Diagrams.ThreeD.Transform:aboutY from diagrams-lib-1.3.0.3

Percentage Accurate: 99.8% → 99.8%

Time: 3.3s

Alternatives: 7

Speedup: 1.0×

Specification

Math

\[x \cdot \cos y + z \cdot \sin y \]

FPCore

(FPCore (x y z)
  :precision binary64
  :pre TRUE
  (+ (* 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)
use fmin_fmax_functions
    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]

ReFlow

f(x, y, z):
	x in [-inf, +inf],
	y in [-inf, +inf],
	z in [-inf, +inf]
code: THEORY
BEGIN
f(x, y, z: real): real =
	(x * (cos(y))) + (z * (sin(y)))
END code

Initial Program: 99.8% accurate, 1.0× speedup

Math

\[x \cdot \cos y + z \cdot \sin y \]

FPCore

(FPCore (x y z)
  :precision binary64
  :pre TRUE
  (+ (* 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)
use fmin_fmax_functions
    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]

ReFlow

f(x, y, z):
	x in [-inf, +inf],
	y in [-inf, +inf],
	z in [-inf, +inf]
code: THEORY
BEGIN
f(x, y, z: real): real =
	(x * (cos(y))) + (z * (sin(y)))
END code

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

ReFlow

f(x, y, z):
	x in [-inf, +inf],
	y in [-inf, +inf],
	z in [-inf, +inf]
code: THEORY
BEGIN
f(x, y, z: real): real =
	(z * (sin(y))) + ((cos(y)) * x)
END code

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

Alternative 2: 86.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(x, 1, \sin y \cdot z\right)\\ \mathbf{if}\;z \leq -1.5998878487717716 \cdot 10^{-87}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1.3906905842088115 \cdot 10^{-61}:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  :pre TRUE
  (let* ((t_0 (fma x 1.0 (* (sin y) z))))
  (if (<= z -1.5998878487717716e-87)
    t_0
    (if (<= z 1.3906905842088115e-61) (* x (cos y)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma(x, 1.0, (sin(y) * z));
	double tmp;
	if (z <= -1.5998878487717716e-87) {
		tmp = t_0;
	} else if (z <= 1.3906905842088115e-61) {
		tmp = x * cos(y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(x, 1.0, Float64(sin(y) * z))
	tmp = 0.0
	if (z <= -1.5998878487717716e-87)
		tmp = t_0;
	elseif (z <= 1.3906905842088115e-61)
		tmp = Float64(x * cos(y));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * 1.0 + N[(N[Sin[y], $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.5998878487717716e-87], t$95$0, If[LessEqual[z, 1.3906905842088115e-61], N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision], t$95$0]]]

ReFlow

f(x, y, z):
	x in [-inf, +inf],
	y in [-inf, +inf],
	z in [-inf, +inf]
code: THEORY
BEGIN
f(x, y, z: real): real =
	LET t_0 = ((x * (1)) + ((sin(y)) * z)) IN
		LET tmp_1 = IF (z <= (139069058420881154967487957630223034015355744194129779251587240099952514292987109389899646542932568675618170646032168982972712282179196417877818123690699010641669275401000049896538257598876953125e-255)) THEN (x * (cos(y))) ELSE t_0 ENDIF IN
		LET tmp = IF (z <= (-159988784877177164685523371169830722870522374072384409119026858684832480050318864005140114871679013601761552483193181404403016128515199488354968070059220853336754264519220820635743157089849151568897638446024723319315030689580225953250192105770111083984375e-341)) THEN t_0 ELSE tmp_1 ENDIF IN
	tmp
END code

Derivation

1. Split input into 2 regimes

2. if z < -1.5998878487717716e-87 or 1.3906905842088115e-61 < z

   1. Initial program 99.8%

      \[x \cdot \cos y + z \cdot \sin y \]

   2. Taylor expanded in y around 0

      \[\leadsto x \cdot 1 + z \cdot \sin y \]
   3. Step-by-step derivation

   4. Applied rewrites 77.3%

      \[\leadsto x \cdot 1 + z \cdot \sin y \]
   5. Step-by-step derivation

   6. Applied rewrites 77.3%

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

   if -1.5998878487717716e-87 < z < 1.3906905842088115e-61

   1. Initial program 99.8%

      \[x \cdot \cos y + z \cdot \sin y \]

