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

Percentage Accurate: 99.8% → 99.8%

Time: 9.1s

Alternatives: 10

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-def 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. Final simplification 99.8%

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

Alternative 2: 85.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -0.102:\\ \;\;\;\;x + z \cdot \sin y\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{-131}:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sin y, z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -0.102)
   (+ x (* z (sin y)))
   (if (<= z 6.5e-131) (* x (cos y)) (fma (sin y) z x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -0.102) {
		tmp = x + (z * sin(y));
	} else if (z <= 6.5e-131) {
		tmp = x * cos(y);
	} else {
		tmp = fma(sin(y), z, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -0.102)
		tmp = Float64(x + Float64(z * sin(y)));
	elseif (z <= 6.5e-131)
		tmp = Float64(x * cos(y));
	else
		tmp = fma(sin(y), z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -0.102], N[(x + N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 6.5e-131], N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision], N[(N[Sin[y], $MachinePrecision] * z + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -0.101999999999999993

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0 90.1%

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

   if -0.101999999999999993 < z < 6.5000000000000002e-131

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf 87.2%

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

   if 6.5000000000000002e-131 < z

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0 88.8%

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

      1. +-commutative 88.8%

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

      2. *-commutative 88.8%

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

      3. fma-def 88.9%

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

   5. Applied egg-rr 88.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sin y, z, x\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.102:\\ \;\;\;\;x + z \cdot \sin y\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{-131}:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sin y, z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 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 4: 75.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \cos y\\ \mathbf{if}\;y \leq -5.4 \cdot 10^{+250}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -5.2 \cdot 10^{+50}:\\ \;\;\;\;z \cdot \sin y\\ \mathbf{elif}\;y \leq -0.012 \lor \neg \left(y \leq 0.0255\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(z + y \cdot \left(x \cdot -0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (cos y))))
   (if (<= y -5.4e+250)
     t_0
     (if (<= y -5.2e+50)
       (* z (sin y))
       (if (or (<= y -0.012) (not (<= y 0.0255)))
         t_0
         (+ x (* y (+ z (* y (* x -0.5))))))))))

C

double code(double x, double y, double z) {
	double t_0 = x * cos(y);
	double tmp;
	if (y <= -5.4e+250) {
		tmp = t_0;
	} else if (y <= -5.2e+50) {
		tmp = z * sin(y);
	} else if ((y <= -0.012) || !(y <= 0.0255)) {
		tmp = t_0;
	} else {
		tmp = x + (y * (z + (y * (x * -0.5))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * cos(y)
    if (y <= (-5.4d+250)) then
        tmp = t_0
    else if (y <= (-5.2d+50)) then
        tmp = z * sin(y)
    else if ((y <= (-0.012d0)) .or. (.not. (y <= 0.0255d0))) then
        tmp = t_0
    else
        tmp = x + (y * (z + (y * (x * (-0.5d0)))))
    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 tmp;
	if (y <= -5.4e+250) {
		tmp = t_0;
	} else if (y <= -5.2e+50) {
		tmp = z * Math.sin(y);
	} else if ((y <= -0.012) || !(y <= 0.0255)) {
		tmp = t_0;
	} else {
		tmp = x + (y * (z + (y * (x * -0.5))));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * math.cos(y)
	tmp = 0
	if y <= -5.4e+250:
		tmp = t_0
	elif y <= -5.2e+50:
		tmp = z * math.sin(y)
	elif (y <= -0.012) or not (y <= 0.0255):
		tmp = t_0
	else:
		tmp = x + (y * (z + (y * (x * -0.5))))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * cos(y))
	tmp = 0.0
	if (y <= -5.4e+250)
		tmp = t_0;
	elseif (y <= -5.2e+50)
		tmp = Float64(z * sin(y));
	elseif ((y <= -0.012) || !(y <= 0.0255))
		tmp = t_0;
	else
		tmp = Float64(x + Float64(y * Float64(z + Float64(y * Float64(x * -0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * cos(y);
	tmp = 0.0;
	if (y <= -5.4e+250)
		tmp = t_0;
	elseif (y <= -5.2e+50)
		tmp = z * sin(y);
	elseif ((y <= -0.012) || ~((y <= 0.0255)))
		tmp = t_0;
	else
		tmp = x + (y * (z + (y * (x * -0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -5.4e+250], t$95$0, If[LessEqual[y, -5.2e+50], N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[y, -0.012], N[Not[LessEqual[y, 0.0255]], $MachinePrecision]], t$95$0, N[(x + N[(y * N[(z + N[(y * N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.4e250 or -5.2000000000000004e50 < y < -0.012 or 0.0254999999999999984 < y

