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

Percentage Accurate: 99.8% → 99.8%

Time: 5.1s

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 99.8%

   \[x \cdot \cos y - z \cdot \sin y \]
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 99.8

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

   \[\leadsto \mathsf{fma}\left(\sin y, -z, x \cdot \cos y\right) \]
6. 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 99.8%

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

Alternative 3: 75.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-z\right) \cdot \sin y\\ \mathbf{if}\;y \leq -1.05 \cdot 10^{+259}:\\ \;\;\;\;\cos y \cdot x\\ \mathbf{elif}\;y \leq -52000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.044:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, z \cdot y, -0.5 \cdot x\right) \cdot y - z, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (- z) (sin y))))
   (if (<= y -1.05e+259)
     (* (cos y) x)
     (if (<= y -52000.0)
       t_0
       (if (<= y 0.044)
         (fma (- (* (fma 0.16666666666666666 (* z y) (* -0.5 x)) y) z) y x)
         t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = -z * sin(y);
	double tmp;
	if (y <= -1.05e+259) {
		tmp = cos(y) * x;
	} else if (y <= -52000.0) {
		tmp = t_0;
	} else if (y <= 0.044) {
		tmp = fma(((fma(0.16666666666666666, (z * y), (-0.5 * x)) * y) - z), y, x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(-z) * sin(y))
	tmp = 0.0
	if (y <= -1.05e+259)
		tmp = Float64(cos(y) * x);
	elseif (y <= -52000.0)
		tmp = t_0;
	elseif (y <= 0.044)
		tmp = fma(Float64(Float64(fma(0.16666666666666666, Float64(z * y), Float64(-0.5 * x)) * y) - z), y, x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[((-z) * N[Sin[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.05e+259], N[(N[Cos[y], $MachinePrecision] * x), $MachinePrecision], If[LessEqual[y, -52000.0], t$95$0, If[LessEqual[y, 0.044], N[(N[(N[(N[(0.16666666666666666 * N[(z * y), $MachinePrecision] + N[(-0.5 * x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] - z), $MachinePrecision] * y + x), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.05000000000000003e259

   1. Initial program 99.3%

      \[x \cdot \cos y - z \cdot \sin y \]
   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. inv-pow N/A

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

      9. lower-pow.f64 99.6

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

   4. Applied rewrites 99.6%

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

      1. lift-pow.f64 N/A

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

      2. unpow-1 N/A

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

      3. lower-/.f64 99.6

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.6

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

   6. Applied rewrites 99.6%

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

   7. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-cos.f64 86.4

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

   9. Applied rewrites 86.4%

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

   10. Taylor expanded in x around inf

       \[\leadsto \cos y \cdot x \]
   11. Step-by-step derivation

   12. Applied rewrites 82.1%

       \[\leadsto \cos y \cdot x \]

   if -1.05000000000000003e259 < y < -52000 or 0.043999999999999997 < y

   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. distribute-lft-neg-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-sin.f64 60.9

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

   5. Applied rewrites 60.9%

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

   if -52000 < y < 0.043999999999999997

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 99.2

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

   5. Applied rewrites 99.2%

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

Alternative 4: 86.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 \cdot x - z \cdot \sin y\\ \mathbf{if}\;z \leq -2.9 \cdot 10^{-62}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{-20}:\\ \;\;\;\;\cos y \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (* 1.0 x) (* z (sin y)))))
   (if (<= z -2.9e-62) t_0 (if (<= z 1.35e-20) (* (cos y) x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (1.0 * x) - (z * sin(y));
	double tmp;
	if (z <= -2.9e-62) {
		tmp = t_0;
	} else if (z <= 1.35e-20) {
		tmp = cos(y) * x;
	} 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 = (1.0d0 * x) - (z * sin(y))
    if (z <= (-2.9d-62)) then
        tmp = t_0
    else if (z <= 1.35d-20) then
        tmp = cos(y) * x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (1.0 * x) - (z * Math.sin(y));
	double tmp;
	if (z <= -2.9e-62) {
		tmp = t_0;
	} else if (z <= 1.35e-20) {
		tmp = Math.cos(y) * x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (1.0 * x) - (z * math.sin(y))
	tmp = 0
	if z <= -2.9e-62:
		tmp = t_0
	elif z <= 1.35e-20:
		tmp = math.cos(y) * x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(1.0 * x) - Float64(z * sin(y)))
	tmp = 0.0
	if (z <= -2.9e-62)
		tmp = t_0;
	elseif (z <= 1.35e-20)
		tmp = Float64(cos(y) * x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (1.0 * x) - (z * sin(y));
	tmp = 0.0;
	if (z <= -2.9e-62)
		tmp = t_0;
	elseif (z <= 1.35e-20)
		tmp = cos(y) * x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(1.0 * x), $MachinePrecision] - N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2.9e-62], t$95$0, If[LessEqual[z, 1.35e-20], N[(N[Cos[y], $MachinePrecision] * x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.89999999999999986e-62 or 1.35e-20 < z

