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

Percentage Accurate: 99.8% → 99.8%

Time: 9.1s

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

   \[x \cdot \sin y + z \cdot \cos y \]
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. fma-define N/A

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

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

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

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

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

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

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

   7. sin-lowering-sin.f64 99.9%

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

4. Applied egg-rr 99.9%

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

Alternative 2: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Final simplification 99.9%

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

Alternative 3: 78.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \sin y\\ t_1 := z \cdot \left(1 + x \cdot \frac{\sin y}{z}\right)\\ \mathbf{if}\;x \leq -1.12 \cdot 10^{+136}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -5.4 \cdot 10^{+17}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-13}:\\ \;\;\;\;\cos y \cdot z\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{+218}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (sin y))) (t_1 (* z (+ 1.0 (* x (/ (sin y) z))))))
   (if (<= x -1.12e+136)
     t_0
     (if (<= x -5.4e+17)
       t_1
       (if (<= x 2.4e-13) (* (cos y) z) (if (<= x 2.1e+218) t_1 t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = x * sin(y);
	double t_1 = z * (1.0 + (x * (sin(y) / z)));
	double tmp;
	if (x <= -1.12e+136) {
		tmp = t_0;
	} else if (x <= -5.4e+17) {
		tmp = t_1;
	} else if (x <= 2.4e-13) {
		tmp = cos(y) * z;
	} else if (x <= 2.1e+218) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_0 = x * sin(y)
    t_1 = z * (1.0d0 + (x * (sin(y) / z)))
    if (x <= (-1.12d+136)) then
        tmp = t_0
    else if (x <= (-5.4d+17)) then
        tmp = t_1
    else if (x <= 2.4d-13) then
        tmp = cos(y) * z
    else if (x <= 2.1d+218) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x * Math.sin(y);
	double t_1 = z * (1.0 + (x * (Math.sin(y) / z)));
	double tmp;
	if (x <= -1.12e+136) {
		tmp = t_0;
	} else if (x <= -5.4e+17) {
		tmp = t_1;
	} else if (x <= 2.4e-13) {
		tmp = Math.cos(y) * z;
	} else if (x <= 2.1e+218) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * math.sin(y)
	t_1 = z * (1.0 + (x * (math.sin(y) / z)))
	tmp = 0
	if x <= -1.12e+136:
		tmp = t_0
	elif x <= -5.4e+17:
		tmp = t_1
	elif x <= 2.4e-13:
		tmp = math.cos(y) * z
	elif x <= 2.1e+218:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * sin(y))
	t_1 = Float64(z * Float64(1.0 + Float64(x * Float64(sin(y) / z))))
	tmp = 0.0
	if (x <= -1.12e+136)
		tmp = t_0;
	elseif (x <= -5.4e+17)
		tmp = t_1;
	elseif (x <= 2.4e-13)
		tmp = Float64(cos(y) * z);
	elseif (x <= 2.1e+218)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * sin(y);
	t_1 = z * (1.0 + (x * (sin(y) / z)));
	tmp = 0.0;
	if (x <= -1.12e+136)
		tmp = t_0;
	elseif (x <= -5.4e+17)
		tmp = t_1;
	elseif (x <= 2.4e-13)
		tmp = cos(y) * z;
	elseif (x <= 2.1e+218)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[Sin[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(z * N[(1.0 + N[(x * N[(N[Sin[y], $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.12e+136], t$95$0, If[LessEqual[x, -5.4e+17], t$95$1, If[LessEqual[x, 2.4e-13], N[(N[Cos[y], $MachinePrecision] * z), $MachinePrecision], If[LessEqual[x, 2.1e+218], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.12000000000000001e136 or 2.0999999999999999e218 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

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

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

      2. sin-lowering-sin.f64 76.8%

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

   5. Simplified 76.8%

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

   if -1.12000000000000001e136 < x < -5.4e17 or 2.3999999999999999e-13 < x < 2.0999999999999999e218

   1. Initial program 99.9%

      \[x \cdot \sin y + z \cdot \cos y \]
   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. fma-define N/A

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

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

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

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

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

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

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

      7. sin-lowering-sin.f64 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in z around inf

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

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

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

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

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

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

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

      4. associate-/l* N/A

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

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

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

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

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

      7. sin-lowering-sin.f64 98.2%

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

   7. Simplified 98.2%

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

   8. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), z\right)\right)\right)\right) \]
   9. Step-by-step derivation

   10. Simplified 86.1%

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

   if -5.4e17 < x < 2.3999999999999999e-13

   1. Initial program 99.9%

      \[x \cdot \sin y + z \cdot \cos y \]
   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 86.7%

