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

Percentage Accurate: 99.8% → 99.8%

Time: 10.2s

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

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

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 99.8%

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

   1. cancel-sign-sub-inv 99.8%

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

   2. +-commutative 99.8%

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

   3. distribute-lft-neg-out 99.8%

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

   4. distribute-rgt-neg-in 99.8%

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

   5. sin-neg 99.8%

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

   6. fma-define 99.8%

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

   7. sin-neg 99.8%

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

3. Simplified 99.8%

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

Alternative 2: 86.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{-123} \lor \neg \left(z \leq 6.8 \cdot 10^{-37}\right):\\ \;\;\;\;\mathsf{fma}\left(z, -\sin y, x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \cos y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -2.3e-123) (not (<= z 6.8e-37)))
   (fma z (- (sin y)) x)
   (* x (cos y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2.3e-123) || !(z <= 6.8e-37)) {
		tmp = fma(z, -sin(y), x);
	} else {
		tmp = x * cos(y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -2.3e-123) || !(z <= 6.8e-37))
		tmp = fma(z, Float64(-sin(y)), x);
	else
		tmp = Float64(x * cos(y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -2.3e-123], N[Not[LessEqual[z, 6.8e-37]], $MachinePrecision]], N[(z * (-N[Sin[y], $MachinePrecision]) + x), $MachinePrecision], N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.29999999999999987e-123 or 6.80000000000000037e-37 < z

   1. Initial program 99.8%

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

      1. cancel-sign-sub-inv 99.8%

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

      2. +-commutative 99.8%

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

      3. distribute-lft-neg-out 99.8%

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

      4. distribute-rgt-neg-in 99.8%

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

      5. sin-neg 99.8%

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

      6. fma-define 99.8%

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

      7. sin-neg 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around 0 87.2%

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

   if -2.29999999999999987e-123 < z < 6.80000000000000037e-37

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 87.9%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{-123} \lor \neg \left(z \leq 6.8 \cdot 10^{-37}\right):\\ \;\;\;\;\mathsf{fma}\left(z, -\sin y, x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 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 99.8%

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

Alternative 4: 75.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(-\sin y\right)\\ \mathbf{if}\;y \leq -1.22 \cdot 10^{+235}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -0.065:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{elif}\;y \leq 0.07:\\ \;\;\;\;x + y \cdot \left(y \cdot \left(x \cdot -0.5 + 0.16666666666666666 \cdot \left(z \cdot y\right)\right) - z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (- (sin y)))))
   (if (<= y -1.22e+235)
     t_0
     (if (<= y -0.065)
       (* x (cos y))
       (if (<= y 0.07)
         (+ x (* y (- (* y (+ (* x -0.5) (* 0.16666666666666666 (* z y)))) z)))
         t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = z * -sin(y);
	double tmp;
	if (y <= -1.22e+235) {
		tmp = t_0;
	} else if (y <= -0.065) {
		tmp = x * cos(y);
	} else if (y <= 0.07) {
		tmp = x + (y * ((y * ((x * -0.5) + (0.16666666666666666 * (z * y)))) - z));
	} 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 * -sin(y)
    if (y <= (-1.22d+235)) then
        tmp = t_0
    else if (y <= (-0.065d0)) then
        tmp = x * cos(y)
    else if (y <= 0.07d0) then
        tmp = x + (y * ((y * ((x * (-0.5d0)) + (0.16666666666666666d0 * (z * y)))) - z))
    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.sin(y);
	double tmp;
	if (y <= -1.22e+235) {
		tmp = t_0;
	} else if (y <= -0.065) {
		tmp = x * Math.cos(y);
	} else if (y <= 0.07) {
		tmp = x + (y * ((y * ((x * -0.5) + (0.16666666666666666 * (z * y)))) - z));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * -math.sin(y)
	tmp = 0
	if y <= -1.22e+235:
		tmp = t_0
	elif y <= -0.065:
		tmp = x * math.cos(y)
	elif y <= 0.07:
		tmp = x + (y * ((y * ((x * -0.5) + (0.16666666666666666 * (z * y)))) - z))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(-sin(y)))
	tmp = 0.0
	if (y <= -1.22e+235)
		tmp = t_0;
	elseif (y <= -0.065)
		tmp = Float64(x * cos(y));
	elseif (y <= 0.07)
		tmp = Float64(x + Float64(y * Float64(Float64(y * Float64(Float64(x * -0.5) + Float64(0.16666666666666666 * Float64(z * y)))) - z)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * -sin(y);
	tmp = 0.0;
	if (y <= -1.22e+235)
		tmp = t_0;
	elseif (y <= -0.065)
		tmp = x * cos(y);
	elseif (y <= 0.07)
		tmp = x + (y * ((y * ((x * -0.5) + (0.16666666666666666 * (z * y)))) - z));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * (-N[Sin[y], $MachinePrecision])), $MachinePrecision]}, If[LessEqual[y, -1.22e+235], t$95$0, If[LessEqual[y, -0.065], N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.07], N[(x + N[(y * N[(N[(y * N[(N[(x * -0.5), $MachinePrecision] + N[(0.16666666666666666 * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.22000000000000003e235 or 0.070000000000000007 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0 61.1%

