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

Percentage Accurate: 99.8% → 99.8%

Time: 10.5s

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

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

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 99.9%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. distribute-rgt-neg-in N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-neg.f64 99.9

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

4. Applied rewrites 99.9%

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

Alternative 2: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x * cos(y)) - (z * sin(y))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

Alternative 3: 74.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -z \cdot \sin y\\ t_1 := x \cdot \cos y\\ \mathbf{if}\;y \leq -3.5 \cdot 10^{+224}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -0.15:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 0.0305:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(x, -0.5, \left(y \cdot z\right) \cdot 0.16666666666666666\right) - z, x\right)\\ \mathbf{elif}\;y \leq 1.06 \cdot 10^{+136}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (* z (sin y)))) (t_1 (* x (cos y))))
   (if (<= y -3.5e+224)
     t_0
     (if (<= y -0.15)
       t_1
       (if (<= y 0.0305)
         (fma y (- (* y (fma x -0.5 (* (* y z) 0.16666666666666666))) z) x)
         (if (<= y 1.06e+136) t_0 t_1))))))

C

double code(double x, double y, double z) {
	double t_0 = -(z * sin(y));
	double t_1 = x * cos(y);
	double tmp;
	if (y <= -3.5e+224) {
		tmp = t_0;
	} else if (y <= -0.15) {
		tmp = t_1;
	} else if (y <= 0.0305) {
		tmp = fma(y, ((y * fma(x, -0.5, ((y * z) * 0.16666666666666666))) - z), x);
	} else if (y <= 1.06e+136) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(-Float64(z * sin(y)))
	t_1 = Float64(x * cos(y))
	tmp = 0.0
	if (y <= -3.5e+224)
		tmp = t_0;
	elseif (y <= -0.15)
		tmp = t_1;
	elseif (y <= 0.0305)
		tmp = fma(y, Float64(Float64(y * fma(x, -0.5, Float64(Float64(y * z) * 0.16666666666666666))) - z), x);
	elseif (y <= 1.06e+136)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = (-N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision])}, Block[{t$95$1 = N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -3.5e+224], t$95$0, If[LessEqual[y, -0.15], t$95$1, If[LessEqual[y, 0.0305], N[(y * N[(N[(y * N[(x * -0.5 + N[(N[(y * z), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[y, 1.06e+136], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.5e224 or 0.030499999999999999 < y < 1.06000000000000003e136

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 77.4

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

   5. Applied rewrites 77.4%

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

   if -3.5e224 < y < -0.149999999999999994 or 1.06000000000000003e136 < y

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 61.0

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

   5. Applied rewrites 61.0%

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

   if -0.149999999999999994 < y < 0.030499999999999999

   1. Initial program 100.0%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-- N/A

         \[\leadsto \frac{1}{\frac{1}{\color{blue}{x \cdot \cos y - z \cdot \sin y}}} \]

      7. lift--.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{\color{blue}{x \cdot \cos y - z \cdot \sin y}}} \]

      8. lower-/.f64 99.7

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{x \cdot \cos y - z \cdot \sin y}}} \]

      9. lift--.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{\color{blue}{x \cdot \cos y - z \cdot \sin y}}} \]

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. distribute-rgt-neg-in N/A

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + y \cdot \left(-1 \cdot z + y \cdot \left(\frac{-1}{2} \cdot x + \frac{1}{6} \cdot \left(y \cdot z\right)\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{y \cdot \left(-1 \cdot z + y \cdot \left(\frac{-1}{2} \cdot x + \frac{1}{6} \cdot \left(y \cdot z\right)\right)\right) + x} \]

      2. mul-1-neg N/A

         \[\leadsto y \cdot \left(\color{blue}{\left(\mathsf{neg}\left(z\right)\right)} + y \cdot \left(\frac{-1}{2} \cdot x + \frac{1}{6} \cdot \left(y \cdot z\right)\right)\right) + x \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, \left(\mathsf{neg}\left(z\right)\right) + y \cdot \left(\frac{-1}{2} \cdot x + \frac{1}{6} \cdot \left(y \cdot z\right)\right), x\right)} \]

      4. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{y \cdot \left(\frac{-1}{2} \cdot x + \frac{1}{6} \cdot \left(y \cdot z\right)\right) + \left(\mathsf{neg}\left(z\right)\right)}, x\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{y \cdot \left(\frac{-1}{2} \cdot x + \frac{1}{6} \cdot \left(y \cdot z\right)\right) - z}, x\right) \]

      6. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{y \cdot \left(\frac{-1}{2} \cdot x + \frac{1}{6} \cdot \left(y \cdot z\right)\right) - z}, x\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{y \cdot \left(\frac{-1}{2} \cdot x + \frac{1}{6} \cdot \left(y \cdot z\right)\right)} - z, x\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, y \cdot \left(\color{blue}{x \cdot \frac{-1}{2}} + \frac{1}{6} \cdot \left(y \cdot z\right)\right) - z, x\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, y \cdot \color{blue}{\mathsf{fma}\left(x, \frac{-1}{2}, \frac{1}{6} \cdot \left(y \cdot z\right)\right)} - z, x\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(x, \frac{-1}{2}, \color{blue}{\left(y \cdot z\right) \cdot \frac{1}{6}}\right) - z, x\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(x, \frac{-1}{2}, \color{blue}{\left(y \cdot z\right) \cdot \frac{1}{6}}\right) - z, x\right) \]

