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

Percentage Accurate: 99.8% → 99.8%

Time: 7.6s

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, 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 99.7%

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

Alternative 2: 84.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{+183} \lor \neg \left(x \leq -5 \cdot 10^{+135} \lor \neg \left(x \leq -1.75 \cdot 10^{+21}\right) \land x \leq 10^{-11}\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \sin y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -6.8e+183)
         (not (or (<= x -5e+135) (and (not (<= x -1.75e+21)) (<= x 1e-11)))))
   (* x (cos y))
   (+ x (* z (sin y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -6.8e+183) || !((x <= -5e+135) || (!(x <= -1.75e+21) && (x <= 1e-11)))) {
		tmp = x * cos(y);
	} else {
		tmp = x + (z * sin(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 ((x <= (-6.8d+183)) .or. (.not. (x <= (-5d+135)) .or. (.not. (x <= (-1.75d+21))) .and. (x <= 1d-11))) then
        tmp = x * cos(y)
    else
        tmp = x + (z * sin(y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -6.8e+183) || !((x <= -5e+135) || (!(x <= -1.75e+21) && (x <= 1e-11)))) {
		tmp = x * Math.cos(y);
	} else {
		tmp = x + (z * Math.sin(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -6.8e+183) or not ((x <= -5e+135) or (not (x <= -1.75e+21) and (x <= 1e-11))):
		tmp = x * math.cos(y)
	else:
		tmp = x + (z * math.sin(y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -6.8e+183) || !((x <= -5e+135) || (!(x <= -1.75e+21) && (x <= 1e-11))))
		tmp = Float64(x * cos(y));
	else
		tmp = Float64(x + Float64(z * sin(y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -6.8e+183) || ~(((x <= -5e+135) || (~((x <= -1.75e+21)) && (x <= 1e-11)))))
		tmp = x * cos(y);
	else
		tmp = x + (z * sin(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -6.8e+183], N[Not[Or[LessEqual[x, -5e+135], And[N[Not[LessEqual[x, -1.75e+21]], $MachinePrecision], LessEqual[x, 1e-11]]]], $MachinePrecision]], N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision], N[(x + N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -6.8e183 or -5.00000000000000029e135 < x < -1.75e21 or 9.99999999999999939e-12 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf 91.4%

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

   if -6.8e183 < x < -5.00000000000000029e135 or -1.75e21 < x < 9.99999999999999939e-12

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0 93.7%

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

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{+183} \lor \neg \left(x \leq -5 \cdot 10^{+135} \lor \neg \left(x \leq -1.75 \cdot 10^{+21}\right) \land x \leq 10^{-11}\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \sin y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \cos y\\ \mathbf{if}\;x \leq -9.5 \cdot 10^{+183}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -2.65 \cdot 10^{+129}:\\ \;\;\;\;x \cdot \left(\sin y \cdot \frac{z}{x} - -1\right)\\ \mathbf{elif}\;x \leq -2.6 \cdot 10^{+22} \lor \neg \left(x \leq 7.4 \cdot 10^{-13}\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \sin y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (cos y))))
   (if (<= x -9.5e+183)
     t_0
     (if (<= x -2.65e+129)
       (* x (- (* (sin y) (/ z x)) -1.0))
       (if (or (<= x -2.6e+22) (not (<= x 7.4e-13)))
         t_0
         (+ x (* z (sin y))))))))

