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

Percentage Accurate: 99.8% → 99.8%

Time: 7.9s

Alternatives: 8

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 99.8

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 99.8

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

4. Applied rewrites 99.8%

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

Alternative 2: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 99.8

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 99.8

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

4. Applied rewrites 99.8%

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

Alternative 3: 85.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.8 \cdot 10^{+118} \lor \neg \left(z \leq 2.7 \cdot 10^{+24}\right):\\ \;\;\;\;\cos y \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, z, x \cdot \sin y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.8e+118) (not (<= z 2.7e+24)))
   (* (cos y) z)
   (fma 1.0 z (* x (sin y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.8e+118) || !(z <= 2.7e+24)) {
		tmp = cos(y) * z;
	} else {
		tmp = fma(1.0, z, (x * sin(y)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.8e+118) || !(z <= 2.7e+24))
		tmp = Float64(cos(y) * z);
	else
		tmp = fma(1.0, z, Float64(x * sin(y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.8e+118], N[Not[LessEqual[z, 2.7e+24]], $MachinePrecision]], N[(N[Cos[y], $MachinePrecision] * z), $MachinePrecision], N[(1.0 * z + N[(x * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.8e118 or 2.7e24 < z

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 89.1

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

   5. Applied rewrites 89.1%

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

   if -1.8e118 < z < 2.7e24

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 91.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 91.2

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 91.2

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

   7. Applied rewrites 91.2%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.8 \cdot 10^{+118} \lor \neg \left(z \leq 2.7 \cdot 10^{+24}\right):\\ \;\;\;\;\cos y \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, z, x \cdot \sin y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.00096 \lor \neg \left(y \leq 0.42\right):\\ \;\;\;\;\cos y \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y \cdot x, -0.5 \cdot z\right), y, x\right), y, z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -0.00096) (not (<= y 0.42)))
   (* (cos y) z)
   (fma (fma (fma -0.16666666666666666 (* y x) (* -0.5 z)) y x) y z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.00096) || !(y <= 0.42)) {
		tmp = cos(y) * z;
	} else {
		tmp = fma(fma(fma(-0.16666666666666666, (y * x), (-0.5 * z)), y, x), y, z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -0.00096) || !(y <= 0.42))
		tmp = Float64(cos(y) * z);
	else
		tmp = fma(fma(fma(-0.16666666666666666, Float64(y * x), Float64(-0.5 * z)), y, x), y, z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -0.00096], N[Not[LessEqual[y, 0.42]], $MachinePrecision]], N[(N[Cos[y], $MachinePrecision] * z), $MachinePrecision], N[(N[(N[(-0.16666666666666666 * N[(y * x), $MachinePrecision] + N[(-0.5 * z), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision] * y + z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -9.60000000000000024e-4 or 0.419999999999999984 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 50.9

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

   5. Applied rewrites 50.9%

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

   if -9.60000000000000024e-4 < y < 0.419999999999999984

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.00096 \lor \neg \left(y \leq 0.42\right):\\ \;\;\;\;\cos y \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y \cdot x, -0.5 \cdot z\right), y, x\right), y, z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.25:\\ \;\;\;\;\sin y \cdot x\\ \mathbf{elif}\;y \leq 0.42:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y \cdot x, -0.5 \cdot z\right), y, x\right), y, z\right)\\ \mathbf{else}:\\ \;\;\;\;\cos y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.25)
   (* (sin y) x)
   (if (<= y 0.42)
     (fma (fma (fma -0.16666666666666666 (* y x) (* -0.5 z)) y x) y z)
     (* (cos y) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.25) {
		tmp = sin(y) * x;
	} else if (y <= 0.42) {
		tmp = fma(fma(fma(-0.16666666666666666, (y * x), (-0.5 * z)), y, x), y, z);
	} else {
		tmp = cos(y) * z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.25)
		tmp = Float64(sin(y) * x);
	elseif (y <= 0.42)
		tmp = fma(fma(fma(-0.16666666666666666, Float64(y * x), Float64(-0.5 * z)), y, x), y, z);
	else
		tmp = Float64(cos(y) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.25], N[(N[Sin[y], $MachinePrecision] * x), $MachinePrecision], If[LessEqual[y, 0.42], N[(N[(N[(-0.16666666666666666 * N[(y * x), $MachinePrecision] + N[(-0.5 * z), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision] * y + z), $MachinePrecision], N[(N[Cos[y], $MachinePrecision] * z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.25

