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

Percentage Accurate: 99.8% → 99.8%

Time: 11.6s

Alternatives: 10

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. accelerator-lowering-fma.f64 N/A

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

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

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

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

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

   5. 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, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_] := N[(N[Cos[y], $MachinePrecision] * z + 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. Step-by-step derivation

   1. +-commutative N/A

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

   2. *-commutative N/A

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

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

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

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

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

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

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

   6. sin-lowering-sin.f64 99.8%

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

4. Applied egg-rr 99.8%

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

5. Final simplification 99.8%

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

Alternative 3: 84.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{+59}:\\ \;\;\;\;z + \sin y \cdot x\\ \mathbf{elif}\;x \leq 7.4 \cdot 10^{-174}:\\ \;\;\;\;z \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sin y, x, z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -3.6e+59)
   (+ z (* (sin y) x))
   (if (<= x 7.4e-174) (* z (cos y)) (fma (sin y) x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -3.6e+59) {
		tmp = z + (sin(y) * x);
	} else if (x <= 7.4e-174) {
		tmp = z * cos(y);
	} else {
		tmp = fma(sin(y), x, z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -3.6e+59)
		tmp = Float64(z + Float64(sin(y) * x));
	elseif (x <= 7.4e-174)
		tmp = Float64(z * cos(y));
	else
		tmp = fma(sin(y), x, z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -3.6e+59], N[(z + N[(N[Sin[y], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 7.4e-174], N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision], N[(N[Sin[y], $MachinePrecision] * x + z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.5999999999999999e59

   1. Initial program 99.7%

      \[x \cdot \sin y + z \cdot \cos y \]
   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 94.0%

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

   if -3.5999999999999999e59 < x < 7.40000000000000019e-174

   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 88.7%

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

   5. Simplified 88.7%

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

   if 7.40000000000000019e-174 < x

   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. accelerator-lowering-fma.f64 N/A

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

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

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

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

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

      5. 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 87.4%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{+59}:\\ \;\;\;\;z + \sin y \cdot x\\ \mathbf{elif}\;x \leq 7.4 \cdot 10^{-174}:\\ \;\;\;\;z \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sin y, x, z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 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 5: 84.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z + \sin y \cdot x\\ \mathbf{if}\;x \leq -5.5 \cdot 10^{+60}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{-171}:\\ \;\;\;\;z \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ z (* (sin y) x))))
   (if (<= x -5.5e+60) t_0 (if (<= x 8.5e-171) (* z (cos y)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = z + (sin(y) * x);
	double tmp;
	if (x <= -5.5e+60) {
		tmp = t_0;
	} else if (x <= 8.5e-171) {
		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 = z + (sin(y) * x)
    if (x <= (-5.5d+60)) then
        tmp = t_0
    else if (x <= 8.5d-171) 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 = z + (Math.sin(y) * x);
	double tmp;
	if (x <= -5.5e+60) {
		tmp = t_0;
	} else if (x <= 8.5e-171) {
		tmp = z * Math.cos(y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z + (math.sin(y) * x)
	tmp = 0
	if x <= -5.5e+60:
		tmp = t_0
	elif x <= 8.5e-171:
		tmp = z * math.cos(y)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z + Float64(sin(y) * x))
	tmp = 0.0
	if (x <= -5.5e+60)
		tmp = t_0;
	elseif (x <= 8.5e-171)
		tmp = Float64(z * cos(y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z + (sin(y) * x);
	tmp = 0.0;
	if (x <= -5.5e+60)
		tmp = t_0;
	elseif (x <= 8.5e-171)
		tmp = z * cos(y);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z + N[(N[Sin[y], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5.5e+60], t$95$0, If[LessEqual[x, 8.5e-171], N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -5.5000000000000001e60 or 8.50000000000000032e-171 < x

   1. Initial program 99.7%

      \[x \cdot \sin y + z \cdot \cos y \]
   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 89.9%

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

   if -5.5000000000000001e60 < x < 8.50000000000000032e-171

   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 88.7%

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

   5. Simplified 88.7%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{+60}:\\ \;\;\;\;z + \sin y \cdot x\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{-171}:\\ \;\;\;\;z \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;z + \sin y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \cos y\\ \mathbf{if}\;z \leq -2.4 \cdot 10^{-16}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{-91}:\\ \;\;\;\;\sin y \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (cos y))))
   (if (<= z -2.4e-16) t_0 (if (<= z 1.7e-91) (* (sin y) x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = z * cos(y);
	double tmp;
	if (z <= -2.4e-16) {
		tmp = t_0;
	} else if (z <= 1.7e-91) {
		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 <= (-2.4d-16)) then
        tmp = t_0
    else if (z <= 1.7d-91) 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 <= -2.4e-16) {
		tmp = t_0;
	} else if (z <= 1.7e-91) {
		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 <= -2.4e-16:
		tmp = t_0
	elif z <= 1.7e-91:
		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 <= -2.4e-16)
		tmp = t_0;
	elseif (z <= 1.7e-91)
		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 <= -2.4e-16)
		tmp = t_0;
	elseif (z <= 1.7e-91)
		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, -2.4e-16], t$95$0, If[LessEqual[z, 1.7e-91], N[(N[Sin[y], $MachinePrecision] * x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.40000000000000005e-16 or 1.70000000000000013e-91 < 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 81.4%