C

double code(double x, double y, double z) {
	double t_0 = x * cos(y);
	double tmp;
	if (x <= -9.5e+183) {
		tmp = t_0;
	} else if (x <= -2.65e+129) {
		tmp = x * ((sin(y) * (z / x)) - -1.0);
	} else if ((x <= -2.6e+22) || !(x <= 7.4e-13)) {
		tmp = t_0;
	} else {
		tmp = x + (z * sin(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) :: t_0
    real(8) :: tmp
    t_0 = x * cos(y)
    if (x <= (-9.5d+183)) then
        tmp = t_0
    else if (x <= (-2.65d+129)) then
        tmp = x * ((sin(y) * (z / x)) - (-1.0d0))
    else if ((x <= (-2.6d+22)) .or. (.not. (x <= 7.4d-13))) then
        tmp = t_0
    else
        tmp = x + (z * sin(y))
    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 (x <= -9.5e+183) {
		tmp = t_0;
	} else if (x <= -2.65e+129) {
		tmp = x * ((Math.sin(y) * (z / x)) - -1.0);
	} else if ((x <= -2.6e+22) || !(x <= 7.4e-13)) {
		tmp = t_0;
	} else {
		tmp = x + (z * Math.sin(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * math.cos(y)
	tmp = 0
	if x <= -9.5e+183:
		tmp = t_0
	elif x <= -2.65e+129:
		tmp = x * ((math.sin(y) * (z / x)) - -1.0)
	elif (x <= -2.6e+22) or not (x <= 7.4e-13):
		tmp = t_0
	else:
		tmp = x + (z * math.sin(y))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * cos(y))
	tmp = 0.0
	if (x <= -9.5e+183)
		tmp = t_0;
	elseif (x <= -2.65e+129)
		tmp = Float64(x * Float64(Float64(sin(y) * Float64(z / x)) - -1.0));
	elseif ((x <= -2.6e+22) || !(x <= 7.4e-13))
		tmp = t_0;
	else
		tmp = Float64(x + Float64(z * sin(y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * cos(y);
	tmp = 0.0;
	if (x <= -9.5e+183)
		tmp = t_0;
	elseif (x <= -2.65e+129)
		tmp = x * ((sin(y) * (z / x)) - -1.0);
	elseif ((x <= -2.6e+22) || ~((x <= 7.4e-13)))
		tmp = t_0;
	else
		tmp = x + (z * sin(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -9.5e+183], t$95$0, If[LessEqual[x, -2.65e+129], N[(x * N[(N[(N[Sin[y], $MachinePrecision] * N[(z / x), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[x, -2.6e+22], N[Not[LessEqual[x, 7.4e-13]], $MachinePrecision]], t$95$0, N[(x + N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -9.5000000000000003e183 or -2.6499999999999999e129 < x < -2.6e22 or 7.39999999999999977e-13 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf 91.4%

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

   if -9.5000000000000003e183 < x < -2.6499999999999999e129

   1. Initial program 99.6%

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

   3. Taylor expanded in y around 0 82.4%

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

   4. Taylor expanded in x around -inf 82.4%

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

      1. mul-1-neg 82.4%

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

      2. *-commutative 82.4%

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

      3. distribute-rgt-neg-in 82.4%

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

      4. sub-neg 82.4%

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

      5. metadata-eval 82.4%

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

      6. +-commutative 82.4%

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

      7. mul-1-neg 82.4%

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

      8. unsub-neg 82.4%

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

      9. *-commutative 82.4%

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

      10. associate-/l* 82.7%

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

   6. Simplified 82.7%

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

   if -2.6e22 < x < 7.39999999999999977e-13

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0 94.8%

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

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{+183}:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{elif}\;x \leq -2.65 \cdot 10^{+129}:\\ \;\;\;\;x \cdot \left(\sin y \cdot \frac{z}{x} - -1\right)\\ \mathbf{elif}\;x \leq -2.6 \cdot 10^{+22} \lor \neg \left(x \leq 7.4 \cdot 10^{-13}\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \sin y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \cos y\\ t_1 := z \cdot \sin y\\ \mathbf{if}\;x \leq -180000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -5.7 \cdot 10^{-92}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-141}:\\ \;\;\;\;\mathsf{fma}\left(y, z, x\right)\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-109}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (cos y))) (t_1 (* z (sin y))))
   (if (<= x -180000000000.0)
     t_0
     (if (<= x -5.7e-92)
       t_1
       (if (<= x -2e-141) (fma y z x) (if (<= x 1.6e-109) t_1 t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = x * cos(y);
	double t_1 = z * sin(y);
	double tmp;
	if (x <= -180000000000.0) {
		tmp = t_0;
	} else if (x <= -5.7e-92) {
		tmp = t_1;
	} else if (x <= -2e-141) {
		tmp = fma(y, z, x);
	} else if (x <= 1.6e-109) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x * cos(y))
	t_1 = Float64(z * sin(y))
	tmp = 0.0
	if (x <= -180000000000.0)
		tmp = t_0;
	elseif (x <= -5.7e-92)
		tmp = t_1;
	elseif (x <= -2e-141)
		tmp = fma(y, z, x);
	elseif (x <= 1.6e-109)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return 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[x, -180000000000.0], t$95$0, If[LessEqual[x, -5.7e-92], t$95$1, If[LessEqual[x, -2e-141], N[(y * z + x), $MachinePrecision], If[LessEqual[x, 1.6e-109], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.8e11 or 1.6000000000000001e-109 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf 84.4%