   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 93.9%

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

   if -2.89999999999999986e-62 < z < 1.35e-20

   1. Initial program 99.8%

      \[x \cdot \cos y - z \cdot \sin y \]
   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. inv-pow N/A

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

      9. lower-pow.f64 99.7

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

   4. Applied rewrites 99.7%

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

      1. lift-pow.f64 N/A

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

      2. unpow-1 N/A

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

      3. lower-/.f64 99.7

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.7

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-cos.f64 99.8

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

   9. Applied rewrites 99.8%

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

   10. Taylor expanded in x around inf

       \[\leadsto \cos y \cdot x \]
   11. Step-by-step derivation

   12. Applied rewrites 91.3%

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

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.9 \cdot 10^{-62}:\\ \;\;\;\;1 \cdot x - z \cdot \sin y\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{-20}:\\ \;\;\;\;\cos y \cdot x\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x - z \cdot \sin y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos y \cdot x\\ \mathbf{if}\;y \leq -0.125:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 7.4 \cdot 10^{-9}:\\ \;\;\;\;1 \cdot x - \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right) \cdot z, y \cdot y, z\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (cos y) x)))
   (if (<= y -0.125)
     t_0
     (if (<= y 7.4e-9)
       (-
        (* 1.0 x)
        (*
         (fma
          (* (fma 0.008333333333333333 (* y y) -0.16666666666666666) z)
          (* y y)
          z)
         y))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = cos(y) * x;
	double tmp;
	if (y <= -0.125) {
		tmp = t_0;
	} else if (y <= 7.4e-9) {
		tmp = (1.0 * x) - (fma((fma(0.008333333333333333, (y * y), -0.16666666666666666) * z), (y * y), z) * y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(cos(y) * x)
	tmp = 0.0
	if (y <= -0.125)
		tmp = t_0;
	elseif (y <= 7.4e-9)
		tmp = Float64(Float64(1.0 * x) - Float64(fma(Float64(fma(0.008333333333333333, Float64(y * y), -0.16666666666666666) * z), Float64(y * y), z) * y));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Cos[y], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[y, -0.125], t$95$0, If[LessEqual[y, 7.4e-9], N[(N[(1.0 * x), $MachinePrecision] - N[(N[(N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * z), $MachinePrecision] * N[(y * y), $MachinePrecision] + z), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.125 or 7.4e-9 < y

   1. Initial program 99.6%

      \[x \cdot \cos y - z \cdot \sin y \]
   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. inv-pow N/A

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

      9. lower-pow.f64 99.6

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

   4. Applied rewrites 99.6%

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

      1. lift-pow.f64 N/A

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

      2. unpow-1 N/A

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

      3. lower-/.f64 99.6

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.6

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

   6. Applied rewrites 99.6%

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

   7. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-cos.f64 83.8

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

   9. Applied rewrites 83.8%

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

   10. Taylor expanded in x around inf

       \[\leadsto \cos y \cdot x \]
   11. Step-by-step derivation

   12. Applied rewrites 44.8%

       \[\leadsto \cos y \cdot x \]

   if -0.125 < y < 7.4e-9

   1. Initial program 100.0%

      \[x \cdot \cos y - z \cdot \sin y \]
   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 100.0%

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

   6. Taylor expanded in y around 0

      \[\leadsto x \cdot 1 - \color{blue}{y \cdot \left(z + {y}^{2} \cdot \left(\frac{-1}{6} \cdot z + \frac{1}{120} \cdot \left({y}^{2} \cdot z\right)\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto x \cdot 1 - \color{blue}{\left(z + {y}^{2} \cdot \left(\frac{-1}{6} \cdot z + \frac{1}{120} \cdot \left({y}^{2} \cdot z\right)\right)\right) \cdot y} \]

