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

Percentage Accurate: 99.8% → 99.8%

Time: 8.4s

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

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_] := N[(N[Sin[y], $MachinePrecision] * (-z) + 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. sub-neg 99.8%

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

   2. +-commutative 99.8%

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

   3. *-commutative 99.8%

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

   4. distribute-rgt-neg-in 99.8%

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

   5. fma-def 99.8%

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

3. Applied egg-rr 99.8%

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

4. Final simplification 99.8%

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

Alternative 2: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

   \[x \cdot \cos y - z \cdot \sin y \]

2. Final simplification 99.8%

   \[\leadsto x \cdot \cos y - \sin y \cdot z \]

Alternative 3: 73.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin y \cdot \left(-z\right)\\ t_1 := x \cdot \cos y\\ \mathbf{if}\;y \leq -1.02 \cdot 10^{+239}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -7.5 \cdot 10^{+209}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -0.45:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{+15}:\\ \;\;\;\;\left(x + x \cdot \left(y \cdot \left(y \cdot -0.5\right)\right)\right) - y \cdot z\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{+160} \lor \neg \left(y \leq 2.6 \cdot 10^{+291}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (sin y) (- z))) (t_1 (* x (cos y))))
   (if (<= y -1.02e+239)
     t_0
     (if (<= y -7.5e+209)
       t_1
       (if (<= y -0.45)
         t_0
         (if (<= y 3.1e+15)
           (- (+ x (* x (* y (* y -0.5)))) (* y z))
           (if (or (<= y 7.2e+160) (not (<= y 2.6e+291))) t_1 t_0)))))))

C

double code(double x, double y, double z) {
	double t_0 = sin(y) * -z;
	double t_1 = x * cos(y);
	double tmp;
	if (y <= -1.02e+239) {
		tmp = t_0;
	} else if (y <= -7.5e+209) {
		tmp = t_1;
	} else if (y <= -0.45) {
		tmp = t_0;
	} else if (y <= 3.1e+15) {
		tmp = (x + (x * (y * (y * -0.5)))) - (y * z);
	} else if ((y <= 7.2e+160) || !(y <= 2.6e+291)) {
		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 = sin(y) * -z
    t_1 = x * cos(y)
    if (y <= (-1.02d+239)) then
        tmp = t_0
    else if (y <= (-7.5d+209)) then
        tmp = t_1
    else if (y <= (-0.45d0)) then
        tmp = t_0
    else if (y <= 3.1d+15) then
        tmp = (x + (x * (y * (y * (-0.5d0))))) - (y * z)
    else if ((y <= 7.2d+160) .or. (.not. (y <= 2.6d+291))) 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 = Math.sin(y) * -z;
	double t_1 = x * Math.cos(y);
	double tmp;
	if (y <= -1.02e+239) {
		tmp = t_0;
	} else if (y <= -7.5e+209) {
		tmp = t_1;
	} else if (y <= -0.45) {
		tmp = t_0;
	} else if (y <= 3.1e+15) {
		tmp = (x + (x * (y * (y * -0.5)))) - (y * z);
	} else if ((y <= 7.2e+160) || !(y <= 2.6e+291)) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.sin(y) * -z
	t_1 = x * math.cos(y)
	tmp = 0
	if y <= -1.02e+239:
		tmp = t_0
	elif y <= -7.5e+209:
		tmp = t_1
	elif y <= -0.45:
		tmp = t_0
	elif y <= 3.1e+15:
		tmp = (x + (x * (y * (y * -0.5)))) - (y * z)
	elif (y <= 7.2e+160) or not (y <= 2.6e+291):
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(sin(y) * Float64(-z))
	t_1 = Float64(x * cos(y))
	tmp = 0.0
	if (y <= -1.02e+239)
		tmp = t_0;
	elseif (y <= -7.5e+209)
		tmp = t_1;
	elseif (y <= -0.45)
		tmp = t_0;
	elseif (y <= 3.1e+15)
		tmp = Float64(Float64(x + Float64(x * Float64(y * Float64(y * -0.5)))) - Float64(y * z));
	elseif ((y <= 7.2e+160) || !(y <= 2.6e+291))
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = sin(y) * -z;
	t_1 = x * cos(y);
	tmp = 0.0;
	if (y <= -1.02e+239)
		tmp = t_0;
	elseif (y <= -7.5e+209)
		tmp = t_1;
	elseif (y <= -0.45)
		tmp = t_0;
	elseif (y <= 3.1e+15)
		tmp = (x + (x * (y * (y * -0.5)))) - (y * z);
	elseif ((y <= 7.2e+160) || ~((y <= 2.6e+291)))
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] * (-z)), $MachinePrecision]}, Block[{t$95$1 = N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.02e+239], t$95$0, If[LessEqual[y, -7.5e+209], t$95$1, If[LessEqual[y, -0.45], t$95$0, If[LessEqual[y, 3.1e+15], N[(N[(x + N[(x * N[(y * N[(y * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(y * z), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[y, 7.2e+160], N[Not[LessEqual[y, 2.6e+291]], $MachinePrecision]], t$95$1, t$95$0]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.02e239 or -7.50000000000000055e209 < y < -0.450000000000000011 or 7.20000000000000042e160 < y < 2.60000000000000006e291