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

   5. Simplified 86.7%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.12 \cdot 10^{+136}:\\ \;\;\;\;x \cdot \sin y\\ \mathbf{elif}\;x \leq -5.4 \cdot 10^{+17}:\\ \;\;\;\;z \cdot \left(1 + x \cdot \frac{\sin y}{z}\right)\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-13}:\\ \;\;\;\;\cos y \cdot z\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{+218}:\\ \;\;\;\;z \cdot \left(1 + x \cdot \frac{\sin y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sin y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos y \cdot z\\ \mathbf{if}\;y \leq -5.1 \cdot 10^{+229}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -0.0005:\\ \;\;\;\;x \cdot \sin y\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-8}:\\ \;\;\;\;z + y \cdot \left(x + y \cdot \left(z \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (cos y) z)))
   (if (<= y -5.1e+229)
     t_0
     (if (<= y -0.0005)
       (* x (sin y))
       (if (<= y 1.3e-8) (+ z (* y (+ x (* y (* z -0.5))))) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = cos(y) * z;
	double tmp;
	if (y <= -5.1e+229) {
		tmp = t_0;
	} else if (y <= -0.0005) {
		tmp = x * sin(y);
	} else if (y <= 1.3e-8) {
		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 = cos(y) * z
    if (y <= (-5.1d+229)) then
        tmp = t_0
    else if (y <= (-0.0005d0)) then
        tmp = x * sin(y)
    else if (y <= 1.3d-8) 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.cos(y) * z;
	double tmp;
	if (y <= -5.1e+229) {
		tmp = t_0;
	} else if (y <= -0.0005) {
		tmp = x * Math.sin(y);
	} else if (y <= 1.3e-8) {
		tmp = z + (y * (x + (y * (z * -0.5))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.cos(y) * z
	tmp = 0
	if y <= -5.1e+229:
		tmp = t_0
	elif y <= -0.0005:
		tmp = x * math.sin(y)
	elif y <= 1.3e-8:
		tmp = z + (y * (x + (y * (z * -0.5))))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(cos(y) * z)
	tmp = 0.0
	if (y <= -5.1e+229)
		tmp = t_0;
	elseif (y <= -0.0005)
		tmp = Float64(x * sin(y));
	elseif (y <= 1.3e-8)
		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 = cos(y) * z;
	tmp = 0.0;
	if (y <= -5.1e+229)
		tmp = t_0;
	elseif (y <= -0.0005)
		tmp = x * sin(y);
	elseif (y <= 1.3e-8)
		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[Cos[y], $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[y, -5.1e+229], t$95$0, If[LessEqual[y, -0.0005], N[(x * N[Sin[y], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.3e-8], 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 3 regimes

2. if y < -5.0999999999999996e229 or 1.3000000000000001e-8 < y

   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 58.4%

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

   5. Simplified 58.4%

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

   if -5.0999999999999996e229 < y < -5.0000000000000001e-4

   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 63.8%

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

   5. Simplified 63.8%

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

   if -5.0000000000000001e-4 < y < 1.3000000000000001e-8

   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 100.0%

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

   5. Simplified 100.0%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.1 \cdot 10^{+229}:\\ \;\;\;\;\cos y \cdot z\\ \mathbf{elif}\;y \leq -0.0005:\\ \;\;\;\;x \cdot \sin y\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-8}:\\ \;\;\;\;z + y \cdot \left(x + y \cdot \left(z \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \sin y\\ \mathbf{if}\;y \leq -0.036:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.1:\\ \;\;\;\;z + y \cdot \left(x + y \cdot \left(z \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (sin y))))
   (if (<= y -0.036)
     t_0
     (if (<= y 0.1) (+ z (* y (+ x (* y (* z -0.5))))) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x * sin(y);
	double tmp;
	if (y <= -0.036) {
		tmp = t_0;
	} else if (y <= 0.1) {
		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 = x * sin(y)
    if (y <= (-0.036d0)) then
        tmp = t_0
    else if (y <= 0.1d0) 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 = x * Math.sin(y);
	double tmp;
	if (y <= -0.036) {
		tmp = t_0;
	} else if (y <= 0.1) {
		tmp = z + (y * (x + (y * (z * -0.5))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * math.sin(y)
	tmp = 0
	if y <= -0.036:
		tmp = t_0
	elif y <= 0.1:
		tmp = z + (y * (x + (y * (z * -0.5))))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * sin(y))
	tmp = 0.0
	if (y <= -0.036)
		tmp = t_0;
	elseif (y <= 0.1)
		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 = x * sin(y);
	tmp = 0.0;
	if (y <= -0.036)
		tmp = t_0;
	elseif (y <= 0.1)
		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[(x * N[Sin[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.036], t$95$0, If[LessEqual[y, 0.1], 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 < -0.0359999999999999973 or 0.10000000000000001 < y

   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 49.0%

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

   5. Simplified 49.0%

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

   if -0.0359999999999999973 < y < 0.10000000000000001

   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 100.0%

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

   5. Simplified 100.0%

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

Alternative 6: 40.7% accurate, 15.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{+72}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{+45}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -7e+72) (* y x) (if (<= x 1.65e+45) z (* y x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -7e+72) {
		tmp = y * x;
	} else if (x <= 1.65e+45) {
		tmp = z;
	} else {
		tmp = y * 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 <= (-7d+72)) then
        tmp = y * x
    else if (x <= 1.65d+45) then
        tmp = z
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -7e+72) {
		tmp = y * x;
	} else if (x <= 1.65e+45) {
		tmp = z;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -7e+72:
		tmp = y * x
	elif x <= 1.65e+45:
		tmp = z
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -7e+72)
		tmp = Float64(y * x);
	elseif (x <= 1.65e+45)
		tmp = z;
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -7e+72)
		tmp = y * x;
	elseif (x <= 1.65e+45)
		tmp = z;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -7e+72], N[(y * x), $MachinePrecision], If[LessEqual[x, 1.65e+45], z, N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -7.0000000000000002e72 or 1.65e45 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

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

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

      2. sin-lowering-sin.f64 68.4%

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

   5. Simplified 68.4%

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

   6. Taylor expanded in y around 0

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

      1. *-lowering-*.f64 35.0%

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

   8. Simplified 35.0%

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

   if -7.0000000000000002e72 < x < 1.65e45

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 52.1%

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

4. Final simplification 45.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{+72}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{+45}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 52.0% 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 99.9%

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

3. Taylor expanded in y around 0

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

   1. +-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 56.7%

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

5. Simplified 56.7%

   \[\leadsto \color{blue}{z + y \cdot x} \]
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 99.9%

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

3. Taylor expanded in y around 0

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

5. Simplified 39.5%

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

Reproduce

herbie shell --seed 2024138 
(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))))