      12. lower-*.f64 99.8

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

   7. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(x, -0.5, \left(y \cdot z\right) \cdot 0.16666666666666666\right) - z, x\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 86.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \cos y\\ \mathbf{if}\;x \leq -2 \cdot 10^{+86}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.92 \cdot 10^{+34}:\\ \;\;\;\;x \cdot 1 - z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (cos y))))
   (if (<= x -2e+86)
     t_0
     (if (<= x 1.92e+34) (- (* x 1.0) (* z (sin y))) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x * cos(y);
	double tmp;
	if (x <= -2e+86) {
		tmp = t_0;
	} else if (x <= 1.92e+34) {
		tmp = (x * 1.0) - (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 = x * cos(y)
    if (x <= (-2d+86)) then
        tmp = t_0
    else if (x <= 1.92d+34) then
        tmp = (x * 1.0d0) - (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 = x * Math.cos(y);
	double tmp;
	if (x <= -2e+86) {
		tmp = t_0;
	} else if (x <= 1.92e+34) {
		tmp = (x * 1.0) - (z * Math.sin(y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * math.cos(y)
	tmp = 0
	if x <= -2e+86:
		tmp = t_0
	elif x <= 1.92e+34:
		tmp = (x * 1.0) - (z * math.sin(y))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * cos(y))
	tmp = 0.0
	if (x <= -2e+86)
		tmp = t_0;
	elseif (x <= 1.92e+34)
		tmp = Float64(Float64(x * 1.0) - Float64(z * sin(y)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * cos(y);
	tmp = 0.0;
	if (x <= -2e+86)
		tmp = t_0;
	elseif (x <= 1.92e+34)
		tmp = (x * 1.0) - (z * sin(y));
	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]}, If[LessEqual[x, -2e+86], t$95$0, If[LessEqual[x, 1.92e+34], N[(N[(x * 1.0), $MachinePrecision] - N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -2e86 or 1.92e34 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 89.0

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

   5. Applied rewrites 89.0%

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

   if -2e86 < x < 1.92e34

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 87.2%

      \[\leadsto x \cdot \color{blue}{1} - z \cdot \sin y \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 74.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \cos y\\ \mathbf{if}\;y \leq -0.15:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 21000:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(x, -0.5, \left(y \cdot z\right) \cdot 0.16666666666666666\right) - z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (cos y))))
   (if (<= y -0.15)
     t_0
     (if (<= y 21000.0)
       (fma y (- (* y (fma x -0.5 (* (* y z) 0.16666666666666666))) z) x)
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x * cos(y);
	double tmp;
	if (y <= -0.15) {
		tmp = t_0;
	} else if (y <= 21000.0) {
		tmp = fma(y, ((y * fma(x, -0.5, ((y * z) * 0.16666666666666666))) - z), x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x * cos(y))
	tmp = 0.0
	if (y <= -0.15)
		tmp = t_0;
	elseif (y <= 21000.0)
		tmp = fma(y, Float64(Float64(y * fma(x, -0.5, Float64(Float64(y * z) * 0.16666666666666666))) - z), x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.15], t$95$0, If[LessEqual[y, 21000.0], N[(y * N[(N[(y * N[(x * -0.5 + N[(N[(y * z), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision] + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.149999999999999994 or 21000 < y

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 49.4

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

   5. Applied rewrites 49.4%

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

   if -0.149999999999999994 < y < 21000

   1. Initial program 100.0%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-- N/A

         \[\leadsto \frac{1}{\frac{1}{\color{blue}{x \cdot \cos y - z \cdot \sin y}}} \]

      7. lift--.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{\color{blue}{x \cdot \cos y - z \cdot \sin y}}} \]

      8. lower-/.f64 99.7

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{x \cdot \cos y - z \cdot \sin y}}} \]

      9. lift--.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{\color{blue}{x \cdot \cos y - z \cdot \sin y}}} \]

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. distribute-rgt-neg-in N/A

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + y \cdot \left(-1 \cdot z + y \cdot \left(\frac{-1}{2} \cdot x + \frac{1}{6} \cdot \left(y \cdot z\right)\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{y \cdot \left(-1 \cdot z + y \cdot \left(\frac{-1}{2} \cdot x + \frac{1}{6} \cdot \left(y \cdot z\right)\right)\right) + x} \]