   1. Initial program 99.6%

      \[x \cdot \cos y - z \cdot \sin y \]

   2. Taylor expanded in x around 0 64.9%

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

      1. associate-*r* 64.9%

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

      2. neg-mul-1 64.9%

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

   4. Simplified 64.9%

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

   if -1.02e239 < y < -7.50000000000000055e209 or 3.1e15 < y < 7.20000000000000042e160 or 2.60000000000000006e291 < y

   1. Initial program 99.6%

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

      1. sub-neg 99.6%

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

      2. +-commutative 99.6%

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

      3. *-commutative 99.6%

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

      4. distribute-rgt-neg-in 99.6%

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

      5. fma-def 99.7%

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

   3. Applied egg-rr 99.7%

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

   4. Taylor expanded in z around 0 67.1%

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

   if -0.450000000000000011 < y < 3.1e15

   1. Initial program 100.0%

      \[x \cdot \cos y - z \cdot \sin y \]

   2. Taylor expanded in y around 0 98.6%

      \[\leadsto \color{blue}{-1 \cdot \left(y \cdot z\right) + \left(-0.5 \cdot \left({y}^{2} \cdot x\right) + x\right)} \]
   3. Step-by-step derivation

      1. +-commutative 98.6%

         \[\leadsto \color{blue}{\left(-0.5 \cdot \left({y}^{2} \cdot x\right) + x\right) + -1 \cdot \left(y \cdot z\right)} \]

      2. mul-1-neg 98.6%

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

      3. unsub-neg 98.6%

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

      4. associate-*r* 98.6%

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

      5. distribute-lft1-in 98.6%

         \[\leadsto \color{blue}{\left(-0.5 \cdot {y}^{2} + 1\right) \cdot x} - y \cdot z \]

      6. fma-def 98.6%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, {y}^{2}, 1\right)} \cdot x - y \cdot z \]

      7. unpow2 98.6%

         \[\leadsto \mathsf{fma}\left(-0.5, \color{blue}{y \cdot y}, 1\right) \cdot x - y \cdot z \]

   4. Simplified 98.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, y \cdot y, 1\right) \cdot x - y \cdot z} \]
   5. Step-by-step derivation

      1. *-commutative 98.6%

         \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(-0.5, y \cdot y, 1\right)} - y \cdot z \]

      2. fma-udef 98.6%

         \[\leadsto x \cdot \color{blue}{\left(-0.5 \cdot \left(y \cdot y\right) + 1\right)} - y \cdot z \]

      3. distribute-rgt-in 98.6%

         \[\leadsto \color{blue}{\left(\left(-0.5 \cdot \left(y \cdot y\right)\right) \cdot x + 1 \cdot x\right)} - y \cdot z \]

      4. associate-*r* 98.6%

         \[\leadsto \left(\color{blue}{\left(\left(-0.5 \cdot y\right) \cdot y\right)} \cdot x + 1 \cdot x\right) - y \cdot z \]

      5. *-un-lft-identity 98.6%

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

   6. Applied egg-rr 98.6%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.02 \cdot 10^{+239}:\\ \;\;\;\;\sin y \cdot \left(-z\right)\\ \mathbf{elif}\;y \leq -7.5 \cdot 10^{+209}:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{elif}\;y \leq -0.45:\\ \;\;\;\;\sin y \cdot \left(-z\right)\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{+15}:\\ \;\;\;\;\left(x + x \cdot \left(y \cdot \left(y \cdot -0.5\right)\right)\right) - y \cdot z\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{+160} \lor \neg \left(y \leq 2.6 \cdot 10^{+291}\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \left(-z\right)\\ \end{array} \]

