Diagrams.ThreeD.Transform:aboutX from diagrams-lib-1.3.0.3, A

Percentage Accurate: 99.8% → 99.8%

Time: 8.9s

Alternatives: 8

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. cancel-sign-sub-inv 99.9%

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

   2. +-commutative 99.9%

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

   3. distribute-lft-neg-out 99.9%

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

   4. distribute-rgt-neg-in 99.9%

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

   5. sin-neg 99.9%

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

   6. fma-define 99.9%

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

   7. sin-neg 99.9%

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

3. Simplified 99.9%

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

Alternative 2: 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]

Derivation

1. Initial program 99.9%

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

Alternative 3: 73.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(-\sin y\right)\\ t_1 := x \cdot \cos y\\ \mathbf{if}\;y \leq -1.05 \cdot 10^{+161}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -0.0077:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.22 \cdot 10^{+33}:\\ \;\;\;\;x + y \cdot \left(y \cdot \left(\left(z \cdot y\right) \cdot 0.16666666666666666\right) - z\right)\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{+149}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (- (sin y)))) (t_1 (* x (cos y))))
   (if (<= y -1.05e+161)
     t_0
     (if (<= y -0.0077)
       t_1
       (if (<= y 1.22e+33)
         (+ x (* y (- (* y (* (* z y) 0.16666666666666666)) z)))
         (if (<= y 1.1e+149) t_0 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 (y <= -1.05e+161) {
		tmp = t_0;
	} else if (y <= -0.0077) {
		tmp = t_1;
	} else if (y <= 1.22e+33) {
		tmp = x + (y * ((y * ((z * y) * 0.16666666666666666)) - z));
	} else if (y <= 1.1e+149) {
		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 = z * -sin(y)
    t_1 = x * cos(y)
    if (y <= (-1.05d+161)) then
        tmp = t_0
    else if (y <= (-0.0077d0)) then
        tmp = t_1
    else if (y <= 1.22d+33) then
        tmp = x + (y * ((y * ((z * y) * 0.16666666666666666d0)) - z))
    else if (y <= 1.1d+149) 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 = z * -Math.sin(y);
	double t_1 = x * Math.cos(y);
	double tmp;
	if (y <= -1.05e+161) {
		tmp = t_0;
	} else if (y <= -0.0077) {
		tmp = t_1;
	} else if (y <= 1.22e+33) {
		tmp = x + (y * ((y * ((z * y) * 0.16666666666666666)) - z));
	} else if (y <= 1.1e+149) {
		tmp = t_0;
	} 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 y <= -1.05e+161:
		tmp = t_0
	elif y <= -0.0077:
		tmp = t_1
	elif y <= 1.22e+33:
		tmp = x + (y * ((y * ((z * y) * 0.16666666666666666)) - z))
	elif y <= 1.1e+149:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(-sin(y)))
	t_1 = Float64(x * cos(y))
	tmp = 0.0
	if (y <= -1.05e+161)
		tmp = t_0;
	elseif (y <= -0.0077)
		tmp = t_1;
	elseif (y <= 1.22e+33)
		tmp = Float64(x + Float64(y * Float64(Float64(y * Float64(Float64(z * y) * 0.16666666666666666)) - z)));
	elseif (y <= 1.1e+149)
		tmp = t_0;
	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 (y <= -1.05e+161)
		tmp = t_0;
	elseif (y <= -0.0077)
		tmp = t_1;
	elseif (y <= 1.22e+33)
		tmp = x + (y * ((y * ((z * y) * 0.16666666666666666)) - z));
	elseif (y <= 1.1e+149)
		tmp = t_0;
	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[y, -1.05e+161], t$95$0, If[LessEqual[y, -0.0077], t$95$1, If[LessEqual[y, 1.22e+33], N[(x + N[(y * N[(N[(y * N[(N[(z * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.1e+149], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.05e161 or 1.22000000000000005e33 < y < 1.1e149