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

   5. Simplified 81.4%

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

   if -2.40000000000000005e-16 < z < 1.70000000000000013e-91

   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 74.5%

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

   5. Simplified 74.5%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.4 \cdot 10^{-16}:\\ \;\;\;\;z \cdot \cos y\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{-91}:\\ \;\;\;\;\sin y \cdot x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin y \cdot x\\ \mathbf{if}\;y \leq -0.27:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 3 \cdot 10^{-18}:\\ \;\;\;\;z + y \cdot \left(x + y \cdot \left(z \cdot -0.5 + x \cdot \left(y \cdot -0.16666666666666666\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (sin y) x)))
   (if (<= y -0.27)
     t_0
     (if (<= y 3e-18)
       (+ z (* y (+ x (* y (+ (* z -0.5) (* x (* y -0.16666666666666666)))))))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = sin(y) * x;
	double tmp;
	if (y <= -0.27) {
		tmp = t_0;
	} else if (y <= 3e-18) {
		tmp = z + (y * (x + (y * ((z * -0.5) + (x * (y * -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 <= (-0.27d0)) then
        tmp = t_0
    else if (y <= 3d-18) then
        tmp = z + (y * (x + (y * ((z * (-0.5d0)) + (x * (y * (-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 <= -0.27) {
		tmp = t_0;
	} else if (y <= 3e-18) {
		tmp = z + (y * (x + (y * ((z * -0.5) + (x * (y * -0.16666666666666666))))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.sin(y) * x
	tmp = 0
	if y <= -0.27:
		tmp = t_0
	elif y <= 3e-18:
		tmp = z + (y * (x + (y * ((z * -0.5) + (x * (y * -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 <= -0.27)
		tmp = t_0;
	elseif (y <= 3e-18)
		tmp = Float64(z + Float64(y * Float64(x + Float64(y * Float64(Float64(z * -0.5) + Float64(x * Float64(y * -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 <= -0.27)
		tmp = t_0;
	elseif (y <= 3e-18)
		tmp = z + (y * (x + (y * ((z * -0.5) + (x * (y * -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, -0.27], t$95$0, If[LessEqual[y, 3e-18], N[(z + N[(y * N[(x + N[(y * N[(N[(z * -0.5), $MachinePrecision] + N[(x * N[(y * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.27000000000000002 or 2.99999999999999983e-18 < 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 54.4%

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

   5. Simplified 54.4%

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

   if -0.27000000000000002 < y < 2.99999999999999983e-18

   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. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

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

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 100.0%

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

   5. Simplified 100.0%

      \[\leadsto \color{blue}{z + y \cdot \left(x + y \cdot \left(z \cdot -0.5 + x \cdot \left(y \cdot -0.16666666666666666\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.27:\\ \;\;\;\;\sin y \cdot x\\ \mathbf{elif}\;y \leq 3 \cdot 10^{-18}:\\ \;\;\;\;z + y \cdot \left(x + y \cdot \left(z \cdot -0.5 + x \cdot \left(y \cdot -0.16666666666666666\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 40.4% accurate, 15.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.8 \cdot 10^{-16}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq 3 \cdot 10^{-234}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.8e-16) z (if (<= z 3e-234) (* y x) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.8e-16) {
		tmp = z;
	} else if (z <= 3e-234) {
		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 <= (-1.8d-16)) then
        tmp = z
    else if (z <= 3d-234) 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 <= -1.8e-16) {
		tmp = z;
	} else if (z <= 3e-234) {
		tmp = y * x;
	} else {
		tmp = z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.8e-16:
		tmp = z
	elif z <= 3e-234:
		tmp = y * x
	else:
		tmp = z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.8e-16)
		tmp = z;
	elseif (z <= 3e-234)
		tmp = Float64(y * x);
	else
		tmp = z;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.8e-16)
		tmp = z;
	elseif (z <= 3e-234)
		tmp = y * x;
	else
		tmp = z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.8e-16], z, If[LessEqual[z, 3e-234], N[(y * x), $MachinePrecision], z]]

Derivation

1. Split input into 2 regimes

2. if z < -1.79999999999999991e-16 or 2.99999999999999987e-234 < 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 48.5%

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

   if -1.79999999999999991e-16 < z < 2.99999999999999987e-234

   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 81.0%

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

   5. Simplified 81.0%

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

   6. Taylor expanded in y around 0

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

   8. Simplified 29.4%

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

4. Final simplification 43.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.8 \cdot 10^{-16}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq 3 \cdot 10^{-234}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 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 51.1%

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

5. Simplified 51.1%

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

Alternative 10: 39.5% 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 38.8%

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

Reproduce

herbie shell --seed 2024191 
(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))))