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

Percentage Accurate: 99.8% → 99.8%

Time: 10.2s

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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Final simplification 99.8%

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

Alternative 2: 85.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos y \cdot x\\ \mathbf{if}\;x \leq -1.05 \cdot 10^{+25}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{+120}:\\ \;\;\;\;1 \cdot x - \sin y \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (cos y) x)))
   (if (<= x -1.05e+25)
     t_0
     (if (<= x 2.4e+120) (- (* 1.0 x) (* (sin y) z)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = cos(y) * x;
	double tmp;
	if (x <= -1.05e+25) {
		tmp = t_0;
	} else if (x <= 2.4e+120) {
		tmp = (1.0 * x) - (sin(y) * z);
	} 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 = cos(y) * x
    if (x <= (-1.05d+25)) then
        tmp = t_0
    else if (x <= 2.4d+120) then
        tmp = (1.0d0 * x) - (sin(y) * z)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.cos(y) * x;
	double tmp;
	if (x <= -1.05e+25) {
		tmp = t_0;
	} else if (x <= 2.4e+120) {
		tmp = (1.0 * x) - (Math.sin(y) * z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.cos(y) * x
	tmp = 0
	if x <= -1.05e+25:
		tmp = t_0
	elif x <= 2.4e+120:
		tmp = (1.0 * x) - (math.sin(y) * z)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(cos(y) * x)
	tmp = 0.0
	if (x <= -1.05e+25)
		tmp = t_0;
	elseif (x <= 2.4e+120)
		tmp = Float64(Float64(1.0 * x) - Float64(sin(y) * z));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = cos(y) * x;
	tmp = 0.0;
	if (x <= -1.05e+25)
		tmp = t_0;
	elseif (x <= 2.4e+120)
		tmp = (1.0 * x) - (sin(y) * z);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Cos[y], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -1.05e+25], t$95$0, If[LessEqual[x, 2.4e+120], N[(N[(1.0 * x), $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.05e25 or 2.40000000000000001e120 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 91.3

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

   5. Applied rewrites 91.3%

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

   if -1.05e25 < x < 2.40000000000000001e120

   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 83.7%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{+25}:\\ \;\;\;\;\cos y \cdot x\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{+120}:\\ \;\;\;\;1 \cdot x - \sin y \cdot z\\ \mathbf{else}:\\ \;\;\;\;\cos y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos y \cdot x\\ \mathbf{if}\;y \leq -3.8 \cdot 10^{+230}:\\ \;\;\;\;\left(-z\right) \cdot \sin y\\ \mathbf{elif}\;y \leq -1.35 \cdot 10^{-5}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(-z, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (cos y) x)))
   (if (<= y -3.8e+230)
     (* (- z) (sin y))
     (if (<= y -1.35e-5) t_0 (if (<= y 2e-9) (fma (- z) y x) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = cos(y) * x;
	double tmp;
	if (y <= -3.8e+230) {
		tmp = -z * sin(y);
	} else if (y <= -1.35e-5) {
		tmp = t_0;
	} else if (y <= 2e-9) {
		tmp = fma(-z, y, x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(cos(y) * x)
	tmp = 0.0
	if (y <= -3.8e+230)
		tmp = Float64(Float64(-z) * sin(y));
	elseif (y <= -1.35e-5)
		tmp = t_0;
	elseif (y <= 2e-9)
		tmp = fma(Float64(-z), y, 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, -3.8e+230], N[((-z) * N[Sin[y], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -1.35e-5], t$95$0, If[LessEqual[y, 2e-9], N[((-z) * y + x), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.8e230

   1. Initial program 99.8%

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

   3. Taylor expanded in z around inf

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

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

      5. lower-sin.f64 70.0

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

   5. Applied rewrites 70.0%

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

   if -3.8e230 < y < -1.3499999999999999e-5 or 2.00000000000000012e-9 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 59.4

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

   5. Applied rewrites 59.4%

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

   if -1.3499999999999999e-5 < y < 2.00000000000000012e-9

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + -1 \cdot \left(y \cdot z\right)} \]
   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 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. distribute-lft-neg-in N/A

