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

Percentage Accurate: 99.8% → 99.8%

Time: 17.8s

Alternatives: 9

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(0 - \sin y, z, x \cdot \cos y\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fma (- 0.0 (sin y)) z (* x (cos y))))

C

double code(double x, double y, double z) {
	return fma((0.0 - sin(y)), z, (x * cos(y)));
}

Julia

function code(x, y, z)
	return fma(Float64(0.0 - sin(y)), z, Float64(x * cos(y)))
end

Wolfram

code[x_, y_, z_] := N[(N[(0.0 - N[Sin[y], $MachinePrecision]), $MachinePrecision] * z + N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. distribute-lft-neg-in N/A

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

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

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

   6. neg-sub0 N/A

      \[\leadsto \mathsf{fma.f64}\left(\left(0 - \sin y\right), z, \left(x \cdot \cos y\right)\right) \]

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

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \sin y\right), z, \left(x \cdot \cos y\right)\right) \]

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

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{sin.f64}\left(y\right)\right), z, \left(x \cdot \cos y\right)\right) \]

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

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{sin.f64}\left(y\right)\right), z, \mathsf{*.f64}\left(x, \cos y\right)\right) \]

   10. cos-lowering-cos.f64 99.8%

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{sin.f64}\left(y\right)\right), z, \mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(y\right)\right)\right) \]

4. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0 - \sin y, z, x \cdot \cos y\right)} \]
5. Add Preprocessing

Alternative 2: 86.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \cos y\\ \mathbf{if}\;x \leq -4 \cdot 10^{+60}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 9 \cdot 10^{+69}:\\ \;\;\;\;\mathsf{fma}\left(0 - \sin y, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (cos y))))
   (if (<= x -4e+60) t_0 (if (<= x 9e+69) (fma (- 0.0 (sin y)) z x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x * cos(y);
	double tmp;
	if (x <= -4e+60) {
		tmp = t_0;
	} else if (x <= 9e+69) {
		tmp = fma((0.0 - sin(y)), z, x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x * cos(y))
	tmp = 0.0
	if (x <= -4e+60)
		tmp = t_0;
	elseif (x <= 9e+69)
		tmp = fma(Float64(0.0 - sin(y)), z, x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4e+60], t$95$0, If[LessEqual[x, 9e+69], N[(N[(0.0 - N[Sin[y], $MachinePrecision]), $MachinePrecision] * z + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.9999999999999998e60 or 8.9999999999999999e69 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf

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

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

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

      2. cos-lowering-cos.f64 91.5%

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

   5. Simplified 91.5%

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

   if -3.9999999999999998e60 < x < 8.9999999999999999e69

   1. Initial program 99.8%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. distribute-lft-neg-in N/A

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

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

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

      6. neg-sub0 N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(0 - \sin y\right), z, \left(x \cdot \cos y\right)\right) \]

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

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \sin y\right), z, \left(x \cdot \cos y\right)\right) \]

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

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{sin.f64}\left(y\right)\right), z, \left(x \cdot \cos y\right)\right) \]

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

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{sin.f64}\left(y\right)\right), z, \mathsf{*.f64}\left(x, \cos y\right)\right) \]

      10. cos-lowering-cos.f64 99.8%

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{sin.f64}\left(y\right)\right), z, \mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(y\right)\right)\right) \]

   4. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0 - \sin y, z, x \cdot \cos y\right)} \]

   5. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{sin.f64}\left(y\right)\right), z, \color{blue}{x}\right) \]
   6. Step-by-step derivation

   7. Simplified 88.0%

      \[\leadsto \mathsf{fma}\left(0 - \sin y, z, \color{blue}{x}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Final simplification 99.8%