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

      1. neg-mul-1 61.1%

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

      2. *-commutative 61.1%

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

      3. distribute-rgt-neg-in 61.1%

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

   5. Simplified 61.1%

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

   if -1.22000000000000003e235 < y < -0.065000000000000002

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf 64.6%

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

   if -0.065000000000000002 < y < 0.070000000000000007

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 99.9%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.22 \cdot 10^{+235}:\\ \;\;\;\;z \cdot \left(-\sin y\right)\\ \mathbf{elif}\;y \leq -0.065:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{elif}\;y \leq 0.07:\\ \;\;\;\;x + y \cdot \left(y \cdot \left(x \cdot -0.5 + 0.16666666666666666 \cdot \left(z \cdot y\right)\right) - z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-\sin y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.25 \cdot 10^{-123}:\\ \;\;\;\;x - z \cdot \sin y\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{-37}:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(\frac{x}{z} - \sin y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -2.25e-123)
   (- x (* z (sin y)))
   (if (<= z 5.5e-37) (* x (cos y)) (* z (- (/ x z) (sin y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.25e-123) {
		tmp = x - (z * sin(y));
	} else if (z <= 5.5e-37) {
		tmp = x * cos(y);
	} else {
		tmp = z * ((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 (z <= (-2.25d-123)) then
        tmp = x - (z * sin(y))
    else if (z <= 5.5d-37) then
        tmp = x * cos(y)
    else
        tmp = z * ((x / z) - sin(y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.25e-123) {
		tmp = x - (z * Math.sin(y));
	} else if (z <= 5.5e-37) {
		tmp = x * Math.cos(y);
	} else {
		tmp = z * ((x / z) - Math.sin(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -2.25e-123:
		tmp = x - (z * math.sin(y))
	elif z <= 5.5e-37:
		tmp = x * math.cos(y)
	else:
		tmp = z * ((x / z) - math.sin(y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -2.25e-123)
		tmp = Float64(x - Float64(z * sin(y)));
	elseif (z <= 5.5e-37)
		tmp = Float64(x * cos(y));
	else
		tmp = Float64(z * Float64(Float64(x / z) - sin(y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -2.25e-123)
		tmp = x - (z * sin(y));
	elseif (z <= 5.5e-37)
		tmp = x * cos(y);
	else
		tmp = z * ((x / z) - sin(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -2.25e-123], N[(x - N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 5.5e-37], N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision], N[(z * N[(N[(x / z), $MachinePrecision] - N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.24999999999999997e-123

   1. Initial program 99.8%

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

      1. log1p-expm1-u 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in y around 0 84.5%

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

   if -2.24999999999999997e-123 < z < 5.4999999999999998e-37

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 87.9%

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

   if 5.4999999999999998e-37 < z

   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. Taylor expanded in y around 0 90.5%

