Diagrams.ThreeD.Transform:aboutY from diagrams-lib-1.3.0.3

Percentage Accurate: 99.8% → 99.8%

Time: 9.2s

Alternatives: 9

Speedup: 1.0×

Specification

Math

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

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: 99.8% accurate, 1.0× speedup

Math

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

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: 99.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. fma-define 99.8%

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

3. Simplified 99.8%

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

Alternative 2: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

   \[x \cdot \cos y + z \cdot \sin y \]
2. Add Preprocessing

3. Final simplification 99.8%

   \[\leadsto z \cdot \sin y + x \cdot \cos y \]
4. Add Preprocessing

Alternative 3: 78.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \sin y\\ t_1 := x \cdot \cos y\\ \mathbf{if}\;x \leq -3.4 \cdot 10^{+106}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -2.3 \cdot 10^{-80}:\\ \;\;\;\;x \cdot \left(1 + \frac{1}{\frac{x}{t\_0}}\right)\\ \mathbf{elif}\;x \leq 1.95 \cdot 10^{-175}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{+49}:\\ \;\;\;\;x \cdot \left(1 + z \cdot \frac{\sin y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (sin y))) (t_1 (* x (cos y))))
   (if (<= x -3.4e+106)
     t_1
     (if (<= x -2.3e-80)
       (* x (+ 1.0 (/ 1.0 (/ x t_0))))
       (if (<= x 1.95e-175)
         t_0
         (if (<= x 6.2e+49) (* x (+ 1.0 (* z (/ (sin y) x)))) t_1))))))

C

double code(double x, double y, double z) {
	double t_0 = z * sin(y);
	double t_1 = x * cos(y);
	double tmp;
	if (x <= -3.4e+106) {
		tmp = t_1;
	} else if (x <= -2.3e-80) {
		tmp = x * (1.0 + (1.0 / (x / t_0)));
	} else if (x <= 1.95e-175) {
		tmp = t_0;
	} else if (x <= 6.2e+49) {
		tmp = x * (1.0 + (z * (sin(y) / x)));
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = z * sin(y)
    t_1 = x * cos(y)
    if (x <= (-3.4d+106)) then
        tmp = t_1
    else if (x <= (-2.3d-80)) then
        tmp = x * (1.0d0 + (1.0d0 / (x / t_0)))
    else if (x <= 1.95d-175) then
        tmp = t_0
    else if (x <= 6.2d+49) then
        tmp = x * (1.0d0 + (z * (sin(y) / x)))
    else
        tmp = t_1
    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 t_1 = x * Math.cos(y);
	double tmp;
	if (x <= -3.4e+106) {
		tmp = t_1;
	} else if (x <= -2.3e-80) {
		tmp = x * (1.0 + (1.0 / (x / t_0)));
	} else if (x <= 1.95e-175) {
		tmp = t_0;
	} else if (x <= 6.2e+49) {
		tmp = x * (1.0 + (z * (Math.sin(y) / x)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * math.sin(y)
	t_1 = x * math.cos(y)
	tmp = 0
	if x <= -3.4e+106:
		tmp = t_1
	elif x <= -2.3e-80:
		tmp = x * (1.0 + (1.0 / (x / t_0)))
	elif x <= 1.95e-175:
		tmp = t_0
	elif x <= 6.2e+49:
		tmp = x * (1.0 + (z * (math.sin(y) / x)))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * sin(y))
	t_1 = Float64(x * cos(y))
	tmp = 0.0
	if (x <= -3.4e+106)
		tmp = t_1;
	elseif (x <= -2.3e-80)
		tmp = Float64(x * Float64(1.0 + Float64(1.0 / Float64(x / t_0))));
	elseif (x <= 1.95e-175)
		tmp = t_0;
	elseif (x <= 6.2e+49)
		tmp = Float64(x * Float64(1.0 + Float64(z * Float64(sin(y) / x))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * sin(y);
	t_1 = x * cos(y);
	tmp = 0.0;
	if (x <= -3.4e+106)
		tmp = t_1;
	elseif (x <= -2.3e-80)
		tmp = x * (1.0 + (1.0 / (x / t_0)));
	elseif (x <= 1.95e-175)
		tmp = t_0;
	elseif (x <= 6.2e+49)
		tmp = x * (1.0 + (z * (sin(y) / x)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.4e+106], t$95$1, If[LessEqual[x, -2.3e-80], N[(x * N[(1.0 + N[(1.0 / N[(x / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.95e-175], t$95$0, If[LessEqual[x, 6.2e+49], N[(x * N[(1.0 + N[(z * N[(N[Sin[y], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -3.39999999999999994e106 or 6.19999999999999985e49 < x