      2. lower-*.f64 N/A

         \[\leadsto x \cdot 1 - \color{blue}{\left(z + {y}^{2} \cdot \left(\frac{-1}{6} \cdot z + \frac{1}{120} \cdot \left({y}^{2} \cdot z\right)\right)\right) \cdot y} \]

   8. Applied rewrites 100.0%

      \[\leadsto x \cdot 1 - \color{blue}{\mathsf{fma}\left(z \cdot \mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), y \cdot y, z\right) \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.125:\\ \;\;\;\;\cos y \cdot x\\ \mathbf{elif}\;y \leq 7.4 \cdot 10^{-9}:\\ \;\;\;\;1 \cdot x - \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right) \cdot z, y \cdot y, z\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\cos y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 40.2% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-z\right) \cdot y\\ \mathbf{if}\;z \leq -2.2 \cdot 10^{+18}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{+151}:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (- z) y)))
   (if (<= z -2.2e+18) t_0 (if (<= z 5.2e+151) (* 1.0 x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = -z * y;
	double tmp;
	if (z <= -2.2e+18) {
		tmp = t_0;
	} else if (z <= 5.2e+151) {
		tmp = 1.0 * x;
	} 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 * y
    if (z <= (-2.2d+18)) then
        tmp = t_0
    else if (z <= 5.2d+151) then
        tmp = 1.0d0 * x
    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 * y;
	double tmp;
	if (z <= -2.2e+18) {
		tmp = t_0;
	} else if (z <= 5.2e+151) {
		tmp = 1.0 * x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = -z * y
	tmp = 0
	if z <= -2.2e+18:
		tmp = t_0
	elif z <= 5.2e+151:
		tmp = 1.0 * x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(-z) * y)
	tmp = 0.0
	if (z <= -2.2e+18)
		tmp = t_0;
	elseif (z <= 5.2e+151)
		tmp = Float64(1.0 * x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = -z * y;
	tmp = 0.0;
	if (z <= -2.2e+18)
		tmp = t_0;
	elseif (z <= 5.2e+151)
		tmp = 1.0 * x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[((-z) * y), $MachinePrecision]}, If[LessEqual[z, -2.2e+18], t$95$0, If[LessEqual[z, 5.2e+151], N[(1.0 * x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.2e18 or 5.20000000000000026e151 < z

   1. Initial program 99.8%

      \[x \cdot \cos y - z \cdot \sin y \]
   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 56.3

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

   5. Applied rewrites 56.3%

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

   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 40.4%

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

   if -2.2e18 < z < 5.20000000000000026e151

   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. 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. inv-pow N/A

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

      9. lower-pow.f64 99.7

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

   4. Applied rewrites 99.7%

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

      1. lift-pow.f64 N/A

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

      2. unpow-1 N/A

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

      3. lower-/.f64 99.7

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.7

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-cos.f64 97.4

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

   9. Applied rewrites 97.4%

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

   10. Taylor expanded in y around 0

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

   12. Applied rewrites 54.5%

       \[\leadsto 1 \cdot x \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 51.8% 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 99.8%

   \[x \cdot \cos y - z \cdot \sin y \]
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 58.8

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

5. Applied rewrites 58.8%

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

7. Applied rewrites 58.8%

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

Alternative 8: 51.8% 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 99.8%

   \[x \cdot \cos y - z \cdot \sin y \]
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 58.8

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

5. Applied rewrites 58.8%

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

Alternative 9: 38.5% accurate, 35.7× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return 1.0 * 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 = 1.0d0 * x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

   \[x \cdot \cos y - z \cdot \sin y \]
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. inv-pow N/A

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

   9. lower-pow.f64 99.6

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

4. Applied rewrites 99.6%

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

   1. lift-pow.f64 N/A

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

   2. unpow-1 N/A

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

   3. lower-/.f64 99.6

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 99.6

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

6. Applied rewrites 99.6%

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

7. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. associate-/l* N/A

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

   5. associate-*r* N/A

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

   6. lower-fma.f64 N/A

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

   7. mul-1-neg N/A

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

   8. lower-neg.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-cos.f64 91.6

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

9. Applied rewrites 91.6%

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

10. Taylor expanded in y around 0

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

12. Applied rewrites 40.5%

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

Reproduce

herbie shell --seed 2024308 
(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))))