Alternative 4: 85.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{-8} \lor \neg \left(x \leq 8.2 \cdot 10^{+92}\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x - \sin y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -5.2e-8) (not (<= x 8.2e+92)))
   (* x (cos y))
   (- x (* (sin y) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -5.2e-8) || !(x <= 8.2e+92)) {
		tmp = x * cos(y);
	} else {
		tmp = x - (sin(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 ((x <= (-5.2d-8)) .or. (.not. (x <= 8.2d+92))) then
        tmp = x * cos(y)
    else
        tmp = x - (sin(y) * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -5.2e-8) || !(x <= 8.2e+92)) {
		tmp = x * Math.cos(y);
	} else {
		tmp = x - (Math.sin(y) * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -5.2e-8) or not (x <= 8.2e+92):
		tmp = x * math.cos(y)
	else:
		tmp = x - (math.sin(y) * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -5.2e-8) || !(x <= 8.2e+92))
		tmp = Float64(x * cos(y));
	else
		tmp = Float64(x - Float64(sin(y) * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -5.2e-8) || ~((x <= 8.2e+92)))
		tmp = x * cos(y);
	else
		tmp = x - (sin(y) * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -5.2e-8], N[Not[LessEqual[x, 8.2e+92]], $MachinePrecision]], N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision], N[(x - N[(N[Sin[y], $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.2000000000000002e-8 or 8.20000000000000047e92 < x

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

      2. +-commutative 99.8%

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

      3. *-commutative 99.8%

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

      4. distribute-rgt-neg-in 99.8%

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

      5. fma-def 99.8%

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

   3. Applied egg-rr 99.8%

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

   4. Taylor expanded in z around 0 89.4%

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

   if -5.2000000000000002e-8 < x < 8.20000000000000047e92

   1. Initial program 99.8%

      \[x \cdot \cos y - z \cdot \sin y \]

   2. Taylor expanded in y around 0 89.0%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{-8} \lor \neg \left(x \leq 8.2 \cdot 10^{+92}\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x - \sin y \cdot z\\ \end{array} \]

Alternative 5: 74.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.014 \lor \neg \left(y \leq 3.1 \cdot 10^{+15}\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;\left(x + x \cdot \left(y \cdot \left(y \cdot -0.5\right)\right)\right) - y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -0.014) (not (<= y 3.1e+15)))
   (* x (cos y))
   (- (+ x (* x (* y (* y -0.5)))) (* y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.014) || !(y <= 3.1e+15)) {
		tmp = x * cos(y);
	} else {
		tmp = (x + (x * (y * (y * -0.5)))) - (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 <= (-0.014d0)) .or. (.not. (y <= 3.1d+15))) then
        tmp = x * cos(y)
    else
        tmp = (x + (x * (y * (y * (-0.5d0))))) - (y * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.014) || !(y <= 3.1e+15)) {
		tmp = x * Math.cos(y);
	} else {
		tmp = (x + (x * (y * (y * -0.5)))) - (y * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -0.014) or not (y <= 3.1e+15):
		tmp = x * math.cos(y)
	else:
		tmp = (x + (x * (y * (y * -0.5)))) - (y * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -0.014) || !(y <= 3.1e+15))
		tmp = Float64(x * cos(y));
	else
		tmp = Float64(Float64(x + Float64(x * Float64(y * Float64(y * -0.5)))) - Float64(y * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -0.014) || ~((y <= 3.1e+15)))
		tmp = x * cos(y);
	else
		tmp = (x + (x * (y * (y * -0.5)))) - (y * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -0.014], N[Not[LessEqual[y, 3.1e+15]], $MachinePrecision]], N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision], N[(N[(x + N[(x * N[(y * N[(y * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(y * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -0.0140000000000000003 or 3.1e15 < y

   1. Initial program 99.6%

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

      1. sub-neg 99.6%

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

      2. +-commutative 99.6%

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

      3. *-commutative 99.6%

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

      4. distribute-rgt-neg-in 99.6%

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

      5. fma-def 99.6%

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

   3. Applied egg-rr 99.6%

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

   4. Taylor expanded in z around 0 49.6%

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

   if -0.0140000000000000003 < y < 3.1e15

   1. Initial program 100.0%

      \[x \cdot \cos y - z \cdot \sin y \]

   2. Taylor expanded in y around 0 99.1%

      \[\leadsto \color{blue}{-1 \cdot \left(y \cdot z\right) + \left(-0.5 \cdot \left({y}^{2} \cdot x\right) + x\right)} \]
   3. Step-by-step derivation

      1. +-commutative 99.1%

         \[\leadsto \color{blue}{\left(-0.5 \cdot \left({y}^{2} \cdot x\right) + x\right) + -1 \cdot \left(y \cdot z\right)} \]

