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

Percentage Accurate: 99.8% → 99.8%

Time: 10.5s

Alternatives: 8

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.7× 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. *-commutative N/A

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

   2. fma-define N/A

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

   3. fma-lowering-fma.f64 N/A

      \[\leadsto \mathsf{fma.f64}\left(\sin y, \color{blue}{x}, \left(z \cdot \cos y\right)\right) \]

   4. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{sin.f64}\left(y\right), x, \left(z \cdot \cos y\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{sin.f64}\left(y\right), x, \mathsf{*.f64}\left(z, \cos y\right)\right) \]

   6. cos-lowering-cos.f64 99.8%

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{sin.f64}\left(y\right), x, \mathsf{*.f64}\left(z, \mathsf{cos.f64}\left(y\right)\right)\right) \]

4. Applied egg-rr 99.8%

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

Alternative 2: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Final simplification 99.8%

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

Alternative 3: 73.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \cos y\\ \mathbf{if}\;z \leq -4.5 \cdot 10^{+31}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq -4.2 \cdot 10^{-172}:\\ \;\;\;\;z + y \cdot x\\ \mathbf{elif}\;z \leq 1.95 \cdot 10^{-123}:\\ \;\;\;\;\sin y \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (cos y))))
   (if (<= z -4.5e+31)
     t_0
     (if (<= z -4.2e-172)
       (+ z (* y x))
       (if (<= z 1.95e-123) (* (sin y) x) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = z * cos(y);
	double tmp;
	if (z <= -4.5e+31) {
		tmp = t_0;
	} else if (z <= -4.2e-172) {
		tmp = z + (y * x);
	} else if (z <= 1.95e-123) {
		tmp = sin(y) * x;
	} 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 = z * cos(y)
    if (z <= (-4.5d+31)) then
        tmp = t_0
    else if (z <= (-4.2d-172)) then
        tmp = z + (y * x)
    else if (z <= 1.95d-123) then
        tmp = sin(y) * x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = z * Math.cos(y);
	double tmp;
	if (z <= -4.5e+31) {
		tmp = t_0;
	} else if (z <= -4.2e-172) {
		tmp = z + (y * x);
	} else if (z <= 1.95e-123) {
		tmp = Math.sin(y) * x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * math.cos(y)
	tmp = 0
	if z <= -4.5e+31:
		tmp = t_0
	elif z <= -4.2e-172:
		tmp = z + (y * x)
	elif z <= 1.95e-123:
		tmp = math.sin(y) * x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * cos(y))
	tmp = 0.0
	if (z <= -4.5e+31)
		tmp = t_0;
	elseif (z <= -4.2e-172)
		tmp = Float64(z + Float64(y * x));
	elseif (z <= 1.95e-123)
		tmp = Float64(sin(y) * x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * cos(y);
	tmp = 0.0;
	if (z <= -4.5e+31)
		tmp = t_0;
	elseif (z <= -4.2e-172)
		tmp = z + (y * x);
	elseif (z <= 1.95e-123)
		tmp = sin(y) * x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -4.5e+31], t$95$0, If[LessEqual[z, -4.2e-172], N[(z + N[(y * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.95e-123], N[(N[Sin[y], $MachinePrecision] * x), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.4999999999999996e31 or 1.94999999999999988e-123 < z

   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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \color{blue}{\cos y}\right) \]

      2. cos-lowering-cos.f64 83.5%

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{cos.f64}\left(y\right)\right) \]

   5. Simplified 83.5%

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

   if -4.4999999999999996e31 < z < -4.1999999999999999e-172

   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. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(z, \color{blue}{\left(x \cdot y\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(z, \left(y \cdot \color{blue}{x}\right)\right) \]

      3. *-lowering-*.f64 71.8%

         \[\leadsto \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(y, \color{blue}{x}\right)\right) \]

   5. Simplified 71.8%

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

   if -4.1999999999999999e-172 < z < 1.94999999999999988e-123

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\sin y}\right) \]

      2. sin-lowering-sin.f64 80.6%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{sin.f64}\left(y\right)\right) \]

