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

Percentage Accurate: 99.8% → 99.8%

Time: 10.3s

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, 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: 85.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \cos y\\ \mathbf{if}\;z \leq -2500000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1.16 \cdot 10^{-37}:\\ \;\;\;\;\mathsf{fma}\left(\sin y, x, z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (cos y))))
   (if (<= z -2500000000.0) t_0 (if (<= z 1.16e-37) (fma (sin y) x z) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = z * cos(y);
	double tmp;
	if (z <= -2500000000.0) {
		tmp = t_0;
	} else if (z <= 1.16e-37) {
		tmp = fma(sin(y), x, z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(z * cos(y))
	tmp = 0.0
	if (z <= -2500000000.0)
		tmp = t_0;
	elseif (z <= 1.16e-37)
		tmp = fma(sin(y), x, z);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2500000000.0], t$95$0, If[LessEqual[z, 1.16e-37], N[(N[Sin[y], $MachinePrecision] * x + z), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.5e9 or 1.15999999999999998e-37 < 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 85.1%

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

   5. Simplified 85.1%

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

   if -2.5e9 < z < 1.15999999999999998e-37

   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. Taylor expanded in y around 0

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

   7. Simplified 91.1%

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

Alternative 3: 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 4: 85.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \cos y\\ \mathbf{if}\;z \leq -9000000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1.22 \cdot 10^{-40}:\\ \;\;\;\;z + \sin y \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (cos y))))
   (if (<= z -9000000000000.0)
     t_0
     (if (<= z 1.22e-40) (+ 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 <= -9000000000000.0) {
		tmp = t_0;
	} else if (z <= 1.22e-40) {
		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 <= (-9000000000000.0d0)) then
        tmp = t_0
    else if (z <= 1.22d-40) 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 <= -9000000000000.0) {
		tmp = t_0;
	} else if (z <= 1.22e-40) {
		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 <= -9000000000000.0:
		tmp = t_0
	elif z <= 1.22e-40:
		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 <= -9000000000000.0)
		tmp = t_0;
	elseif (z <= 1.22e-40)
		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 <= -9000000000000.0)
		tmp = t_0;
	elseif (z <= 1.22e-40)
		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, -9000000000000.0], t$95$0, If[LessEqual[z, 1.22e-40], N[(z + N[(N[Sin[y], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -9e12 or 1.22e-40 < 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 85.1%

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

   5. Simplified 85.1%

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

   if -9e12 < z < 1.22e-40

   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 91.1%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9000000000000:\\ \;\;\;\;z \cdot \cos y\\ \mathbf{elif}\;z \leq 1.22 \cdot 10^{-40}:\\ \;\;\;\;z + \sin y \cdot x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \cos y\\ \mathbf{if}\;z \leq -1.9 \cdot 10^{-137}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{-45}:\\ \;\;\;\;\sin y \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (cos y))))
   (if (<= z -1.9e-137) t_0 (if (<= z 1.25e-45) (* (sin y) x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = z * cos(y);
	double tmp;
	if (z <= -1.9e-137) {
		tmp = t_0;
	} else if (z <= 1.25e-45) {
		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 <= (-1.9d-137)) then
        tmp = t_0
    else if (z <= 1.25d-45) 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 <= -1.9e-137) {
		tmp = t_0;
	} else if (z <= 1.25e-45) {
		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 <= -1.9e-137:
		tmp = t_0
	elif z <= 1.25e-45:
		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 <= -1.9e-137)
		tmp = t_0;
	elseif (z <= 1.25e-45)
		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 <= -1.9e-137)
		tmp = t_0;
	elseif (z <= 1.25e-45)
		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, -1.9e-137], t$95$0, If[LessEqual[z, 1.25e-45], N[(N[Sin[y], $MachinePrecision] * x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.89999999999999999e-137 or 1.24999999999999994e-45 < 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 80.5%