      2. mul-1-neg N/A

         \[\leadsto y \cdot \left(\color{blue}{\left(\mathsf{neg}\left(z\right)\right)} + y \cdot \left(\frac{-1}{2} \cdot x + \frac{1}{6} \cdot \left(y \cdot z\right)\right)\right) + x \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, \left(\mathsf{neg}\left(z\right)\right) + y \cdot \left(\frac{-1}{2} \cdot x + \frac{1}{6} \cdot \left(y \cdot z\right)\right), x\right)} \]

      4. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{y \cdot \left(\frac{-1}{2} \cdot x + \frac{1}{6} \cdot \left(y \cdot z\right)\right) + \left(\mathsf{neg}\left(z\right)\right)}, x\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{y \cdot \left(\frac{-1}{2} \cdot x + \frac{1}{6} \cdot \left(y \cdot z\right)\right) - z}, x\right) \]

      6. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{y \cdot \left(\frac{-1}{2} \cdot x + \frac{1}{6} \cdot \left(y \cdot z\right)\right) - z}, x\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{y \cdot \left(\frac{-1}{2} \cdot x + \frac{1}{6} \cdot \left(y \cdot z\right)\right)} - z, x\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, y \cdot \left(\color{blue}{x \cdot \frac{-1}{2}} + \frac{1}{6} \cdot \left(y \cdot z\right)\right) - z, x\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, y \cdot \color{blue}{\mathsf{fma}\left(x, \frac{-1}{2}, \frac{1}{6} \cdot \left(y \cdot z\right)\right)} - z, x\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(x, \frac{-1}{2}, \color{blue}{\left(y \cdot z\right) \cdot \frac{1}{6}}\right) - z, x\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(x, \frac{-1}{2}, \color{blue}{\left(y \cdot z\right) \cdot \frac{1}{6}}\right) - z, x\right) \]

      12. lower-*.f64 97.6

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

   7. Applied rewrites 97.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(x, -0.5, \left(y \cdot z\right) \cdot 0.16666666666666666\right) - z, x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 41.6% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{-120}:\\ \;\;\;\;x \cdot 1\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{-199}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -6.2e-120) (* x 1.0) (if (<= x 2.7e-199) (* y (- z)) (* x 1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -6.2e-120) {
		tmp = x * 1.0;
	} else if (x <= 2.7e-199) {
		tmp = y * -z;
	} else {
		tmp = x * 1.0;
	}
	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) :: tmp
    if (x <= (-6.2d-120)) then
        tmp = x * 1.0d0
    else if (x <= 2.7d-199) then
        tmp = y * -z
    else
        tmp = x * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -6.2e-120) {
		tmp = x * 1.0;
	} else if (x <= 2.7e-199) {
		tmp = y * -z;
	} else {
		tmp = x * 1.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -6.2e-120:
		tmp = x * 1.0
	elif x <= 2.7e-199:
		tmp = y * -z
	else:
		tmp = x * 1.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -6.2e-120)
		tmp = Float64(x * 1.0);
	elseif (x <= 2.7e-199)
		tmp = Float64(y * Float64(-z));
	else
		tmp = Float64(x * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -6.2e-120)
		tmp = x * 1.0;
	elseif (x <= 2.7e-199)
		tmp = y * -z;
	else
		tmp = x * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -6.2e-120], N[(x * 1.0), $MachinePrecision], If[LessEqual[x, 2.7e-199], N[(y * (-z)), $MachinePrecision], N[(x * 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -6.20000000000000038e-120 or 2.69999999999999989e-199 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 71.2

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

   5. Applied rewrites 71.2%

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

   6. Taylor expanded in y around 0

      \[\leadsto x \cdot 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 45.7%

      \[\leadsto x \cdot 1 \]

   if -6.20000000000000038e-120 < x < 2.69999999999999989e-199

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 52.3

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

   5. Applied rewrites 52.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 41.9%

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

Alternative 7: 51.9% accurate, 23.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in y around 0

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 54.0

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

5. Applied rewrites 54.0%

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

7. Applied rewrites 54.0%

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

Alternative 8: 51.9% accurate, 23.8× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x - (z * y)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in y around 0

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 54.0

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

5. Applied rewrites 54.0%

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

Alternative 9: 39.2% accurate, 35.7× speedup

Math

\[\begin{array}{l} \\ x \cdot 1 \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* x 1.0))

C

double code(double x, double y, double z) {
	return x * 1.0;
}

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 * 1.0d0
end function

Java

public static double code(double x, double y, double z) {
	return x * 1.0;
}

Python

def code(x, y, z):
	return x * 1.0

Julia

function code(x, y, z)
	return Float64(x * 1.0)
end

MATLAB

function tmp = code(x, y, z)
	tmp = x * 1.0;
end

Wolfram

code[x_, y_, z_] := N[(x * 1.0), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around inf

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

   1. lower-*.f64 N/A

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

   2. lower-cos.f64 57.4

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

5. Applied rewrites 57.4%

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

6. Taylor expanded in y around 0

   \[\leadsto x \cdot 1 \]
7. Step-by-step derivation

8. Applied rewrites 37.7%

   \[\leadsto x \cdot 1 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024254 
(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))))