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

Percentage Accurate: 99.8% → 99.8%

Time: 11.6s

Alternatives: 9

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

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

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]

Derivation

1. Initial program 99.8%

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

Alternative 2: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_] := N[(N[Cos[y], $MachinePrecision] * z + N[(x * N[Sin[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-sin.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. +-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 99.8

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

4. Applied egg-rr 99.8%

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

Alternative 3: 84.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{-40}:\\ \;\;\;\;\mathsf{fma}\left(\sin y, x, z\right)\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{+87}:\\ \;\;\;\;z \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sin y + z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.8e-40)
   (fma (sin y) x z)
   (if (<= x 3.3e+87) (* z (cos y)) (+ (* x (sin y)) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.8e-40) {
		tmp = fma(sin(y), x, z);
	} else if (x <= 3.3e+87) {
		tmp = z * cos(y);
	} else {
		tmp = (x * sin(y)) + z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.8e-40)
		tmp = fma(sin(y), x, z);
	elseif (x <= 3.3e+87)
		tmp = Float64(z * cos(y));
	else
		tmp = Float64(Float64(x * sin(y)) + z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.8e-40], N[(N[Sin[y], $MachinePrecision] * x + z), $MachinePrecision], If[LessEqual[x, 3.3e+87], N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision], N[(N[(x * N[Sin[y], $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.8e-40

   1. Initial program 99.9%

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

      1. lift-sin.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in y around 0

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

   7. Simplified 88.9%

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

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-lft-identity N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 88.9

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

   9. Applied egg-rr 88.9%

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

   if -1.8e-40 < x < 3.3000000000000001e87

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 82.1

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

   5. Simplified 82.1%

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

   if 3.3000000000000001e87 < x

   1. Initial program 99.8%

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

      1. lift-sin.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 99.8

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in y around 0

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

   7. Simplified 92.2%

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

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-lft-identity N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 92.2

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 92.2

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

   9. Applied egg-rr 92.2%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{-40}:\\ \;\;\;\;\mathsf{fma}\left(\sin y, x, z\right)\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{+87}:\\ \;\;\;\;z \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sin y + z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\sin y, x, z\right)\\ \mathbf{if}\;x \leq -3.1 \cdot 10^{-40}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{+93}:\\ \;\;\;\;z \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (sin y) x z)))
   (if (<= x -3.1e-40) t_0 (if (<= x 2.7e+93) (* z (cos y)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma(sin(y), x, z);
	double tmp;
	if (x <= -3.1e-40) {
		tmp = t_0;
	} else if (x <= 2.7e+93) {
		tmp = z * cos(y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(sin(y), x, z)
	tmp = 0.0
	if (x <= -3.1e-40)
		tmp = t_0;
	elseif (x <= 2.7e+93)
		tmp = Float64(z * cos(y));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] * x + z), $MachinePrecision]}, If[LessEqual[x, -3.1e-40], t$95$0, If[LessEqual[x, 2.7e+93], N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.10000000000000011e-40 or 2.6999999999999999e93 < x

   1. Initial program 99.9%

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

      1. lift-sin.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in y around 0

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

   7. Simplified 90.3%

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

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-lft-identity N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 90.3

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

   9. Applied egg-rr 90.3%

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

   if -3.10000000000000011e-40 < x < 2.6999999999999999e93

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 82.1

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

   5. Simplified 82.1%

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

Alternative 5: 73.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \sin y\\ \mathbf{if}\;x \leq -2.9 \cdot 10^{+30}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{+97}:\\ \;\;\;\;z \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (sin y))))
   (if (<= x -2.9e+30) t_0 (if (<= x 1.25e+97) (* z (cos y)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x * sin(y);
	double tmp;
	if (x <= -2.9e+30) {
		tmp = t_0;
	} else if (x <= 1.25e+97) {
		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 <= (-2.9d+30)) then
        tmp = t_0
    else if (x <= 1.25d+97) 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 <= -2.9e+30) {
		tmp = t_0;
	} else if (x <= 1.25e+97) {
		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 <= -2.9e+30:
		tmp = t_0
	elif x <= 1.25e+97:
		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 <= -2.9e+30)
		tmp = t_0;
	elseif (x <= 1.25e+97)
		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 <= -2.9e+30)
		tmp = t_0;
	elseif (x <= 1.25e+97)
		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, -2.9e+30], t$95$0, If[LessEqual[x, 1.25e+97], N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.8999999999999998e30 or 1.25e97 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 75.9

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

   5. Simplified 75.9%

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

   if -2.8999999999999998e30 < x < 1.25e97

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 81.4

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

   5. Simplified 81.4%

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

Alternative 6: 75.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \sin y\\ \mathbf{if}\;y \leq -0.92:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.45:\\ \;\;\;\;x \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right) + \mathsf{fma}\left(z \cdot \left(y \cdot y\right), \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot -0.001388888888888889, 0.041666666666666664\right), -0.5\right), z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (sin y))))
   (if (<= y -0.92)
     t_0
     (if (<= y 0.45)
       (+
        (*
         x
         (fma
          (fma
           (* y y)
           (fma (* y y) -0.0001984126984126984 0.008333333333333333)
           -0.16666666666666666)
          (* y (* y y))
          y))
        (fma
         (* z (* y y))
         (fma
          (* y y)
          (fma y (* y -0.001388888888888889) 0.041666666666666664)
          -0.5)
         z))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x * sin(y);
	double tmp;
	if (y <= -0.92) {
		tmp = t_0;
	} else if (y <= 0.45) {
		tmp = (x * fma(fma((y * y), fma((y * y), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666), (y * (y * y)), y)) + fma((z * (y * y)), fma((y * y), fma(y, (y * -0.001388888888888889), 0.041666666666666664), -0.5), z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x * sin(y))
	tmp = 0.0
	if (y <= -0.92)
		tmp = t_0;
	elseif (y <= 0.45)
		tmp = Float64(Float64(x * fma(fma(Float64(y * y), fma(Float64(y * y), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666), Float64(y * Float64(y * y)), y)) + fma(Float64(z * Float64(y * y)), fma(Float64(y * y), fma(y, Float64(y * -0.001388888888888889), 0.041666666666666664), -0.5), z));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[Sin[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.92], t$95$0, If[LessEqual[y, 0.45], N[(N[(x * N[(N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision] + N[(N[(z * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * -0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] + -0.5), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.92000000000000004 or 0.450000000000000011 < y