   5. Simplified 80.6%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.5 \cdot 10^{+31}:\\ \;\;\;\;z \cdot \cos y\\ \mathbf{elif}\;z \leq -4.2 \cdot 10^{-172}:\\ \;\;\;\;z + y \cdot x\\ \mathbf{elif}\;z \leq 1.95 \cdot 10^{-123}:\\ \;\;\;\;\sin y \cdot x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \cos y\\ \mathbf{if}\;z \leq -2.7 \cdot 10^{+36}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 4.3 \cdot 10^{+112}:\\ \;\;\;\;z + \sin y \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (cos y))))
   (if (<= z -2.7e+36) t_0 (if (<= z 4.3e+112) (+ z (* (sin y) x)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = z * cos(y);
	double tmp;
	if (z <= -2.7e+36) {
		tmp = t_0;
	} else if (z <= 4.3e+112) {
		tmp = z + (sin(y) * x);
	} 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 = z * cos(y)
    if (z <= (-2.7d+36)) then
        tmp = t_0
    else if (z <= 4.3d+112) then
        tmp = z + (sin(y) * x)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = z * Math.cos(y);
	double tmp;
	if (z <= -2.7e+36) {
		tmp = t_0;
	} else if (z <= 4.3e+112) {
		tmp = z + (Math.sin(y) * x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * math.cos(y)
	tmp = 0
	if z <= -2.7e+36:
		tmp = t_0
	elif z <= 4.3e+112:
		tmp = z + (math.sin(y) * x)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * cos(y))
	tmp = 0.0
	if (z <= -2.7e+36)
		tmp = t_0;
	elseif (z <= 4.3e+112)
		tmp = Float64(z + Float64(sin(y) * x));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * cos(y);
	tmp = 0.0;
	if (z <= -2.7e+36)
		tmp = t_0;
	elseif (z <= 4.3e+112)
		tmp = z + (sin(y) * x);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2.7e+36], t$95$0, If[LessEqual[z, 4.3e+112], N[(z + N[(N[Sin[y], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.7000000000000001e36 or 4.29999999999999983e112 < z

   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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \color{blue}{\cos y}\right) \]

      2. cos-lowering-cos.f64 89.7%

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{cos.f64}\left(y\right)\right) \]

   5. Simplified 89.7%

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

   if -2.7000000000000001e36 < z < 4.29999999999999983e112

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{sin.f64}\left(y\right)\right), \color{blue}{z}\right) \]
   4. Step-by-step derivation

   5. Simplified 90.3%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.7 \cdot 10^{+36}:\\ \;\;\;\;z \cdot \cos y\\ \mathbf{elif}\;z \leq 4.3 \cdot 10^{+112}:\\ \;\;\;\;z + \sin y \cdot x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin y \cdot x\\ \mathbf{if}\;y \leq -3.55 \cdot 10^{+21}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{+16}:\\ \;\;\;\;z + y \cdot \left(x + y \cdot \left(\left(y \cdot x\right) \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (sin y) x)))
   (if (<= y -3.55e+21)
     t_0
     (if (<= y 2.5e+16)
       (+ z (* y (+ x (* y (* (* y x) -0.16666666666666666)))))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = sin(y) * x;
	double tmp;
	if (y <= -3.55e+21) {
		tmp = t_0;
	} else if (y <= 2.5e+16) {
		tmp = z + (y * (x + (y * ((y * x) * -0.16666666666666666))));
	} 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 = sin(y) * x
    if (y <= (-3.55d+21)) then
        tmp = t_0
    else if (y <= 2.5d+16) then
        tmp = z + (y * (x + (y * ((y * x) * (-0.16666666666666666d0)))))
    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.sin(y) * x;
	double tmp;
	if (y <= -3.55e+21) {
		tmp = t_0;
	} else if (y <= 2.5e+16) {
		tmp = z + (y * (x + (y * ((y * x) * -0.16666666666666666))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.sin(y) * x
	tmp = 0
	if y <= -3.55e+21:
		tmp = t_0
	elif y <= 2.5e+16:
		tmp = z + (y * (x + (y * ((y * x) * -0.16666666666666666))))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(sin(y) * x)
	tmp = 0.0
	if (y <= -3.55e+21)
		tmp = t_0;
	elseif (y <= 2.5e+16)
		tmp = Float64(z + Float64(y * Float64(x + Float64(y * Float64(Float64(y * x) * -0.16666666666666666)))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = sin(y) * x;
	tmp = 0.0;
	if (y <= -3.55e+21)
		tmp = t_0;
	elseif (y <= 2.5e+16)
		tmp = z + (y * (x + (y * ((y * x) * -0.16666666666666666))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[y, -3.55e+21], t$95$0, If[LessEqual[y, 2.5e+16], N[(z + N[(y * N[(x + N[(y * N[(N[(y * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.55e21 or 2.5e16 < 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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\sin y}\right) \]

      2. sin-lowering-sin.f64 50.0%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{sin.f64}\left(y\right)\right) \]