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

   if -1.8e11 < x < -5.70000000000000009e-92 or -2.0000000000000001e-141 < x < 1.6000000000000001e-109

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 74.2%

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

   if -5.70000000000000009e-92 < x < -2.0000000000000001e-141

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 84.5%

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

      1. +-commutative 84.5%

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

      2. fma-define 84.5%

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

   5. Simplified 84.5%

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

Alternative 5: 73.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \cos y\\ t_1 := z \cdot \sin y\\ \mathbf{if}\;x \leq -10000000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -5.7 \cdot 10^{-92}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-141}:\\ \;\;\;\;x + y \cdot \left(z + y \cdot \left(x \cdot -0.5\right)\right)\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-110}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (cos y))) (t_1 (* z (sin y))))
   (if (<= x -10000000000000.0)
     t_0
     (if (<= x -5.7e-92)
       t_1
       (if (<= x -1e-141)
         (+ x (* y (+ z (* y (* x -0.5)))))
         (if (<= x 1.3e-110) t_1 t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = x * cos(y);
	double t_1 = z * sin(y);
	double tmp;
	if (x <= -10000000000000.0) {
		tmp = t_0;
	} else if (x <= -5.7e-92) {
		tmp = t_1;
	} else if (x <= -1e-141) {
		tmp = x + (y * (z + (y * (x * -0.5))));
	} else if (x <= 1.3e-110) {
		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 = z * sin(y)
    if (x <= (-10000000000000.0d0)) then
        tmp = t_0
    else if (x <= (-5.7d-92)) then
        tmp = t_1
    else if (x <= (-1d-141)) then
        tmp = x + (y * (z + (y * (x * (-0.5d0)))))
    else if (x <= 1.3d-110) 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 = z * Math.sin(y);
	double tmp;
	if (x <= -10000000000000.0) {
		tmp = t_0;
	} else if (x <= -5.7e-92) {
		tmp = t_1;
	} else if (x <= -1e-141) {
		tmp = x + (y * (z + (y * (x * -0.5))));
	} else if (x <= 1.3e-110) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * math.cos(y)
	t_1 = z * math.sin(y)
	tmp = 0
	if x <= -10000000000000.0:
		tmp = t_0
	elif x <= -5.7e-92:
		tmp = t_1
	elif x <= -1e-141:
		tmp = x + (y * (z + (y * (x * -0.5))))
	elif x <= 1.3e-110:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * cos(y))
	t_1 = Float64(z * sin(y))
	tmp = 0.0
	if (x <= -10000000000000.0)
		tmp = t_0;
	elseif (x <= -5.7e-92)
		tmp = t_1;
	elseif (x <= -1e-141)
		tmp = Float64(x + Float64(y * Float64(z + Float64(y * Float64(x * -0.5)))));
	elseif (x <= 1.3e-110)
		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 = z * sin(y);
	tmp = 0.0;
	if (x <= -10000000000000.0)
		tmp = t_0;
	elseif (x <= -5.7e-92)
		tmp = t_1;
	elseif (x <= -1e-141)
		tmp = x + (y * (z + (y * (x * -0.5))));
	elseif (x <= 1.3e-110)
		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[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -10000000000000.0], t$95$0, If[LessEqual[x, -5.7e-92], t$95$1, If[LessEqual[x, -1e-141], N[(x + N[(y * N[(z + N[(y * N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.3e-110], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1e13 or 1.29999999999999995e-110 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf 84.4%