   1. Initial program 99.8%

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

      1. fma-define 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around inf 89.4%

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

   if -3.39999999999999994e106 < x < -2.2999999999999998e-80

   1. Initial program 99.7%

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

      1. fma-define 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around inf 99.6%

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

      1. associate-/l* 99.4%

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

   7. Simplified 99.4%

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

      1. associate-*r/ 99.6%

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

      2. clear-num 99.6%

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

   9. Applied egg-rr 99.6%

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

   10. Taylor expanded in y around 0 82.7%

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

   if -2.2999999999999998e-80 < x < 1.94999999999999999e-175

   1. Initial program 99.7%

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

      1. fma-define 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0 76.9%

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

   if 1.94999999999999999e-175 < x < 6.19999999999999985e49

   1. Initial program 99.8%

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

      1. fma-define 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around inf 97.9%

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

      1. associate-/l* 97.8%

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

   7. Simplified 97.8%

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

   8. Taylor expanded in y around 0 85.8%

      \[\leadsto x \cdot \left(\color{blue}{1} + z \cdot \frac{\sin y}{x}\right) \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 4: 78.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \cos y\\ t_1 := x \cdot \left(1 + z \cdot \frac{\sin y}{x}\right)\\ \mathbf{if}\;x \leq -7 \cdot 10^{+92}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -3.8 \cdot 10^{-81}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-175}:\\ \;\;\;\;z \cdot \sin y\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{+51}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (cos y))) (t_1 (* x (+ 1.0 (* z (/ (sin y) x))))))
   (if (<= x -7e+92)
     t_0
     (if (<= x -3.8e-81)
       t_1
       (if (<= x 1.9e-175) (* z (sin y)) (if (<= x 2.7e+51) t_1 t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = x * cos(y);
	double t_1 = x * (1.0 + (z * (sin(y) / x)));
	double tmp;
	if (x <= -7e+92) {
		tmp = t_0;
	} else if (x <= -3.8e-81) {
		tmp = t_1;
	} else if (x <= 1.9e-175) {
		tmp = z * sin(y);
	} else if (x <= 2.7e+51) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_0 = x * cos(y)
    t_1 = x * (1.0d0 + (z * (sin(y) / x)))
    if (x <= (-7d+92)) then
        tmp = t_0
    else if (x <= (-3.8d-81)) then
        tmp = t_1
    else if (x <= 1.9d-175) then
        tmp = z * sin(y)
    else if (x <= 2.7d+51) then
        tmp = t_1
    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 * Math.cos(y);
	double t_1 = x * (1.0 + (z * (Math.sin(y) / x)));
	double tmp;
	if (x <= -7e+92) {
		tmp = t_0;
	} else if (x <= -3.8e-81) {
		tmp = t_1;
	} else if (x <= 1.9e-175) {
		tmp = z * Math.sin(y);
	} else if (x <= 2.7e+51) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * math.cos(y)
	t_1 = x * (1.0 + (z * (math.sin(y) / x)))
	tmp = 0
	if x <= -7e+92:
		tmp = t_0
	elif x <= -3.8e-81:
		tmp = t_1
	elif x <= 1.9e-175:
		tmp = z * math.sin(y)
	elif x <= 2.7e+51:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * cos(y))
	t_1 = Float64(x * Float64(1.0 + Float64(z * Float64(sin(y) / x))))
	tmp = 0.0
	if (x <= -7e+92)
		tmp = t_0;
	elseif (x <= -3.8e-81)
		tmp = t_1;
	elseif (x <= 1.9e-175)
		tmp = Float64(z * sin(y));
	elseif (x <= 2.7e+51)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * cos(y);
	t_1 = x * (1.0 + (z * (sin(y) / x)));
	tmp = 0.0;
	if (x <= -7e+92)
		tmp = t_0;
	elseif (x <= -3.8e-81)
		tmp = t_1;
	elseif (x <= 1.9e-175)
		tmp = z * sin(y);
	elseif (x <= 2.7e+51)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * N[(1.0 + N[(z * N[(N[Sin[y], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -7e+92], t$95$0, If[LessEqual[x, -3.8e-81], t$95$1, If[LessEqual[x, 1.9e-175], N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.7e+51], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.99999999999999972e92 or 2.69999999999999992e51 < x