      2. mul-1-neg 99.1%

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

      3. unsub-neg 99.1%

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

      4. associate-*r* 99.1%

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

      5. distribute-lft1-in 99.1%

         \[\leadsto \color{blue}{\left(-0.5 \cdot {y}^{2} + 1\right) \cdot x} - y \cdot z \]

      6. fma-def 99.1%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, {y}^{2}, 1\right)} \cdot x - y \cdot z \]

      7. unpow2 99.1%

         \[\leadsto \mathsf{fma}\left(-0.5, \color{blue}{y \cdot y}, 1\right) \cdot x - y \cdot z \]

   4. Simplified 99.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, y \cdot y, 1\right) \cdot x - y \cdot z} \]
   5. Step-by-step derivation

      1. *-commutative 99.1%

         \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(-0.5, y \cdot y, 1\right)} - y \cdot z \]

      2. fma-udef 99.1%

         \[\leadsto x \cdot \color{blue}{\left(-0.5 \cdot \left(y \cdot y\right) + 1\right)} - y \cdot z \]

      3. distribute-rgt-in 99.1%

         \[\leadsto \color{blue}{\left(\left(-0.5 \cdot \left(y \cdot y\right)\right) \cdot x + 1 \cdot x\right)} - y \cdot z \]

      4. associate-*r* 99.1%

         \[\leadsto \left(\color{blue}{\left(\left(-0.5 \cdot y\right) \cdot y\right)} \cdot x + 1 \cdot x\right) - y \cdot z \]

      5. *-un-lft-identity 99.1%

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

   6. Applied egg-rr 99.1%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.014 \lor \neg \left(y \leq 3.1 \cdot 10^{+15}\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;\left(x + x \cdot \left(y \cdot \left(y \cdot -0.5\right)\right)\right) - y \cdot z\\ \end{array} \]

Alternative 6: 42.0% accurate, 25.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{+143} \lor \neg \left(z \leq 10^{+94}\right):\\ \;\;\;\;y \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.25e+143) (not (<= z 1e+94))) (* y (- z)) x))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.25e+143) || !(z <= 1e+94)) {
		tmp = y * -z;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-1.25d+143)) .or. (.not. (z <= 1d+94))) then
        tmp = y * -z
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.25e+143) || !(z <= 1e+94)) {
		tmp = y * -z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.25e+143) or not (z <= 1e+94):
		tmp = y * -z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.25e+143) || !(z <= 1e+94))
		tmp = Float64(y * Float64(-z));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.25e+143) || ~((z <= 1e+94)))
		tmp = y * -z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.25e+143], N[Not[LessEqual[z, 1e+94]], $MachinePrecision]], N[(y * (-z)), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if z < -1.25000000000000003e143 or 1e94 < z

   1. Initial program 99.9%

      \[x \cdot \cos y - z \cdot \sin y \]

   2. Taylor expanded in y around 0 48.4%

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

      1. +-commutative 48.4%

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

      2. mul-1-neg 48.4%

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

      3. unsub-neg 48.4%

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

   4. Simplified 48.4%

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

   5. Taylor expanded in x around 0 34.1%

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

      1. associate-*r* 34.1%

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

      2. neg-mul-1 34.1%

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

   7. Simplified 34.1%

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

   if -1.25000000000000003e143 < z < 1e94

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

      2. +-commutative 99.8%

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

      3. *-commutative 99.8%

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

      4. distribute-rgt-neg-in 99.8%

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

      5. fma-def 99.8%

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

   3. Applied egg-rr 99.8%

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

   4. Taylor expanded in y around 0 50.3%

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

4. Final simplification 46.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{+143} \lor \neg \left(z \leq 10^{+94}\right):\\ \;\;\;\;y \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7: 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 99.8%

   \[x \cdot \cos y - z \cdot \sin y \]

2. Taylor expanded in y around 0 53.0%

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

   1. +-commutative 53.0%

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

   2. mul-1-neg 53.0%

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

   3. unsub-neg 53.0%

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

4. Simplified 53.0%

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

5. Final simplification 53.0%

   \[\leadsto x - y \cdot z \]

Alternative 8: 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 99.8%

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

   1. sub-neg 99.8%

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

   2. +-commutative 99.8%

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

   3. *-commutative 99.8%

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

   4. distribute-rgt-neg-in 99.8%

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

   5. fma-def 99.8%

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

3. Applied egg-rr 99.8%

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

4. Taylor expanded in y around 0 41.5%

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

5. Final simplification 41.5%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023213 
(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))))