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

Percentage Accurate: 99.8% → 99.8%

Time: 9.2s

Alternatives: 7

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

Alternative 2: 86.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \cos y\\ \mathbf{if}\;z \leq -8.2 \cdot 10^{+44}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{+68}:\\ \;\;\;\;\mathsf{fma}\left(\sin y, x, 1 \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (cos y))))
   (if (<= z -8.2e+44) t_0 (if (<= z 4.5e+68) (fma (sin y) x (* 1.0 z)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = z * cos(y);
	double tmp;
	if (z <= -8.2e+44) {
		tmp = t_0;
	} else if (z <= 4.5e+68) {
		tmp = fma(sin(y), x, (1.0 * z));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(z * cos(y))
	tmp = 0.0
	if (z <= -8.2e+44)
		tmp = t_0;
	elseif (z <= 4.5e+68)
		tmp = fma(sin(y), x, Float64(1.0 * z));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -8.2e+44], t$95$0, If[LessEqual[z, 4.5e+68], N[(N[Sin[y], $MachinePrecision] * x + N[(1.0 * z), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -8.1999999999999993e44 or 4.5000000000000003e68 < z

   1. Initial program 99.8%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \cos y} \]
   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 91.3

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

   5. Applied rewrites 91.3%

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

   if -8.1999999999999993e44 < z < 4.5000000000000003e68

   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. Taylor expanded in y around 0

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

   7. Applied rewrites 87.0%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8.2 \cdot 10^{+44}:\\ \;\;\;\;z \cdot \cos y\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{+68}:\\ \;\;\;\;\mathsf{fma}\left(\sin y, x, 1 \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \sin y\\ \mathbf{if}\;x \leq -2.8 \cdot 10^{+122}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 5.4 \cdot 10^{+146}:\\ \;\;\;\;z \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (sin y))))
   (if (<= x -2.8e+122) t_0 (if (<= x 5.4e+146) (* z (cos y)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x * sin(y);
	double tmp;
	if (x <= -2.8e+122) {
		tmp = t_0;
	} else if (x <= 5.4e+146) {
		tmp = z * cos(y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * sin(y)
    if (x <= (-2.8d+122)) then
        tmp = t_0
    else if (x <= 5.4d+146) then
        tmp = z * cos(y)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x * Math.sin(y);
	double tmp;
	if (x <= -2.8e+122) {
		tmp = t_0;
	} else if (x <= 5.4e+146) {
		tmp = z * Math.cos(y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * math.sin(y)
	tmp = 0
	if x <= -2.8e+122:
		tmp = t_0
	elif x <= 5.4e+146:
		tmp = z * math.cos(y)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * sin(y))
	tmp = 0.0
	if (x <= -2.8e+122)
		tmp = t_0;
	elseif (x <= 5.4e+146)
		tmp = Float64(z * cos(y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * sin(y);
	tmp = 0.0;
	if (x <= -2.8e+122)
		tmp = t_0;
	elseif (x <= 5.4e+146)
		tmp = z * cos(y);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[Sin[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.8e+122], t$95$0, If[LessEqual[x, 5.4e+146], N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.8e122 or 5.39999999999999977e146 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 70.8

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

   5. Applied rewrites 70.8%

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

   if -2.8e122 < x < 5.39999999999999977e146

   1. Initial program 99.8%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \cos y} \]
   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 81.9

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

   5. Applied rewrites 81.9%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{+122}:\\ \;\;\;\;x \cdot \sin y\\ \mathbf{elif}\;x \leq 5.4 \cdot 10^{+146}:\\ \;\;\;\;z \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sin y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \cos y\\ \mathbf{if}\;y \leq -2200000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.000155:\\ \;\;\;\;\mathsf{fma}\left(y, x, z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (cos y))))
   (if (<= y -2200000.0) t_0 (if (<= y 0.000155) (fma y x z) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = z * cos(y);
	double tmp;
	if (y <= -2200000.0) {
		tmp = t_0;
	} else if (y <= 0.000155) {
		tmp = fma(y, x, z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(z * cos(y))
	tmp = 0.0
	if (y <= -2200000.0)
		tmp = t_0;
	elseif (y <= 0.000155)
		tmp = fma(y, x, z);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2200000.0], t$95$0, If[LessEqual[y, 0.000155], N[(y * x + z), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.2e6 or 1.55e-4 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \cos y} \]
   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 56.8