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 65.0%

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

      1. neg-mul-1 65.0%

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

      2. distribute-rgt-neg-in 65.0%

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

   5. Simplified 65.0%

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

   if -1.05e161 < y < -0.0077000000000000002 or 1.1e149 < y

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf 71.1%

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

   if -0.0077000000000000002 < y < 1.22000000000000005e33

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 97.9%

      \[\leadsto \color{blue}{x + y \cdot \left(y \cdot \left(-0.5 \cdot x + 0.16666666666666666 \cdot \left(y \cdot z\right)\right) - z\right)} \]

   4. Taylor expanded in x around 0 98.0%

      \[\leadsto x + y \cdot \left(y \cdot \color{blue}{\left(0.16666666666666666 \cdot \left(y \cdot z\right)\right)} - z\right) \]
   5. Step-by-step derivation

      1. *-commutative 98.0%

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

      2. *-commutative 98.0%

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

   6. Simplified 98.0%

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

Alternative 4: 72.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(-\sin y\right)\\ t_1 := x \cdot \cos y\\ \mathbf{if}\;y \leq -5.9 \cdot 10^{+159}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 10^{+81}:\\ \;\;\;\;t\_1 - z \cdot y\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{+154}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (- (sin y)))) (t_1 (* x (cos y))))
   (if (<= y -5.9e+159)
     t_0
     (if (<= y 1e+81) (- t_1 (* z y)) (if (<= y 1.1e+154) t_0 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 (y <= -5.9e+159) {
		tmp = t_0;
	} else if (y <= 1e+81) {
		tmp = t_1 - (z * y);
	} else if (y <= 1.1e+154) {
		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 = z * -sin(y)
    t_1 = x * cos(y)
    if (y <= (-5.9d+159)) then
        tmp = t_0
    else if (y <= 1d+81) then
        tmp = t_1 - (z * y)
    else if (y <= 1.1d+154) 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 = z * -Math.sin(y);
	double t_1 = x * Math.cos(y);
	double tmp;
	if (y <= -5.9e+159) {
		tmp = t_0;
	} else if (y <= 1e+81) {
		tmp = t_1 - (z * y);
	} else if (y <= 1.1e+154) {
		tmp = t_0;
	} 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 y <= -5.9e+159:
		tmp = t_0
	elif y <= 1e+81:
		tmp = t_1 - (z * y)
	elif y <= 1.1e+154:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(-sin(y)))
	t_1 = Float64(x * cos(y))
	tmp = 0.0
	if (y <= -5.9e+159)
		tmp = t_0;
	elseif (y <= 1e+81)
		tmp = Float64(t_1 - Float64(z * y));
	elseif (y <= 1.1e+154)
		tmp = t_0;
	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 (y <= -5.9e+159)
		tmp = t_0;
	elseif (y <= 1e+81)
		tmp = t_1 - (z * y);
	elseif (y <= 1.1e+154)
		tmp = t_0;
	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[y, -5.9e+159], t$95$0, If[LessEqual[y, 1e+81], N[(t$95$1 - N[(z * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.1e+154], t$95$0, t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.89999999999999993e159 or 9.99999999999999921e80 < y < 1.1000000000000001e154