   1. Initial program 99.5%

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

   3. Taylor expanded in x around inf 65.2%

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

   if -5.4e250 < y < -5.2000000000000004e50

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0 65.0%

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

   if -0.012 < y < 0.0254999999999999984

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 99.5%

      \[\leadsto \color{blue}{x + \left(-0.5 \cdot \left(x \cdot {y}^{2}\right) + y \cdot z\right)} \]
   4. Step-by-step derivation

      1. +-commutative 99.5%

         \[\leadsto x + \color{blue}{\left(y \cdot z + -0.5 \cdot \left(x \cdot {y}^{2}\right)\right)} \]

      2. *-commutative 99.5%

         \[\leadsto x + \left(\color{blue}{z \cdot y} + -0.5 \cdot \left(x \cdot {y}^{2}\right)\right) \]

      3. associate-*r* 99.5%

         \[\leadsto x + \left(z \cdot y + \color{blue}{\left(-0.5 \cdot x\right) \cdot {y}^{2}}\right) \]

      4. unpow2 99.5%

         \[\leadsto x + \left(z \cdot y + \left(-0.5 \cdot x\right) \cdot \color{blue}{\left(y \cdot y\right)}\right) \]

      5. associate-*r* 99.5%

         \[\leadsto x + \left(z \cdot y + \color{blue}{\left(\left(-0.5 \cdot x\right) \cdot y\right) \cdot y}\right) \]

      6. distribute-rgt-out 99.5%

         \[\leadsto x + \color{blue}{y \cdot \left(z + \left(-0.5 \cdot x\right) \cdot y\right)} \]

      7. *-commutative 99.5%

         \[\leadsto x + y \cdot \left(z + \color{blue}{\left(x \cdot -0.5\right)} \cdot y\right) \]

   5. Simplified 99.5%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.4 \cdot 10^{+250}:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{elif}\;y \leq -5.2 \cdot 10^{+50}:\\ \;\;\;\;z \cdot \sin y\\ \mathbf{elif}\;y \leq -0.012 \lor \neg \left(y \leq 0.0255\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(z + y \cdot \left(x \cdot -0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \cos y\\ \mathbf{if}\;y \leq -5.5 \cdot 10^{+250}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -9.5 \cdot 10^{+50}:\\ \;\;\;\;z \cdot \sin y\\ \mathbf{elif}\;y \leq -0.00014 \lor \neg \left(y \leq 0.0195\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (cos y))))
   (if (<= y -5.5e+250)
     t_0
     (if (<= y -9.5e+50)
       (* z (sin y))
       (if (or (<= y -0.00014) (not (<= y 0.0195))) t_0 (fma y z x))))))

C

double code(double x, double y, double z) {
	double t_0 = x * cos(y);
	double tmp;
	if (y <= -5.5e+250) {
		tmp = t_0;
	} else if (y <= -9.5e+50) {
		tmp = z * sin(y);
	} else if ((y <= -0.00014) || !(y <= 0.0195)) {
		tmp = t_0;
	} else {
		tmp = fma(y, z, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x * cos(y))
	tmp = 0.0
	if (y <= -5.5e+250)
		tmp = t_0;
	elseif (y <= -9.5e+50)
		tmp = Float64(z * sin(y));
	elseif ((y <= -0.00014) || !(y <= 0.0195))
		tmp = t_0;
	else
		tmp = fma(y, z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -5.5e+250], t$95$0, If[LessEqual[y, -9.5e+50], N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[y, -0.00014], N[Not[LessEqual[y, 0.0195]], $MachinePrecision]], t$95$0, N[(y * z + x), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.4999999999999999e250 or -9.4999999999999993e50 < y < -1.3999999999999999e-4 or 0.0195 < y