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

   5. Simplified 80.5%

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

   if -1.89999999999999999e-137 < z < 1.24999999999999994e-45

   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 79.6%

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

   5. Simplified 79.6%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{-137}:\\ \;\;\;\;z \cdot \cos y\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{-45}:\\ \;\;\;\;\sin y \cdot x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin y \cdot x\\ \mathbf{if}\;y \leq -48:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.0078:\\ \;\;\;\;z + y \cdot \left(x + y \cdot \left(z \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (sin y) x)))
   (if (<= y -48.0)
     t_0
     (if (<= y 0.0078) (+ z (* y (+ x (* y (* z -0.5))))) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = sin(y) * x;
	double tmp;
	if (y <= -48.0) {
		tmp = t_0;
	} else if (y <= 0.0078) {
		tmp = z + (y * (x + (y * (z * -0.5))));
	} 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 <= (-48.0d0)) then
        tmp = t_0
    else if (y <= 0.0078d0) then
        tmp = z + (y * (x + (y * (z * (-0.5d0)))))
    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 <= -48.0) {
		tmp = t_0;
	} else if (y <= 0.0078) {
		tmp = z + (y * (x + (y * (z * -0.5))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.sin(y) * x
	tmp = 0
	if y <= -48.0:
		tmp = t_0
	elif y <= 0.0078:
		tmp = z + (y * (x + (y * (z * -0.5))))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(sin(y) * x)
	tmp = 0.0
	if (y <= -48.0)
		tmp = t_0;
	elseif (y <= 0.0078)
		tmp = Float64(z + Float64(y * Float64(x + Float64(y * Float64(z * -0.5)))));
	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 <= -48.0)
		tmp = t_0;
	elseif (y <= 0.0078)
		tmp = z + (y * (x + (y * (z * -0.5))));
	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, -48.0], t$95$0, If[LessEqual[y, 0.0078], N[(z + N[(y * N[(x + N[(y * N[(z * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -48 or 0.0077999999999999996 < y

   1. Initial program 99.6%

      \[x \cdot \sin y + z \cdot \cos y \]
   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 45.7%

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

   5. Simplified 45.7%

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

   if -48 < y < 0.0077999999999999996

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

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

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

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

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

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

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 99.3%

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

   5. Simplified 99.3%

      \[\leadsto \color{blue}{z + y \cdot \left(x + y \cdot \left(z \cdot -0.5\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -48:\\ \;\;\;\;\sin y \cdot x\\ \mathbf{elif}\;y \leq 0.0078:\\ \;\;\;\;z + y \cdot \left(x + y \cdot \left(z \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 41.6% accurate, 15.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.7 \cdot 10^{-158}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq 9.2 \cdot 10^{-57}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.7e-158) z (if (<= z 9.2e-57) (* y x) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.7e-158) {
		tmp = z;
	} else if (z <= 9.2e-57) {
		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 <= (-1.7d-158)) then
        tmp = z
    else if (z <= 9.2d-57) 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 <= -1.7e-158) {
		tmp = z;
	} else if (z <= 9.2e-57) {
		tmp = y * x;
	} else {
		tmp = z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.7e-158:
		tmp = z
	elif z <= 9.2e-57:
		tmp = y * x
	else:
		tmp = z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.7e-158)
		tmp = z;
	elseif (z <= 9.2e-57)
		tmp = Float64(y * x);
	else
		tmp = z;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.7e-158)
		tmp = z;
	elseif (z <= 9.2e-57)
		tmp = y * x;
	else
		tmp = z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.7e-158], z, If[LessEqual[z, 9.2e-57], N[(y * x), $MachinePrecision], z]]

Derivation

1. Split input into 2 regimes

2. if z < -1.7e-158 or 9.2000000000000001e-57 < 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 44.3%

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

   if -1.7e-158 < z < 9.2000000000000001e-57

   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 79.6%

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

   5. Simplified 79.6%

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

   6. Taylor expanded in y around 0

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

   8. Simplified 44.5%

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

4. Final simplification 44.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.7 \cdot 10^{-158}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq 9.2 \cdot 10^{-57}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 52.8% 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 52.6%

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

5. Simplified 52.6%

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

Alternative 9: 39.6% 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 35.2%

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

Reproduce

herbie shell --seed 2024158 
(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))))