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

Percentage Accurate: 99.8% → 99.8%

Time: 9.6s

Alternatives: 8

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. fma-define 99.8%

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

3. Simplified 99.8%

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

Alternative 2: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Final simplification 99.8%

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

Alternative 3: 79.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \cos y\\ \mathbf{if}\;z \leq -2.8 \cdot 10^{-18}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq -3.3 \cdot 10^{-184}:\\ \;\;\;\;z \cdot \left(1 + x \cdot \frac{\sin y}{z}\right)\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{-267}:\\ \;\;\;\;x \cdot \sin y\\ \mathbf{elif}\;z \leq 1.5 \cdot 10^{+61}:\\ \;\;\;\;z \cdot \left(1 + \frac{x}{\frac{z}{\sin y}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (cos y))))
   (if (<= z -2.8e-18)
     t_0
     (if (<= z -3.3e-184)
       (* z (+ 1.0 (* x (/ (sin y) z))))
       (if (<= z 1.3e-267)
         (* x (sin y))
         (if (<= z 1.5e+61) (* z (+ 1.0 (/ x (/ z (sin y))))) t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = z * cos(y);
	double tmp;
	if (z <= -2.8e-18) {
		tmp = t_0;
	} else if (z <= -3.3e-184) {
		tmp = z * (1.0 + (x * (sin(y) / z)));
	} else if (z <= 1.3e-267) {
		tmp = x * sin(y);
	} else if (z <= 1.5e+61) {
		tmp = z * (1.0 + (x / (z / sin(y))));
	} 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) :: tmp
    t_0 = z * cos(y)
    if (z <= (-2.8d-18)) then
        tmp = t_0
    else if (z <= (-3.3d-184)) then
        tmp = z * (1.0d0 + (x * (sin(y) / z)))
    else if (z <= 1.3d-267) then
        tmp = x * sin(y)
    else if (z <= 1.5d+61) then
        tmp = z * (1.0d0 + (x / (z / sin(y))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = z * Math.cos(y);
	double tmp;
	if (z <= -2.8e-18) {
		tmp = t_0;
	} else if (z <= -3.3e-184) {
		tmp = z * (1.0 + (x * (Math.sin(y) / z)));
	} else if (z <= 1.3e-267) {
		tmp = x * Math.sin(y);
	} else if (z <= 1.5e+61) {
		tmp = z * (1.0 + (x / (z / Math.sin(y))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * math.cos(y)
	tmp = 0
	if z <= -2.8e-18:
		tmp = t_0
	elif z <= -3.3e-184:
		tmp = z * (1.0 + (x * (math.sin(y) / z)))
	elif z <= 1.3e-267:
		tmp = x * math.sin(y)
	elif z <= 1.5e+61:
		tmp = z * (1.0 + (x / (z / math.sin(y))))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * cos(y))
	tmp = 0.0
	if (z <= -2.8e-18)
		tmp = t_0;
	elseif (z <= -3.3e-184)
		tmp = Float64(z * Float64(1.0 + Float64(x * Float64(sin(y) / z))));
	elseif (z <= 1.3e-267)
		tmp = Float64(x * sin(y));
	elseif (z <= 1.5e+61)
		tmp = Float64(z * Float64(1.0 + Float64(x / Float64(z / sin(y)))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * cos(y);
	tmp = 0.0;
	if (z <= -2.8e-18)
		tmp = t_0;
	elseif (z <= -3.3e-184)
		tmp = z * (1.0 + (x * (sin(y) / z)));
	elseif (z <= 1.3e-267)
		tmp = x * sin(y);
	elseif (z <= 1.5e+61)
		tmp = z * (1.0 + (x / (z / sin(y))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2.8e-18], t$95$0, If[LessEqual[z, -3.3e-184], N[(z * N[(1.0 + N[(x * N[(N[Sin[y], $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.3e-267], N[(x * N[Sin[y], $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.5e+61], N[(z * N[(1.0 + N[(x / N[(z / N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -2.80000000000000012e-18 or 1.5e61 < z

   1. Initial program 99.8%

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

      1. fma-define 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 90.5%

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

   if -2.80000000000000012e-18 < z < -3.2999999999999997e-184

   1. Initial program 99.8%

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

      1. fma-define 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in z around inf 88.6%

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

      1. associate-/l* 88.5%

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

   7. Simplified 88.5%

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

   8. Taylor expanded in y around 0 80.8%

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

   if -3.2999999999999997e-184 < z < 1.3000000000000001e-267

   1. Initial program 99.7%

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

      1. fma-define 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around inf 85.6%

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

   if 1.3000000000000001e-267 < z < 1.5e61

   1. Initial program 99.7%

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

      1. fma-define 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in z around inf 93.5%