   1. Initial program 99.5%

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

   3. Taylor expanded in x around inf 65.2%

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

   if -5.4999999999999999e250 < y < -9.4999999999999993e50

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0 65.0%

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

   if -1.3999999999999999e-4 < y < 0.0195

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 99.5%

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

      1. +-commutative 99.5%

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

      2. fma-def 99.5%

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

   5. Simplified 99.5%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{+250}:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{elif}\;y \leq -9.5 \cdot 10^{+50}:\\ \;\;\;\;z \cdot \sin y\\ \mathbf{elif}\;y \leq -0.00014 \lor \neg \left(y \leq 0.0195\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -0.35 \lor \neg \left(z \leq 2.15 \cdot 10^{-131}\right):\\ \;\;\;\;x + z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \cos y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -0.35) (not (<= z 2.15e-131)))
   (+ x (* z (sin y)))
   (* x (cos y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -0.35) || !(z <= 2.15e-131)) {
		tmp = x + (z * sin(y));
	} else {
		tmp = x * cos(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 <= (-0.35d0)) .or. (.not. (z <= 2.15d-131))) then
        tmp = x + (z * sin(y))
    else
        tmp = x * cos(y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -0.35) || !(z <= 2.15e-131)) {
		tmp = x + (z * Math.sin(y));
	} else {
		tmp = x * Math.cos(y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -0.35) or not (z <= 2.15e-131):
		tmp = x + (z * math.sin(y))
	else:
		tmp = x * math.cos(y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -0.35) || !(z <= 2.15e-131))
		tmp = Float64(x + Float64(z * sin(y)));
	else
		tmp = Float64(x * cos(y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -0.35) || ~((z <= 2.15e-131)))
		tmp = x + (z * sin(y));
	else
		tmp = x * cos(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -0.35], N[Not[LessEqual[z, 2.15e-131]], $MachinePrecision]], N[(x + N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -0.34999999999999998 or 2.15000000000000009e-131 < z

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0 89.4%

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

   if -0.34999999999999998 < z < 2.15000000000000009e-131

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf 87.2%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.35 \lor \neg \left(z \leq 2.15 \cdot 10^{-131}\right):\\ \;\;\;\;x + z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.0165 \lor \neg \left(y \leq 0.035\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(z + y \cdot \left(x \cdot -0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -0.0165) (not (<= y 0.035)))
   (* x (cos y))
   (+ x (* y (+ z (* y (* x -0.5)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.0165) || !(y <= 0.035)) {
		tmp = x * cos(y);
	} else {
		tmp = x + (y * (z + (y * (x * -0.5))));
	}
	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 <= (-0.0165d0)) .or. (.not. (y <= 0.035d0))) then
        tmp = x * cos(y)
    else
        tmp = x + (y * (z + (y * (x * (-0.5d0)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.0165) || !(y <= 0.035)) {
		tmp = x * Math.cos(y);
	} else {
		tmp = x + (y * (z + (y * (x * -0.5))));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -0.0165) or not (y <= 0.035):
		tmp = x * math.cos(y)
	else:
		tmp = x + (y * (z + (y * (x * -0.5))))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -0.0165) || !(y <= 0.035))
		tmp = Float64(x * cos(y));
	else
		tmp = Float64(x + Float64(y * Float64(z + Float64(y * Float64(x * -0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -0.0165) || ~((y <= 0.035)))
		tmp = x * cos(y);
	else
		tmp = x + (y * (z + (y * (x * -0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -0.0165], N[Not[LessEqual[y, 0.035]], $MachinePrecision]], N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision], N[(x + N[(y * N[(z + N[(y * N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -0.016500000000000001 or 0.035000000000000003 < y