   1. Initial program 99.8%

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

      1. fma-define 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around inf 89.4%

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

   if -6.99999999999999972e92 < x < -3.7999999999999999e-81 or 1.9e-175 < x < 2.69999999999999992e51

   1. Initial program 99.8%

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

      1. fma-define 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around inf 98.6%

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

      1. associate-/l* 98.5%

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

   7. Simplified 98.5%

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

   8. Taylor expanded in y around 0 84.3%

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

   if -3.7999999999999999e-81 < x < 1.9e-175

   1. Initial program 99.7%

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

      1. fma-define 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0 76.9%

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

Alternative 5: 74.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \cos y\\ t_1 := z \cdot \sin y\\ \mathbf{if}\;y \leq -1 \cdot 10^{+195}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -2.7 \cdot 10^{+113}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -1650000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 4.3 \cdot 10^{-11}:\\ \;\;\;\;x + y \cdot \left(z + y \cdot \left(x \cdot -0.5 + -0.16666666666666666 \cdot \left(y \cdot z\right)\right)\right)\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{+156}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (cos y))) (t_1 (* z (sin y))))
   (if (<= y -1e+195)
     t_0
     (if (<= y -2.7e+113)
       t_1
       (if (<= y -1650000000000.0)
         t_0
         (if (<= y 4.3e-11)
           (+
            x
            (* y (+ z (* y (+ (* x -0.5) (* -0.16666666666666666 (* y z)))))))
           (if (<= y 1.2e+156) t_0 t_1)))))))