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

      1. associate-/l* 93.3%

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

   7. Simplified 93.3%

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

      1. clear-num 93.3%

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

      2. un-div-inv 93.5%

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

   9. Applied egg-rr 93.5%

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

   10. Taylor expanded in y around 0 90.2%

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

Alternative 4: 79.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \cos y\\ t_1 := z \cdot \left(1 + x \cdot \frac{\sin y}{z}\right)\\ \mathbf{if}\;z \leq -2.5 \cdot 10^{-18}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq -6.6 \cdot 10^{-184}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{-268}:\\ \;\;\;\;x \cdot \sin y\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{+61}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (cos y))) (t_1 (* z (+ 1.0 (* x (/ (sin y) z))))))
   (if (<= z -2.5e-18)
     t_0
     (if (<= z -6.6e-184)
       t_1
       (if (<= z 1.8e-268) (* x (sin y)) (if (<= z 1.4e+61) t_1 t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = z * cos(y);
	double t_1 = z * (1.0 + (x * (sin(y) / z)));
	double tmp;
	if (z <= -2.5e-18) {
		tmp = t_0;
	} else if (z <= -6.6e-184) {
		tmp = t_1;
	} else if (z <= 1.8e-268) {
		tmp = x * sin(y);
	} else if (z <= 1.4e+61) {
		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 = z * cos(y)
    t_1 = z * (1.0d0 + (x * (sin(y) / z)))
    if (z <= (-2.5d-18)) then
        tmp = t_0
    else if (z <= (-6.6d-184)) then
        tmp = t_1
    else if (z <= 1.8d-268) then
        tmp = x * sin(y)
    else if (z <= 1.4d+61) 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 = z * Math.cos(y);
	double t_1 = z * (1.0 + (x * (Math.sin(y) / z)));
	double tmp;
	if (z <= -2.5e-18) {
		tmp = t_0;
	} else if (z <= -6.6e-184) {
		tmp = t_1;
	} else if (z <= 1.8e-268) {
		tmp = x * Math.sin(y);
	} else if (z <= 1.4e+61) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * math.cos(y)
	t_1 = z * (1.0 + (x * (math.sin(y) / z)))
	tmp = 0
	if z <= -2.5e-18:
		tmp = t_0
	elif z <= -6.6e-184:
		tmp = t_1
	elif z <= 1.8e-268:
		tmp = x * math.sin(y)
	elif z <= 1.4e+61:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * cos(y))
	t_1 = Float64(z * Float64(1.0 + Float64(x * Float64(sin(y) / z))))
	tmp = 0.0
	if (z <= -2.5e-18)
		tmp = t_0;
	elseif (z <= -6.6e-184)
		tmp = t_1;
	elseif (z <= 1.8e-268)
		tmp = Float64(x * sin(y));
	elseif (z <= 1.4e+61)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * cos(y);
	t_1 = z * (1.0 + (x * (sin(y) / z)));
	tmp = 0.0;
	if (z <= -2.5e-18)
		tmp = t_0;
	elseif (z <= -6.6e-184)
		tmp = t_1;
	elseif (z <= 1.8e-268)
		tmp = x * sin(y);
	elseif (z <= 1.4e+61)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(z * N[(1.0 + N[(x * N[(N[Sin[y], $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2.5e-18], t$95$0, If[LessEqual[z, -6.6e-184], t$95$1, If[LessEqual[z, 1.8e-268], N[(x * N[Sin[y], $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.4e+61], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.50000000000000018e-18 or 1.4000000000000001e61 < z

   1. Initial program 99.8%

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

      1. fma-define 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 90.5%

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

   if -2.50000000000000018e-18 < z < -6.5999999999999995e-184 or 1.8000000000000001e-268 < z < 1.4000000000000001e61

   1. Initial program 99.8%

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

      1. fma-define 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in z around inf 91.8%

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

      1. associate-/l* 91.6%

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

   7. Simplified 91.6%

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

   8. Taylor expanded in y around 0 86.8%

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

   if -6.5999999999999995e-184 < z < 1.8000000000000001e-268

   1. Initial program 99.7%

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

      1. fma-define 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around inf 85.6%