   1. Initial program 99.5%

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

   3. Taylor expanded in x around inf 54.2%

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

   if -0.016500000000000001 < y < 0.035000000000000003

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 99.5%

      \[\leadsto \color{blue}{x + \left(-0.5 \cdot \left(x \cdot {y}^{2}\right) + y \cdot z\right)} \]
   4. Step-by-step derivation

      1. +-commutative 99.5%

         \[\leadsto x + \color{blue}{\left(y \cdot z + -0.5 \cdot \left(x \cdot {y}^{2}\right)\right)} \]

      2. *-commutative 99.5%

         \[\leadsto x + \left(\color{blue}{z \cdot y} + -0.5 \cdot \left(x \cdot {y}^{2}\right)\right) \]

      3. associate-*r* 99.5%

         \[\leadsto x + \left(z \cdot y + \color{blue}{\left(-0.5 \cdot x\right) \cdot {y}^{2}}\right) \]

      4. unpow2 99.5%

         \[\leadsto x + \left(z \cdot y + \left(-0.5 \cdot x\right) \cdot \color{blue}{\left(y \cdot y\right)}\right) \]

      5. associate-*r* 99.5%

         \[\leadsto x + \left(z \cdot y + \color{blue}{\left(\left(-0.5 \cdot x\right) \cdot y\right) \cdot y}\right) \]

      6. distribute-rgt-out 99.5%

         \[\leadsto x + \color{blue}{y \cdot \left(z + \left(-0.5 \cdot x\right) \cdot y\right)} \]

      7. *-commutative 99.5%

         \[\leadsto x + y \cdot \left(z + \color{blue}{\left(x \cdot -0.5\right)} \cdot y\right) \]

   5. Simplified 99.5%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.0165 \lor \neg \left(y \leq 0.035\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(z + y \cdot \left(x \cdot -0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 40.5% accurate, 15.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.15 \cdot 10^{+222} \lor \neg \left(z \leq 3.65 \cdot 10^{+130}\right):\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.15e+222) (not (<= z 3.65e+130))) (* y z) x))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.15e+222) || !(z <= 3.65e+130)) {
		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 <= (-1.15d+222)) .or. (.not. (z <= 3.65d+130))) 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 <= -1.15e+222) || !(z <= 3.65e+130)) {
		tmp = y * z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.15e+222) or not (z <= 3.65e+130):
		tmp = y * z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.15e+222) || !(z <= 3.65e+130))
		tmp = Float64(y * z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.15e+222) || ~((z <= 3.65e+130)))
		tmp = y * z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.15e+222], N[Not[LessEqual[z, 3.65e+130]], $MachinePrecision]], N[(y * z), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if z < -1.15000000000000005e222 or 3.65e130 < z

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0 53.5%

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

      1. +-commutative 53.5%

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

   5. Simplified 53.5%

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

   6. Taylor expanded in y around inf 39.6%

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

      1. *-commutative 39.6%

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

   8. Simplified 39.6%

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

   if -1.15000000000000005e222 < z < 3.65e130

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0 55.8%

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

      1. +-commutative 55.8%

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

   5. Simplified 55.8%

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

   6. Taylor expanded in y around 0 49.8%

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

4. Final simplification 47.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.15 \cdot 10^{+222} \lor \neg \left(z \leq 3.65 \cdot 10^{+130}\right):\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 51.6% 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. Add Preprocessing

3. Taylor expanded in y around 0 55.3%

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

   1. +-commutative 55.3%

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

5. Simplified 55.3%

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

6. Final simplification 55.3%

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

Alternative 10: 38.7% accurate, 207.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in y around 0 55.3%

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

   1. +-commutative 55.3%

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

5. Simplified 55.3%

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

6. Taylor expanded in y around 0 43.0%

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

7. Final simplification 43.0%

   \[\leadsto x \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024026 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Transform:aboutY from diagrams-lib-1.3.0.3"
  :precision binary64
  (+ (* x (cos y)) (* z (sin y))))