   1. Initial program 99.7%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 99.7

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-sin.f64 94.4

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

   7. Applied rewrites 94.4%

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

   8. Taylor expanded in x around inf

      \[\leadsto \sin y \cdot x \]
   9. Step-by-step derivation

   10. Applied rewrites 56.7%

       \[\leadsto \sin y \cdot x \]

   if -1.25 < y < 0.419999999999999984

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 99.4

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

   5. Applied rewrites 99.4%

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

   if 0.419999999999999984 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 54.4

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

   5. Applied rewrites 54.4%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.25:\\ \;\;\;\;\sin y \cdot x\\ \mathbf{elif}\;y \leq 0.42:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y \cdot x, -0.5 \cdot z\right), y, x\right), y, z\right)\\ \mathbf{else}:\\ \;\;\;\;\cos y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 40.2% accurate, 11.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.2 \cdot 10^{-10} \lor \neg \left(z \leq 8.5 \cdot 10^{-14}\right):\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -3.2e-10) (not (<= z 8.5e-14))) z (* x y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -3.2e-10) || !(z <= 8.5e-14)) {
		tmp = z;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-3.2d-10)) .or. (.not. (z <= 8.5d-14))) then
        tmp = z
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -3.2e-10) || !(z <= 8.5e-14)) {
		tmp = z;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -3.2e-10) or not (z <= 8.5e-14):
		tmp = z
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -3.2e-10) || !(z <= 8.5e-14))
		tmp = z;
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -3.2e-10) || ~((z <= 8.5e-14)))
		tmp = z;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -3.2e-10], N[Not[LessEqual[z, 8.5e-14]], $MachinePrecision]], z, N[(x * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.19999999999999981e-10 or 8.50000000000000038e-14 < z

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow1 N/A

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

      4. metadata-eval N/A

         \[\leadsto {\left(z \cdot \cos y\right)}^{\color{blue}{\left(\frac{2}{2}\right)}} + x \cdot \sin y \]

      5. sqrt-pow1 N/A

         \[\leadsto \color{blue}{\sqrt{{\left(z \cdot \cos y\right)}^{2}}} + x \cdot \sin y \]

      6. pow2 N/A

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

      7. sqr-neg-rev N/A

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

      8. lift-*.f64 N/A

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

      9. distribute-lft-neg-out N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-lft-neg-out N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. sqrt-prod N/A

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

      15. lower-fma.f64 N/A

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

   4. Applied rewrites 19.4%

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

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{z \cdot {\left(\sqrt{-1}\right)}^{2}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot z} \]

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 5.3

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

   7. Applied rewrites 5.3%

      \[\leadsto \color{blue}{-z} \]
   8. Step-by-step derivation

   9. Applied rewrites 53.2%

      \[\leadsto z \]

   if -3.19999999999999981e-10 < z < 8.50000000000000038e-14

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 60.4

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

   5. Applied rewrites 60.4%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 39.4%

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

4. Final simplification 46.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.2 \cdot 10^{-10} \lor \neg \left(z \leq 8.5 \cdot 10^{-14}\right):\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 52.9% accurate, 30.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 57.9

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

5. Applied rewrites 57.9%

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

Alternative 8: 39.3% accurate, 214.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := z

Derivation

1. Initial program 99.8%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. unpow1 N/A

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

   4. metadata-eval N/A

      \[\leadsto {\left(z \cdot \cos y\right)}^{\color{blue}{\left(\frac{2}{2}\right)}} + x \cdot \sin y \]

   5. sqrt-pow1 N/A

      \[\leadsto \color{blue}{\sqrt{{\left(z \cdot \cos y\right)}^{2}}} + x \cdot \sin y \]

   6. pow2 N/A

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

   7. sqr-neg-rev N/A

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

   8. lift-*.f64 N/A

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

   9. distribute-lft-neg-out N/A

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

   10. lift-*.f64 N/A

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

   11. distribute-lft-neg-out N/A

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

   12. *-commutative N/A

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

   13. associate-*r* N/A

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

   14. sqrt-prod N/A

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

   15. lower-fma.f64 N/A

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

4. Applied rewrites 31.9%

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

5. Taylor expanded in y around 0

   \[\leadsto \color{blue}{z \cdot {\left(\sqrt{-1}\right)}^{2}} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot z} \]

   2. unpow2 N/A

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

   3. rem-square-sqrt N/A

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

   4. mul-1-neg N/A

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

   5. lower-neg.f64 4.8

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

7. Applied rewrites 4.8%

   \[\leadsto \color{blue}{-z} \]
8. Step-by-step derivation

9. Applied rewrites 39.5%

   \[\leadsto z \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024340 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Transform:aboutX from diagrams-lib-1.3.0.3, B"
  :precision binary64
  (+ (* x (sin y)) (* z (cos y))))