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

Alternative 5: 74.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7.8 \cdot 10^{-81} \lor \neg \left(z \leq 2.1 \cdot 10^{-43}\right):\\ \;\;\;\;z \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sin y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -7.8e-81) (not (<= z 2.1e-43))) (* z (cos y)) (* x (sin y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -7.8e-81) || !(z <= 2.1e-43)) {
		tmp = z * cos(y);
	} else {
		tmp = x * 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 ((z <= (-7.8d-81)) .or. (.not. (z <= 2.1d-43))) then
        tmp = z * cos(y)
    else
        tmp = x * sin(y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -7.8e-81) || !(z <= 2.1e-43)) {
		tmp = z * Math.cos(y);
	} else {
		tmp = x * Math.sin(y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -7.8e-81) or not (z <= 2.1e-43):
		tmp = z * math.cos(y)
	else:
		tmp = x * math.sin(y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -7.8e-81) || !(z <= 2.1e-43))
		tmp = Float64(z * cos(y));
	else
		tmp = Float64(x * sin(y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -7.8e-81) || ~((z <= 2.1e-43)))
		tmp = z * cos(y);
	else
		tmp = x * sin(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -7.8e-81], N[Not[LessEqual[z, 2.1e-43]], $MachinePrecision]], N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision], N[(x * N[Sin[y], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -7.7999999999999997e-81 or 2.1000000000000001e-43 < z

   1. Initial program 99.8%

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

      1. fma-define 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 82.7%

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

   if -7.7999999999999997e-81 < z < 2.1000000000000001e-43

   1. Initial program 99.7%

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

      1. fma-define 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around inf 73.5%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.8 \cdot 10^{-81} \lor \neg \left(z \leq 2.1 \cdot 10^{-43}\right):\\ \;\;\;\;z \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sin y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.04 \lor \neg \left(y \leq 0.17\right):\\ \;\;\;\;x \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;z + y \cdot \left(x + y \cdot \left(z \cdot -0.5 + -0.16666666666666666 \cdot \left(x \cdot y\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -0.04) (not (<= y 0.17)))
   (* x (sin y))
   (+ z (* y (+ x (* y (+ (* z -0.5) (* -0.16666666666666666 (* x y)))))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.04) || !(y <= 0.17)) {
		tmp = x * sin(y);
	} else {
		tmp = z + (y * (x + (y * ((z * -0.5) + (-0.16666666666666666 * (x * 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 ((y <= (-0.04d0)) .or. (.not. (y <= 0.17d0))) then
        tmp = x * sin(y)
    else
        tmp = z + (y * (x + (y * ((z * (-0.5d0)) + ((-0.16666666666666666d0) * (x * y))))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.04) || !(y <= 0.17)) {
		tmp = x * Math.sin(y);
	} else {
		tmp = z + (y * (x + (y * ((z * -0.5) + (-0.16666666666666666 * (x * y))))));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -0.04) or not (y <= 0.17):
		tmp = x * math.sin(y)
	else:
		tmp = z + (y * (x + (y * ((z * -0.5) + (-0.16666666666666666 * (x * y))))))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -0.04) || !(y <= 0.17))
		tmp = Float64(x * sin(y));
	else
		tmp = Float64(z + Float64(y * Float64(x + Float64(y * Float64(Float64(z * -0.5) + Float64(-0.16666666666666666 * Float64(x * y)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -0.04) || ~((y <= 0.17)))
		tmp = x * sin(y);
	else
		tmp = z + (y * (x + (y * ((z * -0.5) + (-0.16666666666666666 * (x * y))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -0.04], N[Not[LessEqual[y, 0.17]], $MachinePrecision]], N[(x * N[Sin[y], $MachinePrecision]), $MachinePrecision], N[(z + N[(y * N[(x + N[(y * N[(N[(z * -0.5), $MachinePrecision] + N[(-0.16666666666666666 * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -0.0400000000000000008 or 0.170000000000000012 < y

   1. Initial program 99.5%

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

      1. fma-define 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in x around inf 53.0%

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

   if -0.0400000000000000008 < y < 0.170000000000000012

   1. Initial program 100.0%

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

      1. fma-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 99.1%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.04 \lor \neg \left(y \leq 0.17\right):\\ \;\;\;\;x \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;z + y \cdot \left(x + y \cdot \left(z \cdot -0.5 + -0.16666666666666666 \cdot \left(x \cdot y\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 51.2% accurate, 41.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. fma-define 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in y around 0 52.6%

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

   1. +-commutative 52.6%

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

7. Simplified 52.6%

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

8. Final simplification 52.6%

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

Alternative 8: 16.5% accurate, 69.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. fma-define 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in y around 0 52.6%

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

   1. +-commutative 52.6%

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

7. Simplified 52.6%

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

8. Taylor expanded in x around inf 14.5%

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

   1. *-commutative 14.5%

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

10. Simplified 14.5%

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

11. Final simplification 14.5%

    \[\leadsto x \cdot y \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024123 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Transform:aboutX from diagrams-lib-1.3.0.3, B"
  :precision binary64
  (+ (* x (sin y)) (* z (cos y))))