   5. Simplified 50.0%

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

   if -3.55e21 < y < 2.5e16

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(z, \color{blue}{\left(y \cdot \left(x + y \cdot \left(\frac{-1}{2} \cdot z + \frac{-1}{6} \cdot \left(x \cdot y\right)\right)\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(y, \color{blue}{\left(x + y \cdot \left(\frac{-1}{2} \cdot z + \frac{-1}{6} \cdot \left(x \cdot y\right)\right)\right)}\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \color{blue}{\left(y \cdot \left(\frac{-1}{2} \cdot z + \frac{-1}{6} \cdot \left(x \cdot y\right)\right)\right)}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{-1}{2} \cdot z + \frac{-1}{6} \cdot \left(x \cdot y\right)\right)}\right)\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{-1}{2} \cdot z\right), \color{blue}{\left(\frac{-1}{6} \cdot \left(x \cdot y\right)\right)}\right)\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\left(z \cdot \frac{-1}{2}\right), \left(\color{blue}{\frac{-1}{6}} \cdot \left(x \cdot y\right)\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \frac{-1}{2}\right), \left(\color{blue}{\frac{-1}{6}} \cdot \left(x \cdot y\right)\right)\right)\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \frac{-1}{2}\right), \left(\left(x \cdot y\right) \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \frac{-1}{2}\right), \mathsf{*.f64}\left(\left(x \cdot y\right), \color{blue}{\frac{-1}{6}}\right)\right)\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \frac{-1}{2}\right), \mathsf{*.f64}\left(\left(y \cdot x\right), \frac{-1}{6}\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 95.1%

          \[\leadsto \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, x\right), \frac{-1}{6}\right)\right)\right)\right)\right)\right) \]

   5. Simplified 95.1%

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

   6. Taylor expanded in z around 0

      \[\leadsto \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{-1}{6} \cdot \left(x \cdot y\right)\right)}\right)\right)\right)\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \left(\left(x \cdot y\right) \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left(x \cdot y\right), \color{blue}{\frac{-1}{6}}\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f64 95.8%

         \[\leadsto \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), \frac{-1}{6}\right)\right)\right)\right)\right) \]

   8. Simplified 95.8%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.55 \cdot 10^{+21}:\\ \;\;\;\;\sin y \cdot x\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{+16}:\\ \;\;\;\;z + y \cdot \left(x + y \cdot \left(\left(y \cdot x\right) \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 41.3% accurate, 15.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7.2 \cdot 10^{-160}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{-214}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -7.2e-160) z (if (<= z 2.5e-214) (* y x) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -7.2e-160) {
		tmp = z;
	} else if (z <= 2.5e-214) {
		tmp = y * x;
	} else {
		tmp = z;
	}
	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 <= (-7.2d-160)) then
        tmp = z
    else if (z <= 2.5d-214) then
        tmp = y * x
    else
        tmp = z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -7.2e-160) {
		tmp = z;
	} else if (z <= 2.5e-214) {
		tmp = y * x;
	} else {
		tmp = z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -7.2e-160:
		tmp = z
	elif z <= 2.5e-214:
		tmp = y * x
	else:
		tmp = z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -7.2e-160)
		tmp = z;
	elseif (z <= 2.5e-214)
		tmp = Float64(y * x);
	else
		tmp = z;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -7.2e-160)
		tmp = z;
	elseif (z <= 2.5e-214)
		tmp = y * x;
	else
		tmp = z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -7.2e-160], z, If[LessEqual[z, 2.5e-214], N[(y * x), $MachinePrecision], z]]

Derivation

1. Split input into 2 regimes

2. if z < -7.1999999999999994e-160 or 2.4999999999999999e-214 < z

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 48.7%

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

   if -7.1999999999999994e-160 < z < 2.4999999999999999e-214

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\sin y}\right) \]

      2. sin-lowering-sin.f64 84.3%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{sin.f64}\left(y\right)\right) \]

   5. Simplified 84.3%

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

   6. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{y}\right) \]
   7. Step-by-step derivation

   8. Simplified 37.1%

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

4. Final simplification 46.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.2 \cdot 10^{-160}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{-214}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 52.5% accurate, 41.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

      \[\leadsto \mathsf{+.f64}\left(z, \color{blue}{\left(x \cdot y\right)}\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(z, \left(y \cdot \color{blue}{x}\right)\right) \]

   3. *-lowering-*.f64 53.7%

      \[\leadsto \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(y, \color{blue}{x}\right)\right) \]

5. Simplified 53.7%

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

Alternative 8: 38.8% accurate, 207.0× 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 y around 0

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

5. Simplified 41.8%

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

Reproduce

herbie shell --seed 2024160 
(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))))