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

Percentage Accurate: 99.8% → 99.8%

Time: 11.9s

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, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_] := N[(N[Cos[y], $MachinePrecision] * x + (-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. Step-by-step derivation

   1. lift-cos.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. sub-neg N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-neg.f64 99.8

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

4. Applied egg-rr 99.8%

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

Alternative 2: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(N[(N[Cos[y], $MachinePrecision] * x), $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. Final simplification 99.8%

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

Alternative 3: 74.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -z \cdot \sin y\\ \mathbf{if}\;y \leq -0.048:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.08:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(z, y \cdot 0.16666666666666666, x \cdot -0.5\right), -z\right), x\right)\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{+105}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\cos y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (* z (sin y)))))
   (if (<= y -0.048)
     t_0
     (if (<= y 0.08)
       (fma y (fma y (fma z (* y 0.16666666666666666) (* x -0.5)) (- z)) x)
       (if (<= y 6.2e+105) t_0 (* (cos y) x))))))

C

double code(double x, double y, double z) {
	double t_0 = -(z * sin(y));
	double tmp;
	if (y <= -0.048) {
		tmp = t_0;
	} else if (y <= 0.08) {
		tmp = fma(y, fma(y, fma(z, (y * 0.16666666666666666), (x * -0.5)), -z), x);
	} else if (y <= 6.2e+105) {
		tmp = t_0;
	} else {
		tmp = cos(y) * x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(-Float64(z * sin(y)))
	tmp = 0.0
	if (y <= -0.048)
		tmp = t_0;
	elseif (y <= 0.08)
		tmp = fma(y, fma(y, fma(z, Float64(y * 0.16666666666666666), Float64(x * -0.5)), Float64(-z)), x);
	elseif (y <= 6.2e+105)
		tmp = t_0;
	else
		tmp = Float64(cos(y) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = (-N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision])}, If[LessEqual[y, -0.048], t$95$0, If[LessEqual[y, 0.08], N[(y * N[(y * N[(z * N[(y * 0.16666666666666666), $MachinePrecision] + N[(x * -0.5), $MachinePrecision]), $MachinePrecision] + (-z)), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[y, 6.2e+105], t$95$0, N[(N[Cos[y], $MachinePrecision] * x), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.048000000000000001 or 0.0800000000000000017 < y < 6.20000000000000008e105

   1. Initial program 99.6%

      \[x \cdot \cos y - z \cdot \sin y \]
   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 60.7

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

   5. Simplified 60.7%

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

   if -0.048000000000000001 < y < 0.0800000000000000017

   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. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 99.3

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

   5. Simplified 99.3%

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

   if 6.20000000000000008e105 < y

   1. Initial program 99.5%

      \[x \cdot \cos y - z \cdot \sin y \]
   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 59.1

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

   5. Simplified 59.1%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.048:\\ \;\;\;\;-z \cdot \sin y\\ \mathbf{elif}\;y \leq 0.08:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(z, y \cdot 0.16666666666666666, x \cdot -0.5\right), -z\right), x\right)\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{+105}:\\ \;\;\;\;-z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;\cos y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\sin y, -z, x\right)\\ \mathbf{if}\;z \leq -3 \cdot 10^{-16}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{-147}:\\ \;\;\;\;\cos y \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (sin y) (- z) x)))
   (if (<= z -3e-16) t_0 (if (<= z 8.5e-147) (* (cos y) x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma(sin(y), -z, x);
	double tmp;
	if (z <= -3e-16) {
		tmp = t_0;
	} else if (z <= 8.5e-147) {
		tmp = cos(y) * x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(sin(y), Float64(-z), x)
	tmp = 0.0
	if (z <= -3e-16)
		tmp = t_0;
	elseif (z <= 8.5e-147)
		tmp = Float64(cos(y) * x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] * (-z) + x), $MachinePrecision]}, If[LessEqual[z, -3e-16], t$95$0, If[LessEqual[z, 8.5e-147], N[(N[Cos[y], $MachinePrecision] * x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.99999999999999994e-16 or 8.5000000000000002e-147 < z

   1. Initial program 99.8%

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

      1. lift-cos.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. sub-neg N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-neg.f64 99.8