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

Alternative 4: 86.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \cos y\\ \mathbf{if}\;x \leq -1.7 \cdot 10^{+60}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.18 \cdot 10^{+70}:\\ \;\;\;\;x - \sin y \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (cos y))))
   (if (<= x -1.7e+60) t_0 (if (<= x 1.18e+70) (- x (* (sin y) z)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x * cos(y);
	double tmp;
	if (x <= -1.7e+60) {
		tmp = t_0;
	} else if (x <= 1.18e+70) {
		tmp = x - (sin(y) * z);
	} 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 * cos(y)
    if (x <= (-1.7d+60)) then
        tmp = t_0
    else if (x <= 1.18d+70) then
        tmp = x - (sin(y) * z)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x * Math.cos(y);
	double tmp;
	if (x <= -1.7e+60) {
		tmp = t_0;
	} else if (x <= 1.18e+70) {
		tmp = x - (Math.sin(y) * z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * math.cos(y)
	tmp = 0
	if x <= -1.7e+60:
		tmp = t_0
	elif x <= 1.18e+70:
		tmp = x - (math.sin(y) * z)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * cos(y))
	tmp = 0.0
	if (x <= -1.7e+60)
		tmp = t_0;
	elseif (x <= 1.18e+70)
		tmp = Float64(x - Float64(sin(y) * z));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * cos(y);
	tmp = 0.0;
	if (x <= -1.7e+60)
		tmp = t_0;
	elseif (x <= 1.18e+70)
		tmp = x - (sin(y) * z);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.7e+60], t$95$0, If[LessEqual[x, 1.18e+70], N[(x - N[(N[Sin[y], $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.7e60 or 1.18000000000000001e70 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf

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

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

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

      2. cos-lowering-cos.f64 91.5%

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

   5. Simplified 91.5%

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

   if -1.7e60 < x < 1.18000000000000001e70

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 88.0%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{+60}:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{elif}\;x \leq 1.18 \cdot 10^{+70}:\\ \;\;\;\;x - \sin y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \cos y\\ \mathbf{if}\;x \leq -8 \cdot 10^{-39}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{-69}:\\ \;\;\;\;0 - \sin y \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (cos y))))
   (if (<= x -8e-39) t_0 (if (<= x 3.8e-69) (- 0.0 (* (sin y) z)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x * cos(y);
	double tmp;
	if (x <= -8e-39) {
		tmp = t_0;
	} else if (x <= 3.8e-69) {
		tmp = 0.0 - (sin(y) * z);
	} 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 * cos(y)
    if (x <= (-8d-39)) then
        tmp = t_0
    else if (x <= 3.8d-69) then
        tmp = 0.0d0 - (sin(y) * z)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x * Math.cos(y);
	double tmp;
	if (x <= -8e-39) {
		tmp = t_0;
	} else if (x <= 3.8e-69) {
		tmp = 0.0 - (Math.sin(y) * z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * math.cos(y)
	tmp = 0
	if x <= -8e-39:
		tmp = t_0
	elif x <= 3.8e-69:
		tmp = 0.0 - (math.sin(y) * z)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * cos(y))
	tmp = 0.0
	if (x <= -8e-39)
		tmp = t_0;
	elseif (x <= 3.8e-69)
		tmp = Float64(0.0 - Float64(sin(y) * z));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * cos(y);
	tmp = 0.0;
	if (x <= -8e-39)
		tmp = t_0;
	elseif (x <= 3.8e-69)
		tmp = 0.0 - (sin(y) * z);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -8e-39], t$95$0, If[LessEqual[x, 3.8e-69], N[(0.0 - N[(N[Sin[y], $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -7.99999999999999943e-39 or 3.7999999999999998e-69 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf

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

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

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

      2. cos-lowering-cos.f64 82.6%

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

   5. Simplified 82.6%

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

   if -7.99999999999999943e-39 < x < 3.7999999999999998e-69

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(z \cdot \sin y\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{z \cdot \sin y} \]

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

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(z \cdot \sin y\right)}\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(z, \color{blue}{\sin y}\right)\right) \]

      5. sin-lowering-sin.f64 79.4%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(z, \mathsf{sin.f64}\left(y\right)\right)\right) \]

   5. Simplified 79.4%

      \[\leadsto \color{blue}{0 - z \cdot \sin y} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(z \cdot \sin y\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\left(z \cdot \sin y\right)\right) \]