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

   if -1e13 < x < -5.70000000000000009e-92 or -1e-141 < x < 1.29999999999999995e-110

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 74.2%

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

   if -5.70000000000000009e-92 < x < -1e-141

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. add-cube-cbrt 98.7%

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

      3. associate-*r* 98.7%

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

      4. fma-define 98.7%

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

      5. pow2 98.7%

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

   4. Applied egg-rr 98.7%

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

   5. Taylor expanded in y around 0 84.5%

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

      1. associate-*r* 84.5%

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

      2. *-commutative 84.5%

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

   7. Simplified 84.5%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -10000000000000:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{elif}\;x \leq -5.7 \cdot 10^{-92}:\\ \;\;\;\;z \cdot \sin y\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-141}:\\ \;\;\;\;x + y \cdot \left(z + y \cdot \left(x \cdot -0.5\right)\right)\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-110}:\\ \;\;\;\;z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.00075 \lor \neg \left(y \leq 33000\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(z + y \cdot \left(x \cdot -0.5 + -0.16666666666666666 \cdot \left(y \cdot z\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -0.00075) (not (<= y 33000.0)))
   (* 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 <= -0.00075) || !(y <= 33000.0)) {
		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 <= (-0.00075d0)) .or. (.not. (y <= 33000.0d0))) 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 <= -0.00075) || !(y <= 33000.0)) {
		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 <= -0.00075) or not (y <= 33000.0):
		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 <= -0.00075) || !(y <= 33000.0))
		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 <= -0.00075) || ~((y <= 33000.0)))
		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, -0.00075], N[Not[LessEqual[y, 33000.0]], $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 < -7.5000000000000002e-4 or 33000 < y

   1. Initial program 99.5%

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

   3. Taylor expanded in x around inf 54.0%

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

   if -7.5000000000000002e-4 < y < 33000

   1. Initial program 100.0%

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

   3. 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 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.00075 \lor \neg \left(y \leq 33000\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(z + y \cdot \left(x \cdot -0.5 + -0.16666666666666666 \cdot \left(y \cdot z\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 41.4% accurate, 15.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.5 \cdot 10^{+212} \lor \neg \left(z \leq 6 \cdot 10^{+112}\right):\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -2.5e+212) (not (<= z 6e+112))) (* y z) x))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2.5e+212) || !(z <= 6e+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 <= (-2.5d+212)) .or. (.not. (z <= 6d+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 <= -2.5e+212) || !(z <= 6e+112)) {
		tmp = y * z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -2.5e+212) or not (z <= 6e+112):
		tmp = y * z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -2.5e+212) || !(z <= 6e+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 <= -2.5e+212) || ~((z <= 6e+112)))
		tmp = y * z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -2.5e+212], N[Not[LessEqual[z, 6e+112]], $MachinePrecision]], N[(y * z), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if z < -2.49999999999999996e212 or 5.99999999999999958e112 < z

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0 50.3%

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

      1. +-commutative 50.3%

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

   5. Simplified 50.3%

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

   6. Taylor expanded in y around inf 41.4%

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

   if -2.49999999999999996e212 < z < 5.99999999999999958e112

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0 52.2%

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

      1. +-commutative 52.2%

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

   5. Simplified 52.2%

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

   6. Taylor expanded in z around inf 47.0%

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

   7. Taylor expanded in z around 0 44.3%

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

4. Final simplification 43.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.5 \cdot 10^{+212} \lor \neg \left(z \leq 6 \cdot 10^{+112}\right):\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 52.1% 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 99.7%

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

3. Taylor expanded in y around 0 51.8%

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

   1. +-commutative 51.8%

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

5. Simplified 51.8%

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

6. Final simplification 51.8%

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

Alternative 9: 38.5% 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 99.7%

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

3. Taylor expanded in y around 0 51.8%

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

   1. +-commutative 51.8%

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

5. Simplified 51.8%

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

6. Taylor expanded in z around inf 47.7%

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

7. Taylor expanded in z around 0 37.4%

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

Reproduce

herbie shell --seed 2024103 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Transform:aboutY from diagrams-lib-1.3.0.3"
  :precision binary64
  (+ (* x (cos y)) (* z (sin y))))