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 92.2%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 61.0%

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

Alternative 3: 74.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} t_0 := x \cdot \cos y\\ \mathbf{if}\;x \leq -1.2014353339475216 \cdot 10^{-53}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.3907246641268288 \cdot 10^{-133}:\\ \;\;\;\;z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  :pre TRUE
  (let* ((t_0 (* x (cos y))))
  (if (<= x -1.2014353339475216e-53)
    t_0
    (if (<= x 1.3907246641268288e-133) (* z (sin y)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x * cos(y);
	double tmp;
	if (x <= -1.2014353339475216e-53) {
		tmp = t_0;
	} else if (x <= 1.3907246641268288e-133) {
		tmp = z * sin(y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
use fmin_fmax_functions
    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 * cos(y)
    if (x <= (-1.2014353339475216d-53)) then
        tmp = t_0
    else if (x <= 1.3907246641268288d-133) then
        tmp = z * sin(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.cos(y);
	double tmp;
	if (x <= -1.2014353339475216e-53) {
		tmp = t_0;
	} else if (x <= 1.3907246641268288e-133) {
		tmp = z * Math.sin(y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * math.cos(y)
	tmp = 0
	if x <= -1.2014353339475216e-53:
		tmp = t_0
	elif x <= 1.3907246641268288e-133:
		tmp = z * math.sin(y)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * cos(y))
	tmp = 0.0
	if (x <= -1.2014353339475216e-53)
		tmp = t_0;
	elseif (x <= 1.3907246641268288e-133)
		tmp = Float64(z * sin(y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * cos(y);
	tmp = 0.0;
	if (x <= -1.2014353339475216e-53)
		tmp = t_0;
	elseif (x <= 1.3907246641268288e-133)
		tmp = z * sin(y);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.2014353339475216e-53], t$95$0, If[LessEqual[x, 1.3907246641268288e-133], N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision], t$95$0]]]

ReFlow

f(x, y, z):
	x in [-inf, +inf],
	y in [-inf, +inf],
	z in [-inf, +inf]
code: THEORY
BEGIN
f(x, y, z: real): real =
	LET t_0 = (x * (cos(y))) IN
		LET tmp_1 = IF (x <= (13907246641268287719553288589367698732274009277678938853341083963550402305825730272558285284260707557226592286720658421468340914383077588215032524302117800378662076704866440675793204111827833749549383931163536547600078549247750976353053708772891633938598045310131695191357079148094045872458853184477607279647119117250357415993544663024295005016028881072998046875e-494)) THEN (z * (sin(y))) ELSE t_0 ENDIF IN
		LET tmp = IF (x <= (-1201435333947521567179383940228922831410793169252770766996959593329040310462231836384689301534482831918011668944531164721304996863097958037513990348088555037975311279296875e-224)) THEN t_0 ELSE tmp_1 ENDIF IN
	tmp
END code

Derivation

1. Split input into 2 regimes

2. if x < -1.2014353339475216e-53 or 1.3907246641268288e-133 < x

   1. Initial program 99.8%

      \[x \cdot \cos y + z \cdot \sin y \]

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 92.2%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 61.0%

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

   if -1.2014353339475216e-53 < x < 1.3907246641268288e-133

   1. Initial program 99.8%

      \[x \cdot \cos y + z \cdot \sin y \]

   2. Taylor expanded in x around 0

      \[\leadsto z \cdot \sin y \]
   3. Step-by-step derivation

   4. Applied rewrites 40.4%

      \[\leadsto z \cdot \sin y \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 73.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} t_0 := x \cdot \cos y\\ \mathbf{if}\;y \leq -0.0031420532090521234:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2.23898532760956:\\ \;\;\;\;x + y \cdot \left(z + y \cdot \mathsf{fma}\left(-0.5, x, -0.16666666666666666 \cdot \left(y \cdot z\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  :pre TRUE
  (let* ((t_0 (* x (cos y))))
  (if (<= y -0.0031420532090521234)
    t_0
    (if (<= y 2.23898532760956)
      (+
       x
       (*
        y
        (+ z (* y (fma -0.5 x (* -0.16666666666666666 (* y z)))))))
      t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x * cos(y);
	double tmp;
	if (y <= -0.0031420532090521234) {
		tmp = t_0;
	} else if (y <= 2.23898532760956) {
		tmp = x + (y * (z + (y * fma(-0.5, x, (-0.16666666666666666 * (y * z))))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x * cos(y))
	tmp = 0.0
	if (y <= -0.0031420532090521234)
		tmp = t_0;
	elseif (y <= 2.23898532760956)
		tmp = Float64(x + Float64(y * Float64(z + Float64(y * fma(-0.5, x, Float64(-0.16666666666666666 * Float64(y * z)))))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.0031420532090521234], t$95$0, If[LessEqual[y, 2.23898532760956], N[(x + N[(y * N[(z + N[(y * N[(-0.5 * x + N[(-0.16666666666666666 * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