C

double code(double x, double y, double z) {
	double t_0 = x * cos(y);
	double t_1 = z * sin(y);
	double tmp;
	if (y <= -1e+195) {
		tmp = t_0;
	} else if (y <= -2.7e+113) {
		tmp = t_1;
	} else if (y <= -1650000000000.0) {
		tmp = t_0;
	} else if (y <= 4.3e-11) {
		tmp = x + (y * (z + (y * ((x * -0.5) + (-0.16666666666666666 * (y * z))))));
	} else if (y <= 1.2e+156) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = x * cos(y)
    t_1 = z * sin(y)
    if (y <= (-1d+195)) then
        tmp = t_0
    else if (y <= (-2.7d+113)) then
        tmp = t_1
    else if (y <= (-1650000000000.0d0)) then
        tmp = t_0
    else if (y <= 4.3d-11) then
        tmp = x + (y * (z + (y * ((x * (-0.5d0)) + ((-0.16666666666666666d0) * (y * z))))))
    else if (y <= 1.2d+156) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x * Math.cos(y);
	double t_1 = z * Math.sin(y);
	double tmp;
	if (y <= -1e+195) {
		tmp = t_0;
	} else if (y <= -2.7e+113) {
		tmp = t_1;
	} else if (y <= -1650000000000.0) {
		tmp = t_0;
	} else if (y <= 4.3e-11) {
		tmp = x + (y * (z + (y * ((x * -0.5) + (-0.16666666666666666 * (y * z))))));
	} else if (y <= 1.2e+156) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * math.cos(y)
	t_1 = z * math.sin(y)
	tmp = 0
	if y <= -1e+195:
		tmp = t_0
	elif y <= -2.7e+113:
		tmp = t_1
	elif y <= -1650000000000.0:
		tmp = t_0
	elif y <= 4.3e-11:
		tmp = x + (y * (z + (y * ((x * -0.5) + (-0.16666666666666666 * (y * z))))))
	elif y <= 1.2e+156:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * cos(y))
	t_1 = Float64(z * sin(y))
	tmp = 0.0
	if (y <= -1e+195)
		tmp = t_0;
	elseif (y <= -2.7e+113)
		tmp = t_1;
	elseif (y <= -1650000000000.0)
		tmp = t_0;
	elseif (y <= 4.3e-11)
		tmp = Float64(x + Float64(y * Float64(z + Float64(y * Float64(Float64(x * -0.5) + Float64(-0.16666666666666666 * Float64(y * z)))))));
	elseif (y <= 1.2e+156)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * cos(y);
	t_1 = z * sin(y);
	tmp = 0.0;
	if (y <= -1e+195)
		tmp = t_0;
	elseif (y <= -2.7e+113)
		tmp = t_1;
	elseif (y <= -1650000000000.0)
		tmp = t_0;
	elseif (y <= 4.3e-11)
		tmp = x + (y * (z + (y * ((x * -0.5) + (-0.16666666666666666 * (y * z))))));
	elseif (y <= 1.2e+156)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1e+195], t$95$0, If[LessEqual[y, -2.7e+113], t$95$1, If[LessEqual[y, -1650000000000.0], t$95$0, If[LessEqual[y, 4.3e-11], N[(x + N[(y * N[(z + N[(y * N[(N[(x * -0.5), $MachinePrecision] + N[(-0.16666666666666666 * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.2e+156], t$95$0, t$95$1]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -9.99999999999999977e194 or -2.70000000000000011e113 < y < -1.65e12 or 4.30000000000000001e-11 < y < 1.2000000000000001e156

   1. Initial program 99.5%

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

      1. fma-define 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in x around inf 63.6%

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

   if -9.99999999999999977e194 < y < -2.70000000000000011e113 or 1.2000000000000001e156 < y

   1. Initial program 99.5%

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

      1. fma-define 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in x around 0 69.4%