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sin.f64 90.6

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

   7. Simplified 90.6%

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

   8. Taylor expanded in y around 0

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

   10. Simplified 79.8%

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

   11. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

       4. lower-fma.f64 N/A

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

       5. lower-sin.f64 N/A

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

       6. mul-1-neg N/A

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

       7. lower-neg.f64 89.0

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

   13. Simplified 89.0%

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

   if -2.99999999999999994e-16 < z < 8.5000000000000002e-147

   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 89.5

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

   5. Simplified 89.5%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3 \cdot 10^{-16}:\\ \;\;\;\;\mathsf{fma}\left(\sin y, -z, x\right)\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{-147}:\\ \;\;\;\;\cos y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sin y, -z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos y \cdot x\\ \mathbf{if}\;y \leq -170000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.045:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(z, y \cdot 0.16666666666666666, x \cdot -0.5\right), -z\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (cos y) x)))
   (if (<= y -170000.0)
     t_0
     (if (<= y 0.045)
       (fma y (fma y (fma z (* y 0.16666666666666666) (* x -0.5)) (- z)) x)
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = cos(y) * x;
	double tmp;
	if (y <= -170000.0) {
		tmp = t_0;
	} else if (y <= 0.045) {
		tmp = fma(y, fma(y, fma(z, (y * 0.16666666666666666), (x * -0.5)), -z), x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(cos(y) * x)
	tmp = 0.0
	if (y <= -170000.0)
		tmp = t_0;
	elseif (y <= 0.045)
		tmp = fma(y, fma(y, fma(z, Float64(y * 0.16666666666666666), Float64(x * -0.5)), Float64(-z)), x);
	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, -170000.0], t$95$0, If[LessEqual[y, 0.045], N[(y * N[(y * N[(z * N[(y * 0.16666666666666666), $MachinePrecision] + N[(x * -0.5), $MachinePrecision]), $MachinePrecision] + (-z)), $MachinePrecision] + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.7e5 or 0.044999999999999998 < y

   1. Initial program 99.6%

      \[x \cdot \cos y - z \cdot \sin y \]
   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 46.9

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

   5. Simplified 46.9%

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

   if -1.7e5 < y < 0.044999999999999998

   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. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 99.0

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

   5. Simplified 99.0%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -170000:\\ \;\;\;\;\cos y \cdot x\\ \mathbf{elif}\;y \leq 0.045:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(z, y \cdot 0.16666666666666666, x \cdot -0.5\right), -z\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\cos y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 41.7% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(-z\right)\\ \mathbf{if}\;z \leq -1.18 \cdot 10^{+185}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 4.6 \cdot 10^{+194}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (- z))))
   (if (<= z -1.18e+185) t_0 (if (<= z 4.6e+194) x t_0))))

C

double code(double x, double y, double z) {
	double t_0 = y * -z;
	double tmp;
	if (z <= -1.18e+185) {
		tmp = t_0;
	} else if (z <= 4.6e+194) {
		tmp = 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 = y * -z
    if (z <= (-1.18d+185)) then
        tmp = t_0
    else if (z <= 4.6d+194) then
        tmp = 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 = y * -z;
	double tmp;
	if (z <= -1.18e+185) {
		tmp = t_0;
	} else if (z <= 4.6e+194) {
		tmp = x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * -z
	tmp = 0
	if z <= -1.18e+185:
		tmp = t_0
	elif z <= 4.6e+194:
		tmp = x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * Float64(-z))
	tmp = 0.0
	if (z <= -1.18e+185)
		tmp = t_0;
	elseif (z <= 4.6e+194)
		tmp = x;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y * -z;
	tmp = 0.0;
	if (z <= -1.18e+185)
		tmp = t_0;
	elseif (z <= 4.6e+194)
		tmp = x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * (-z)), $MachinePrecision]}, If[LessEqual[z, -1.18e+185], t$95$0, If[LessEqual[z, 4.6e+194], x, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.18e185 or 4.6000000000000001e194 < z

   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 80.7

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

   5. Simplified 80.7%

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

   6. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 37.7

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

   8. Simplified 37.7%

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

   if -1.18e185 < z < 4.6000000000000001e194

   1. Initial program 99.8%

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

      1. lift-cos.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. sub-neg N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-neg.f64 99.8

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sin.f64 96.5

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

   7. Simplified 96.5%

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

   8. Taylor expanded in y around 0

      \[\leadsto x \cdot \color{blue}{1} \]
   9. Step-by-step derivation

   10. Simplified 50.1%

       \[\leadsto x \cdot \color{blue}{1} \]
   11. Step-by-step derivation

       1. *-rgt-identity 50.1

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

   12. Applied egg-rr 50.1%

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

Alternative 7: 52.2% accurate, 23.8× 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. 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 55.2

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

5. Simplified 55.2%

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

6. Final simplification 55.2%

   \[\leadsto x - y \cdot z \]
7. Add Preprocessing

Alternative 8: 38.7% accurate, 214.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 99.8%

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

   1. lift-cos.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. sub-neg N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-neg.f64 99.8

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

4. Applied egg-rr 99.8%

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

5. Taylor expanded in x around inf

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

   1. lower-*.f64 N/A

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

   2. mul-1-neg N/A

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

   3. unsub-neg N/A

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

   4. lower--.f64 N/A

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

   5. lower-cos.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-sin.f64 94.2

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

7. Simplified 94.2%

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

8. Taylor expanded in y around 0

   \[\leadsto x \cdot \color{blue}{1} \]
9. Step-by-step derivation

10. Simplified 42.6%

    \[\leadsto x \cdot \color{blue}{1} \]
11. Step-by-step derivation

    1. *-rgt-identity 42.6

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

12. Applied egg-rr 42.6%

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

Reproduce

herbie shell --seed 2024208 
(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))))