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 100.0

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

   8. Applied rewrites 100.0%

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

Alternative 4: 74.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos y \cdot x\\ \mathbf{if}\;y \leq -1.35 \cdot 10^{-5}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(-z, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (cos y) x)))
   (if (<= y -1.35e-5) t_0 (if (<= y 2e-9) (fma (- z) y x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = cos(y) * x;
	double tmp;
	if (y <= -1.35e-5) {
		tmp = t_0;
	} else if (y <= 2e-9) {
		tmp = fma(-z, y, x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(cos(y) * x)
	tmp = 0.0
	if (y <= -1.35e-5)
		tmp = t_0;
	elseif (y <= 2e-9)
		tmp = fma(Float64(-z), y, 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, -1.35e-5], t$95$0, If[LessEqual[y, 2e-9], N[((-z) * y + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.3499999999999999e-5 or 2.00000000000000012e-9 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 56.7

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

   5. Applied rewrites 56.7%

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

   if -1.3499999999999999e-5 < y < 2.00000000000000012e-9

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + -1 \cdot \left(y \cdot z\right)} \]
   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 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. distribute-lft-neg-in N/A

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 100.0

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

   8. Applied rewrites 100.0%

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

Alternative 5: 38.5% accurate, 15.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.8 \cdot 10^{+25}:\\ \;\;\;\;\left(-y\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= z -4.8e+25) (* (- y) z) (* 1.0 x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -4.8e+25) {
		tmp = -y * z;
	} else {
		tmp = 1.0 * x;
	}
	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 <= (-4.8d+25)) then
        tmp = -y * z
    else
        tmp = 1.0d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -4.8e+25) {
		tmp = -y * z;
	} else {
		tmp = 1.0 * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -4.8e+25:
		tmp = -y * z
	else:
		tmp = 1.0 * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -4.8e+25)
		tmp = Float64(Float64(-y) * z);
	else
		tmp = Float64(1.0 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -4.8e+25)
		tmp = -y * z;
	else
		tmp = 1.0 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -4.8e+25], N[((-y) * z), $MachinePrecision], N[(1.0 * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.79999999999999992e25

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 51.3

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

   5. Applied rewrites 51.3%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 38.8%

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

   if -4.79999999999999992e25 < 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 52.8

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

   5. Applied rewrites 52.8%

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

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\cos y + -1 \cdot \frac{z \cdot \sin y}{x}\right)} \]
   7. 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. mul-1-neg N/A

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

      5. associate-/l* N/A

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

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

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

      7. mul-1-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. lower-cos.f64 94.6

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

   8. Applied rewrites 94.6%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 44.5%

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

Alternative 6: 51.9% accurate, 23.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

   \[x \cdot \cos y - z \cdot \sin y \]
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 52.5

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

5. Applied rewrites 52.5%

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

6. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. *-commutative N/A

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

   4. distribute-lft-neg-in N/A

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

   5. mul-1-neg N/A

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

   6. lower-fma.f64 N/A

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

   7. mul-1-neg N/A

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

   8. lower-neg.f64 52.5

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

8. Applied rewrites 52.5%

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

Alternative 7: 51.9% accurate, 23.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

   \[x \cdot \cos y - z \cdot \sin y \]
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 52.5

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

5. Applied rewrites 52.5%

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

Alternative 8: 38.6% 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. Taylor expanded in y around 0

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 52.5

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

5. Applied rewrites 52.5%

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

6. Taylor expanded in x around inf

   \[\leadsto \color{blue}{x \cdot \left(\cos y + -1 \cdot \frac{z \cdot \sin y}{x}\right)} \]
7. 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. mul-1-neg N/A

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

   5. associate-/l* N/A

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

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

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

   7. mul-1-neg N/A

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

   8. lower-fma.f64 N/A

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

   9. mul-1-neg N/A

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

   10. lower-neg.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-sin.f64 N/A

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

   13. lower-cos.f64 92.7

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

8. Applied rewrites 92.7%

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

9. Taylor expanded in y around 0

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

11. Applied rewrites 38.5%

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

Reproduce

herbie shell --seed 2024235 
(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))))