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

   if -1.65e12 < y < 4.30000000000000001e-11

   1. Initial program 100.0%

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

      1. fma-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 99.3%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{+195}:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{elif}\;y \leq -2.7 \cdot 10^{+113}:\\ \;\;\;\;z \cdot \sin y\\ \mathbf{elif}\;y \leq -1650000000000:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{elif}\;y \leq 4.3 \cdot 10^{-11}:\\ \;\;\;\;x + y \cdot \left(z + y \cdot \left(x \cdot -0.5 + -0.16666666666666666 \cdot \left(y \cdot z\right)\right)\right)\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{+156}:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \sin y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1650000000000 \lor \neg \left(y \leq 4.3 \cdot 10^{-11}\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(z + y \cdot \left(x \cdot -0.5 + -0.16666666666666666 \cdot \left(y \cdot z\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1650000000000.0) (not (<= y 4.3e-11)))
   (* x (cos y))
   (+ x (* y (+ z (* y (+ (* x -0.5) (* -0.16666666666666666 (* y z)))))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1650000000000.0) || !(y <= 4.3e-11)) {
		tmp = x * cos(y);
	} else {
		tmp = x + (y * (z + (y * ((x * -0.5) + (-0.16666666666666666 * (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 <= (-1650000000000.0d0)) .or. (.not. (y <= 4.3d-11))) then
        tmp = x * cos(y)
    else
        tmp = x + (y * (z + (y * ((x * (-0.5d0)) + ((-0.16666666666666666d0) * (y * z))))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1650000000000.0) || !(y <= 4.3e-11)) {
		tmp = x * Math.cos(y);
	} else {
		tmp = x + (y * (z + (y * ((x * -0.5) + (-0.16666666666666666 * (y * z))))));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -1650000000000.0) or not (y <= 4.3e-11):
		tmp = x * math.cos(y)
	else:
		tmp = x + (y * (z + (y * ((x * -0.5) + (-0.16666666666666666 * (y * z))))))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1650000000000.0) || !(y <= 4.3e-11))
		tmp = Float64(x * cos(y));
	else
		tmp = Float64(x + Float64(y * Float64(z + Float64(y * Float64(Float64(x * -0.5) + Float64(-0.16666666666666666 * Float64(y * z)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -1650000000000.0) || ~((y <= 4.3e-11)))
		tmp = x * cos(y);
	else
		tmp = x + (y * (z + (y * ((x * -0.5) + (-0.16666666666666666 * (y * z))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1650000000000.0], N[Not[LessEqual[y, 4.3e-11]], $MachinePrecision]], N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision], N[(x + N[(y * N[(z + N[(y * N[(N[(x * -0.5), $MachinePrecision] + N[(-0.16666666666666666 * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.65e12 or 4.30000000000000001e-11 < y

   1. Initial program 99.5%

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

      1. fma-define 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in x around inf 50.0%

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

   if -1.65e12 < y < 4.30000000000000001e-11

   1. Initial program 100.0%

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

      1. fma-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 99.3%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1650000000000 \lor \neg \left(y \leq 4.3 \cdot 10^{-11}\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(z + y \cdot \left(x \cdot -0.5 + -0.16666666666666666 \cdot \left(y \cdot z\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 41.8% accurate, 15.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7.2 \cdot 10^{+119} \lor \neg \left(z \leq 1.32 \cdot 10^{+112}\right):\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -7.2e+119) (not (<= z 1.32e+112))) (* y z) x))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -7.2e+119) || !(z <= 1.32e+112)) {
		tmp = 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 ((z <= (-7.2d+119)) .or. (.not. (z <= 1.32d+112))) then
        tmp = 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 ((z <= -7.2e+119) || !(z <= 1.32e+112)) {
		tmp = y * z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -7.2e+119) or not (z <= 1.32e+112):
		tmp = y * z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -7.2e+119) || !(z <= 1.32e+112))
		tmp = Float64(y * z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -7.2e+119) || ~((z <= 1.32e+112)))
		tmp = y * z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -7.2e+119], N[Not[LessEqual[z, 1.32e+112]], $MachinePrecision]], N[(y * z), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if z < -7.20000000000000003e119 or 1.32e112 < z

   1. Initial program 99.7%

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

      1. fma-define 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 50.6%

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

   6. Taylor expanded in x around 0 37.1%

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

   if -7.20000000000000003e119 < z < 1.32e112

   1. Initial program 99.8%

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

      1. log1p-expm1-u 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in y around 0 52.9%

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

4. Final simplification 47.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.2 \cdot 10^{+119} \lor \neg \left(z \leq 1.32 \cdot 10^{+112}\right):\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 52.0% accurate, 41.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. fma-define 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in y around 0 56.5%

   \[\leadsto \color{blue}{x + y \cdot z} \]
6. Add Preprocessing

Alternative 9: 38.9% 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 99.8%

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

   1. log1p-expm1-u 99.7%

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

4. Applied egg-rr 99.7%

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

5. Taylor expanded in y around 0 40.4%

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

Reproduce

herbie shell --seed 2024182 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Transform:aboutY from diagrams-lib-1.3.0.3"
  :precision binary64
  (+ (* x (cos y)) (* z (sin y))))