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 66.7%

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

      1. neg-mul-1 66.7%

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

      2. distribute-rgt-neg-in 66.7%

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

   5. Simplified 66.7%

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

   if -5.89999999999999993e159 < y < 9.99999999999999921e80

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 90.7%

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

   if 1.1000000000000001e154 < y

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf 71.9%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.9 \cdot 10^{+159}:\\ \;\;\;\;z \cdot \left(-\sin y\right)\\ \mathbf{elif}\;y \leq 10^{+81}:\\ \;\;\;\;x \cdot \cos y - z \cdot y\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{+154}:\\ \;\;\;\;z \cdot \left(-\sin y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.044 \lor \neg \left(y \leq 0.145\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(y \cdot \left(x \cdot -0.5 + \left(z \cdot y\right) \cdot 0.16666666666666666\right) - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -0.044) (not (<= y 0.145)))
   (* x (cos y))
   (+ x (* y (- (* y (+ (* x -0.5) (* (* z y) 0.16666666666666666))) z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.044) || !(y <= 0.145)) {
		tmp = x * cos(y);
	} else {
		tmp = x + (y * ((y * ((x * -0.5) + ((z * y) * 0.16666666666666666))) - 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 <= (-0.044d0)) .or. (.not. (y <= 0.145d0))) then
        tmp = x * cos(y)
    else
        tmp = x + (y * ((y * ((x * (-0.5d0)) + ((z * y) * 0.16666666666666666d0))) - z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.044) || !(y <= 0.145)) {
		tmp = x * Math.cos(y);
	} else {
		tmp = x + (y * ((y * ((x * -0.5) + ((z * y) * 0.16666666666666666))) - z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -0.044) or not (y <= 0.145):
		tmp = x * math.cos(y)
	else:
		tmp = x + (y * ((y * ((x * -0.5) + ((z * y) * 0.16666666666666666))) - z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -0.044) || !(y <= 0.145))
		tmp = Float64(x * cos(y));
	else
		tmp = Float64(x + Float64(y * Float64(Float64(y * Float64(Float64(x * -0.5) + Float64(Float64(z * y) * 0.16666666666666666))) - z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -0.044) || ~((y <= 0.145)))
		tmp = x * cos(y);
	else
		tmp = x + (y * ((y * ((x * -0.5) + ((z * y) * 0.16666666666666666))) - z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -0.044], N[Not[LessEqual[y, 0.145]], $MachinePrecision]], N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision], N[(x + N[(y * N[(N[(y * N[(N[(x * -0.5), $MachinePrecision] + N[(N[(z * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -0.043999999999999997 or 0.14499999999999999 < y

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf 55.7%

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

   if -0.043999999999999997 < y < 0.14499999999999999

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 99.7%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.044 \lor \neg \left(y \leq 0.145\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(y \cdot \left(x \cdot -0.5 + \left(z \cdot y\right) \cdot 0.16666666666666666\right) - z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 39.3% accurate, 23.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1.3 \cdot 10^{+106}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= z 1.3e+106) x (* z (- y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.3e+106) {
		tmp = x;
	} 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 <= 1.3d+106) then
        tmp = x
    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 <= 1.3e+106) {
		tmp = x;
	} else {
		tmp = z * -y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 1.3e+106:
		tmp = x
	else:
		tmp = z * -y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 1.3e+106)
		tmp = x;
	else
		tmp = Float64(z * Float64(-y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 1.3e+106)
		tmp = x;
	else
		tmp = z * -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 1.3e+106], x, N[(z * (-y)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.3000000000000001e106

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 55.7%

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

      1. mul-1-neg 55.7%

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

      2. unsub-neg 55.7%

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

   5. Simplified 55.7%

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

   6. Taylor expanded in x around inf 45.9%

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

   if 1.3000000000000001e106 < z

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0 53.7%

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

      1. mul-1-neg 53.7%

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

      2. unsub-neg 53.7%

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

   5. Simplified 53.7%

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

   6. Taylor expanded in x around 0 43.9%

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

      1. associate-*r* 43.9%

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

      2. neg-mul-1 43.9%

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

      3. *-commutative 43.9%

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

   8. Simplified 43.9%

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

Alternative 7: 51.2% accurate, 41.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in y around 0 55.4%

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

   1. mul-1-neg 55.4%

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

   2. unsub-neg 55.4%

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

5. Simplified 55.4%

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

6. Final simplification 55.4%

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

Alternative 8: 37.9% 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 99.9%

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

3. Taylor expanded in y around 0 55.4%

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

   1. mul-1-neg 55.4%

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

   2. unsub-neg 55.4%

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

5. Simplified 55.4%

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

6. Taylor expanded in x around inf 40.3%

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

Reproduce

herbie shell --seed 2024101 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Transform:aboutX from diagrams-lib-1.3.0.3, A"
  :precision binary64
  (- (* x (cos y)) (* z (sin y))))