ReFlow

f(x, y, z):
	x in [-inf, +inf],
	y in [-inf, +inf],
	z in [-inf, +inf]
code: THEORY
BEGIN
f(x, y, z: real): real =
	LET t_0 = (x * (cos(y))) IN
		LET tmp_1 = IF (y <= (2238985327609559927708460236317478120326995849609375e-51)) THEN (x + (y * (z + (y * (((-5e-1) * x) + ((-1666666666666666574148081281236954964697360992431640625e-55) * (y * z))))))) ELSE t_0 ENDIF IN
		LET tmp = IF (y <= (-314205320905212338866707710849368595518171787261962890625e-59)) THEN t_0 ELSE tmp_1 ENDIF IN
	tmp
END code

Derivation

1. Split input into 2 regimes

2. if y < -0.0031420532090521234 or 2.2389853276095599 < y

   1. Initial program 99.8%

      \[x \cdot \cos y + z \cdot \sin y \]

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 92.2%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 61.0%

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

   if -0.0031420532090521234 < y < 2.2389853276095599

   1. Initial program 99.8%

      \[x \cdot \cos y + z \cdot \sin y \]

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 51.5%

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

Alternative 5: 52.7% accurate, 12.8× speedup

Math

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

FPCore

(FPCore (x y z)
  :precision binary64
  :pre TRUE
  (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(z, y, x)
end

Wolfram

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

ReFlow

f(x, y, z):
	x in [-inf, +inf],
	y in [-inf, +inf],
	z in [-inf, +inf]
code: THEORY
BEGIN
f(x, y, z: real): real =
	(z * y) + x
END code

Derivation

1. Initial program 99.8%

   \[x \cdot \cos y + z \cdot \sin y \]

2. Taylor expanded in y around 0

   \[\leadsto x + y \cdot z \]
3. Step-by-step derivation

4. Applied rewrites 52.7%

   \[\leadsto x + y \cdot z \]
5. Step-by-step derivation

6. Applied rewrites 52.7%

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

Alternative 6: 39.6% accurate, 19.2× speedup

Math

\[x \cdot 1 \]

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x * 1.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

ReFlow

f(x, y, z):
	x in [-inf, +inf],
	y in [-inf, +inf],
	z in [-inf, +inf]
code: THEORY
BEGIN
f(x, y, z: real): real =
	x * (1)
END code

Derivation

1. Initial program 99.8%

   \[x \cdot \cos y + z \cdot \sin y \]

2. Taylor expanded in x around inf

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

4. Applied rewrites 92.2%

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

5. Taylor expanded in x around inf

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

7. Applied rewrites 61.0%

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

8. Taylor expanded in y around 0

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

10. Applied rewrites 39.6%

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

Alternative 7: 16.5% accurate, 19.2× speedup

Math

\[y \cdot z \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

ReFlow

f(x, y, z):
	x in [-inf, +inf],
	y in [-inf, +inf],
	z in [-inf, +inf]
code: THEORY
BEGIN
f(x, y, z: real): real =
	y * z
END code

Derivation

1. Initial program 99.8%

   \[x \cdot \cos y + z \cdot \sin y \]

2. Taylor expanded in y around 0

   \[\leadsto x + y \cdot z \]
3. Step-by-step derivation

4. Applied rewrites 52.7%

   \[\leadsto x + y \cdot z \]

5. Taylor expanded in x around 0

   \[\leadsto y \cdot z \]
6. Step-by-step derivation

7. Applied rewrites 16.5%

   \[\leadsto y \cdot z \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2026092 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Transform:aboutY from diagrams-lib-1.3.0.3"
  :precision binary64
  (+ (* x (cos y)) (* z (sin y))))