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

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

      4. sin-lowering-sin.f64 79.4%

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

   7. Applied egg-rr 79.4%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{-39}:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{-69}:\\ \;\;\;\;0 - \sin y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \cos y\\ \mathbf{if}\;y \leq -1.35 \cdot 10^{-7}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 3 \cdot 10^{-18}:\\ \;\;\;\;x - y \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (cos y))))
   (if (<= y -1.35e-7) t_0 (if (<= y 3e-18) (- x (* y z)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x * cos(y);
	double tmp;
	if (y <= -1.35e-7) {
		tmp = t_0;
	} else if (y <= 3e-18) {
		tmp = x - (y * z);
	} 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 * cos(y)
    if (y <= (-1.35d-7)) then
        tmp = t_0
    else if (y <= 3d-18) then
        tmp = x - (y * z)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x * Math.cos(y);
	double tmp;
	if (y <= -1.35e-7) {
		tmp = t_0;
	} else if (y <= 3e-18) {
		tmp = x - (y * z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * math.cos(y)
	tmp = 0
	if y <= -1.35e-7:
		tmp = t_0
	elif y <= 3e-18:
		tmp = x - (y * z)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * cos(y))
	tmp = 0.0
	if (y <= -1.35e-7)
		tmp = t_0;
	elseif (y <= 3e-18)
		tmp = Float64(x - Float64(y * z));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * cos(y);
	tmp = 0.0;
	if (y <= -1.35e-7)
		tmp = t_0;
	elseif (y <= 3e-18)
		tmp = x - (y * z);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.35e-7], t$95$0, If[LessEqual[y, 3e-18], N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.35000000000000004e-7 or 2.99999999999999983e-18 < y

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf

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

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

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

      2. cos-lowering-cos.f64 55.2%

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

   5. Simplified 55.2%

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

   if -1.35000000000000004e-7 < y < 2.99999999999999983e-18

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

         \[\leadsto x + \left(\mathsf{neg}\left(y \cdot z\right)\right) \]

      2. unsub-neg N/A

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 100.0%

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

   5. Simplified 100.0%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.35 \cdot 10^{-7}:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{elif}\;y \leq 3 \cdot 10^{-18}:\\ \;\;\;\;x - y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 41.2% accurate, 13.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{-97}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{-109}:\\ \;\;\;\;0 - y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -7.5e-97) x (if (<= x 3.2e-109) (- 0.0 (* y z)) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -7.5e-97) {
		tmp = x;
	} else if (x <= 3.2e-109) {
		tmp = 0.0 - (y * z);
	} else {
		tmp = x;
	}
	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 <= (-7.5d-97)) then
        tmp = x
    else if (x <= 3.2d-109) then
        tmp = 0.0d0 - (y * z)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -7.5e-97) {
		tmp = x;
	} else if (x <= 3.2e-109) {
		tmp = 0.0 - (y * z);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -7.5e-97:
		tmp = x
	elif x <= 3.2e-109:
		tmp = 0.0 - (y * z)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -7.5e-97)
		tmp = x;
	elseif (x <= 3.2e-109)
		tmp = Float64(0.0 - Float64(y * z));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -7.5e-97)
		tmp = x;
	elseif (x <= 3.2e-109)
		tmp = 0.0 - (y * z);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -7.5e-97], x, If[LessEqual[x, 3.2e-109], N[(0.0 - N[(y * z), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if x < -7.5e-97 or 3.2000000000000002e-109 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 44.6%

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

   if -7.5e-97 < x < 3.2000000000000002e-109

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

         \[\leadsto x + \left(\mathsf{neg}\left(y \cdot z\right)\right) \]

      2. unsub-neg N/A

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 53.3%

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

   5. Simplified 53.3%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-1 \cdot \left(y \cdot z\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(y \cdot z\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{y \cdot z} \]

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

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(y \cdot z\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \left(z \cdot \color{blue}{y}\right)\right) \]

      5. *-lowering-*.f64 43.3%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(z, \color{blue}{y}\right)\right) \]

   8. Simplified 43.3%

      \[\leadsto \color{blue}{0 - z \cdot y} \]
   9. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(z \cdot y\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(y \cdot z\right) \]

      3. distribute-lft-neg-in N/A

         \[\leadsto \left(\mathsf{neg}\left(y\right)\right) \cdot \color{blue}{z} \]

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

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

      5. neg-lowering-neg.f64 43.3%

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

   10. Applied egg-rr 43.3%

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

4. Final simplification 44.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{-97}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{-109}:\\ \;\;\;\;0 - y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 52.4% accurate, 41.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in y around 0

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

   1. mul-1-neg N/A

      \[\leadsto x + \left(\mathsf{neg}\left(y \cdot z\right)\right) \]

   2. unsub-neg N/A

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

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

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

   4. *-commutative N/A

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

   5. *-lowering-*.f64 51.0%

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

5. Simplified 51.0%

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

6. Final simplification 51.0%

   \[\leadsto x - y \cdot z \]
7. Add Preprocessing

Alternative 9: 38.8% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in y around 0

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

5. Simplified 33.6%

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

Reproduce

herbie shell --seed 2024191 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Transform:aboutX from diagrams-lib-1.3.0.3, A"
  :precision binary64
  (- (* x (cos y)) (* z (sin y))))