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

   6. Taylor expanded in z around inf 90.5%

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

      1. +-commutative 90.5%

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

      2. mul-1-neg 90.5%

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

      3. unsub-neg 90.5%

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

   8. Simplified 90.5%

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

Alternative 6: 86.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.55 \cdot 10^{-123} \lor \neg \left(z \leq 1.06 \cdot 10^{-36}\right):\\ \;\;\;\;x - z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \cos y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -2.55e-123) (not (<= z 1.06e-36)))
   (- x (* z (sin y)))
   (* x (cos y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2.55e-123) || !(z <= 1.06e-36)) {
		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 <= (-2.55d-123)) .or. (.not. (z <= 1.06d-36))) 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 <= -2.55e-123) || !(z <= 1.06e-36)) {
		tmp = x - (z * Math.sin(y));
	} else {
		tmp = x * Math.cos(y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -2.55e-123) or not (z <= 1.06e-36):
		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 <= -2.55e-123) || !(z <= 1.06e-36))
		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 <= -2.55e-123) || ~((z <= 1.06e-36)))
		tmp = x - (z * sin(y));
	else
		tmp = x * cos(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -2.55e-123], N[Not[LessEqual[z, 1.06e-36]], $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 < -2.55000000000000005e-123 or 1.05999999999999999e-36 < z

   1. Initial program 99.8%

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

      1. log1p-expm1-u 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in y around 0 87.2%

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

   if -2.55000000000000005e-123 < z < 1.05999999999999999e-36

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 87.9%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.55 \cdot 10^{-123} \lor \neg \left(z \leq 1.06 \cdot 10^{-36}\right):\\ \;\;\;\;x - z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.046 \lor \neg \left(y \leq 2.5\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(-0.5 \cdot \left(y \cdot x\right) - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -0.046) (not (<= y 2.5)))
   (* x (cos y))
   (+ x (* y (- (* -0.5 (* y x)) z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.046) || !(y <= 2.5)) {
		tmp = x * cos(y);
	} else {
		tmp = x + (y * ((-0.5 * (y * x)) - 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 <= (-0.046d0)) .or. (.not. (y <= 2.5d0))) then
        tmp = x * cos(y)
    else
        tmp = x + (y * (((-0.5d0) * (y * x)) - z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.046) || !(y <= 2.5)) {
		tmp = x * Math.cos(y);
	} else {
		tmp = x + (y * ((-0.5 * (y * x)) - z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -0.046) or not (y <= 2.5):
		tmp = x * math.cos(y)
	else:
		tmp = x + (y * ((-0.5 * (y * x)) - z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -0.046) || !(y <= 2.5))
		tmp = Float64(x * cos(y));
	else
		tmp = Float64(x + Float64(y * Float64(Float64(-0.5 * Float64(y * x)) - z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -0.046) || ~((y <= 2.5)))
		tmp = x * cos(y);
	else
		tmp = x + (y * ((-0.5 * (y * x)) - z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -0.046], N[Not[LessEqual[y, 2.5]], $MachinePrecision]], N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision], N[(x + N[(y * N[(N[(-0.5 * N[(y * x), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -0.045999999999999999 or 2.5 < y

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf 49.6%

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

   if -0.045999999999999999 < y < 2.5

   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%

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

      1. sub-neg 99.3%

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

      2. sub-neg 99.3%

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

      3. *-commutative 99.3%

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

   5. Simplified 99.3%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.046 \lor \neg \left(y \leq 2.5\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(-0.5 \cdot \left(y \cdot x\right) - z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 41.3% accurate, 23.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{+192}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= z -2.1e+192) (* z (- y)) x))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.1e+192) {
		tmp = z * -y;
	} 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 <= (-2.1d+192)) then
        tmp = z * -y
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.1e+192) {
		tmp = z * -y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -2.1e+192:
		tmp = z * -y
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -2.1e+192)
		tmp = Float64(z * Float64(-y));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -2.1e+192)
		tmp = z * -y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -2.1e+192], N[(z * (-y)), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if z < -2.09999999999999995e192

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0 55.2%

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

      1. mul-1-neg 55.2%

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

      2. unsub-neg 55.2%

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

      3. *-commutative 55.2%

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

   5. Simplified 55.2%

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

   6. Taylor expanded in x around 0 38.9%

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

      1. neg-mul-1 38.9%

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

      2. distribute-rgt-neg-in 38.9%

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

   8. Simplified 38.9%

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

   if -2.09999999999999995e192 < z

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0 47.2%

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

4. Final simplification 46.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{+192}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 52.8% 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 99.8%

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

3. Taylor expanded in y around 0 57.6%

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

   1. mul-1-neg 57.6%

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

   2. unsub-neg 57.6%

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

   3. *-commutative 57.6%

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

5. Simplified 57.6%

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

Alternative 10: 39.8% 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 44.4%

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

Reproduce

herbie shell --seed 2024165 
(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))))