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

   5. Applied rewrites 56.8%

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

   if -2.2e6 < y < 1.55e-4

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{z + x \cdot y} \]
   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 99.3

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

   5. Applied rewrites 99.3%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2200000:\\ \;\;\;\;z \cdot \cos y\\ \mathbf{elif}\;y \leq 0.000155:\\ \;\;\;\;\mathsf{fma}\left(y, x, z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 40.5% accurate, 17.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.2 \cdot 10^{+147}:\\ \;\;\;\;1 \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= x 3.2e+147) (* 1.0 z) (* x y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 3.2e+147) {
		tmp = 1.0 * 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 (x <= 3.2d+147) then
        tmp = 1.0d0 * 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 (x <= 3.2e+147) {
		tmp = 1.0 * z;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 3.2e+147:
		tmp = 1.0 * z
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 3.2e+147)
		tmp = Float64(1.0 * z);
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 3.2e+147)
		tmp = 1.0 * z;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 3.2e+147], N[(1.0 * z), $MachinePrecision], N[(x * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.19999999999999979e147

   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. flip-+ N/A

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

      3. clear-num N/A

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

      4. associate-/r/ N/A

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

      5. flip3-- N/A

         \[\leadsto \frac{1}{x \cdot \sin y - z \cdot \cos y} \cdot \color{blue}{\frac{{\left(\left(x \cdot \sin y\right) \cdot \left(x \cdot \sin y\right)\right)}^{3} - {\left(\left(z \cdot \cos y\right) \cdot \left(z \cdot \cos y\right)\right)}^{3}}{\left(\left(x \cdot \sin y\right) \cdot \left(x \cdot \sin y\right)\right) \cdot \left(\left(x \cdot \sin y\right) \cdot \left(x \cdot \sin y\right)\right) + \left(\left(\left(z \cdot \cos y\right) \cdot \left(z \cdot \cos y\right)\right) \cdot \left(\left(z \cdot \cos y\right) \cdot \left(z \cdot \cos y\right)\right) + \left(\left(x \cdot \sin y\right) \cdot \left(x \cdot \sin y\right)\right) \cdot \left(\left(z \cdot \cos y\right) \cdot \left(z \cdot \cos y\right)\right)\right)}} \]

      6. clear-num N/A

         \[\leadsto \frac{1}{x \cdot \sin y - z \cdot \cos y} \cdot \color{blue}{\frac{1}{\frac{\left(\left(x \cdot \sin y\right) \cdot \left(x \cdot \sin y\right)\right) \cdot \left(\left(x \cdot \sin y\right) \cdot \left(x \cdot \sin y\right)\right) + \left(\left(\left(z \cdot \cos y\right) \cdot \left(z \cdot \cos y\right)\right) \cdot \left(\left(z \cdot \cos y\right) \cdot \left(z \cdot \cos y\right)\right) + \left(\left(x \cdot \sin y\right) \cdot \left(x \cdot \sin y\right)\right) \cdot \left(\left(z \cdot \cos y\right) \cdot \left(z \cdot \cos y\right)\right)\right)}{{\left(\left(x \cdot \sin y\right) \cdot \left(x \cdot \sin y\right)\right)}^{3} - {\left(\left(z \cdot \cos y\right) \cdot \left(z \cdot \cos y\right)\right)}^{3}}}} \]

   4. Applied rewrites 54.1%

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

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-cos.f64 95.8

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

   7. Applied rewrites 95.8%

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

   8. Taylor expanded in y around 0

      \[\leadsto 1 \cdot z \]
   9. Step-by-step derivation

   10. Applied rewrites 42.6%

       \[\leadsto 1 \cdot z \]

   if 3.19999999999999979e147 < x

   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 49.2

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

   5. Applied rewrites 49.2%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 34.6%

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

4. Final simplification 41.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.2 \cdot 10^{+147}:\\ \;\;\;\;1 \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 52.5% 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 48.7

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

5. Applied rewrites 48.7%

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

Alternative 7: 17.2% accurate, 35.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(x * y), $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 48.7

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

5. Applied rewrites 48.7%

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

6. Taylor expanded in z around 0

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

8. Applied rewrites 13.6%

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

9. Final simplification 13.6%

   \[\leadsto x \cdot y \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024267 
(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))))