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 50.4

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

   5. Simplified 50.4%

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

   if -0.92000000000000004 < y < 0.450000000000000011

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto x \cdot \sin y + \color{blue}{\left(z + {y}^{2} \cdot \left(\frac{-1}{2} \cdot z + {y}^{2} \cdot \left(\frac{-1}{720} \cdot \left({y}^{2} \cdot z\right) + \frac{1}{24} \cdot z\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \sin y + \color{blue}{\left({y}^{2} \cdot \left(\frac{-1}{2} \cdot z + {y}^{2} \cdot \left(\frac{-1}{720} \cdot \left({y}^{2} \cdot z\right) + \frac{1}{24} \cdot z\right)\right) + z\right)} \]

   5. Simplified 100.0%

      \[\leadsto x \cdot \sin y + \color{blue}{\mathsf{fma}\left(z \cdot \left(y \cdot y\right), \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot -0.001388888888888889, 0.041666666666666664\right), -0.5\right), z\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto x \cdot \color{blue}{\left(y \cdot \left(1 + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}\right)\right)\right)} + \mathsf{fma}\left(z \cdot \left(y \cdot y\right), \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2}\right), z\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \left(y \cdot \color{blue}{\left({y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}\right) + 1\right)}\right) + \mathsf{fma}\left(z \cdot \left(y \cdot y\right), \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2}\right), z\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto x \cdot \color{blue}{\left(\left({y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}\right)\right) \cdot y + 1 \cdot y\right)} + \mathsf{fma}\left(z \cdot \left(y \cdot y\right), \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2}\right), z\right) \]

      3. *-commutative N/A

         \[\leadsto x \cdot \left(\color{blue}{\left(\left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}\right) \cdot {y}^{2}\right)} \cdot y + 1 \cdot y\right) + \mathsf{fma}\left(z \cdot \left(y \cdot y\right), \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2}\right), z\right) \]

      4. associate-*l* N/A

         \[\leadsto x \cdot \left(\color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}\right) \cdot \left({y}^{2} \cdot y\right)} + 1 \cdot y\right) + \mathsf{fma}\left(z \cdot \left(y \cdot y\right), \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2}\right), z\right) \]

      5. *-lft-identity N/A

         \[\leadsto x \cdot \left(\left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}\right) \cdot \left({y}^{2} \cdot y\right) + \color{blue}{y}\right) + \mathsf{fma}\left(z \cdot \left(y \cdot y\right), \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2}\right), z\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}, {y}^{2} \cdot y, y\right)} + \mathsf{fma}\left(z \cdot \left(y \cdot y\right), \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2}\right), z\right) \]

   8. Simplified 99.9%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)} + \mathsf{fma}\left(z \cdot \left(y \cdot y\right), \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot -0.001388888888888889, 0.041666666666666664\right), -0.5\right), z\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 40.3% accurate, 11.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.14 \cdot 10^{+55}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{+85}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.14e+55) (* x y) (if (<= x 3.8e+85) z (* x y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.14e+55) {
		tmp = x * y;
	} else if (x <= 3.8e+85) {
		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 (x <= (-1.14d+55)) then
        tmp = x * y
    else if (x <= 3.8d+85) 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 (x <= -1.14e+55) {
		tmp = x * y;
	} else if (x <= 3.8e+85) {
		tmp = z;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.14e+55:
		tmp = x * y
	elif x <= 3.8e+85:
		tmp = z
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.14e+55)
		tmp = Float64(x * y);
	elseif (x <= 3.8e+85)
		tmp = z;
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.14e+55)
		tmp = x * y;
	elseif (x <= 3.8e+85)
		tmp = z;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.14e+55], N[(x * y), $MachinePrecision], If[LessEqual[x, 3.8e+85], z, N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.1399999999999999e55 or 3.79999999999999992e85 < x

   1. Initial program 99.9%

      \[x \cdot \sin y + z \cdot \cos y \]
   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 45.8

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

   5. Simplified 45.8%

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

   6. Taylor expanded in y around inf

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

      1. lower-*.f64 32.9

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

   8. Simplified 32.9%

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

   if -1.1399999999999999e55 < x < 3.79999999999999992e85

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 80.8

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

   5. Simplified 80.8%

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

   6. Taylor expanded in y around 0

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

   8. Simplified 48.6%

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

      1. *-rgt-identity 48.6

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

   10. Applied egg-rr 48.6%

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

Alternative 8: 51.9% 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 47.9

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

5. Simplified 47.9%

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

Alternative 9: 38.3% 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. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

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

   2. lower-cos.f64 59.1

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

5. Simplified 59.1%

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

6. Taylor expanded in y around 0

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

8. Simplified 35.8%

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

   1. *-rgt-identity 35.8

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

10. Applied egg-rr 35.8%

    \[\leadsto \color{blue}{z} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024212 
(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))))