	return 1.0

Julia

function code(x, y, z)
	return 1.0
end

MATLAB

function tmp = code(x, y, z)
	tmp = 1.0;
end

Wolfram

code[x_, y_, z_] := 1.0

Derivation

1. Initial program 99.9%

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

   1. cancel-sign-sub-inv 99.9%

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

   2. +-commutative 99.9%

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

   3. associate-+l+ 99.9%

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

   4. +-commutative 99.9%

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

   5. distribute-lft-neg-out 99.9%

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

   6. distribute-rgt-neg-in 99.9%

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

   7. sin-neg 99.9%

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

   8. fma-define 99.9%

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

   9. sin-neg 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in x around 0 61.4%

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

   1. neg-mul-1 61.4%

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

   2. sub-neg 61.4%

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

7. Simplified 61.4%

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

8. Taylor expanded in y around 0 22.8%

   \[\leadsto \color{blue}{1} \]

9. Final simplification 22.8%

   \[\leadsto 1 \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024079 
(FPCore (x y z)
  :name "Graphics.Rasterific.Svg.PathConverter:segmentToBezier from rasterific-svg-0.2.3.1, B"
  :precision binary64
  (- (+ x (cos y)) (* z (sin y))))