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

Percentage Accurate: 99.8% → 99.8%

Time: 14.8s

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. fma-def 99.8%

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

3. Simplified 99.8%

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

4. Final simplification 99.8%

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

Alternative 2: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

2. Final simplification 99.8%

   \[\leadsto z \cdot \cos y + x \cdot \sin y \]

Alternative 3: 74.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \cos y\\ t_1 := x \cdot \sin y\\ \mathbf{if}\;y \leq -3.2 \cdot 10^{+206}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -3 \cdot 10^{+162}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -7.2 \cdot 10^{+28}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -0.011:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{-7}:\\ \;\;\;\;z + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (cos y))) (t_1 (* x (sin y))))
   (if (<= y -3.2e+206)
     t_0
     (if (<= y -3e+162)
       t_1
       (if (<= y -7.2e+28)
         t_0
         (if (<= y -0.011) t_1 (if (<= y 2.4e-7) (+ z (* x y)) t_0)))))))

C

double code(double x, double y, double z) {
	double t_0 = z * cos(y);
	double t_1 = x * sin(y);
	double tmp;
	if (y <= -3.2e+206) {
		tmp = t_0;
	} else if (y <= -3e+162) {
		tmp = t_1;
	} else if (y <= -7.2e+28) {
		tmp = t_0;
	} else if (y <= -0.011) {
		tmp = t_1;
	} else if (y <= 2.4e-7) {
		tmp = z + (x * 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) :: t_1
    real(8) :: tmp
    t_0 = z * cos(y)
    t_1 = x * sin(y)
    if (y <= (-3.2d+206)) then
        tmp = t_0
    else if (y <= (-3d+162)) then
        tmp = t_1
    else if (y <= (-7.2d+28)) then
        tmp = t_0
    else if (y <= (-0.011d0)) then
        tmp = t_1
    else if (y <= 2.4d-7) then
        tmp = z + (x * 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.cos(y);
	double t_1 = x * Math.sin(y);
	double tmp;
	if (y <= -3.2e+206) {
		tmp = t_0;
	} else if (y <= -3e+162) {
		tmp = t_1;
	} else if (y <= -7.2e+28) {
		tmp = t_0;
	} else if (y <= -0.011) {
		tmp = t_1;
	} else if (y <= 2.4e-7) {
		tmp = z + (x * y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * math.cos(y)
	t_1 = x * math.sin(y)
	tmp = 0
	if y <= -3.2e+206:
		tmp = t_0
	elif y <= -3e+162:
		tmp = t_1
	elif y <= -7.2e+28:
		tmp = t_0
	elif y <= -0.011:
		tmp = t_1
	elif y <= 2.4e-7:
		tmp = z + (x * y)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * cos(y))
	t_1 = Float64(x * sin(y))
	tmp = 0.0
	if (y <= -3.2e+206)
		tmp = t_0;
	elseif (y <= -3e+162)
		tmp = t_1;
	elseif (y <= -7.2e+28)
		tmp = t_0;
	elseif (y <= -0.011)
		tmp = t_1;
	elseif (y <= 2.4e-7)
		tmp = Float64(z + Float64(x * y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * cos(y);
	t_1 = x * sin(y);
	tmp = 0.0;
	if (y <= -3.2e+206)
		tmp = t_0;
	elseif (y <= -3e+162)
		tmp = t_1;
	elseif (y <= -7.2e+28)
		tmp = t_0;
	elseif (y <= -0.011)
		tmp = t_1;
	elseif (y <= 2.4e-7)
		tmp = z + (x * y);
	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]}, Block[{t$95$1 = N[(x * N[Sin[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -3.2e+206], t$95$0, If[LessEqual[y, -3e+162], t$95$1, If[LessEqual[y, -7.2e+28], t$95$0, If[LessEqual[y, -0.011], t$95$1, If[LessEqual[y, 2.4e-7], N[(z + N[(x * y), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.20000000000000005e206 or -2.9999999999999998e162 < y < -7.1999999999999999e28 or 2.39999999999999979e-7 < y

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. fma-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around 0 63.9%

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

   if -3.20000000000000005e206 < y < -2.9999999999999998e162 or -7.1999999999999999e28 < y < -0.010999999999999999

   1. Initial program 99.7%

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

      1. add-cube-cbrt 98.6%

         \[\leadsto \color{blue}{\left(\sqrt[3]{x \cdot \sin y + z \cdot \cos y} \cdot \sqrt[3]{x \cdot \sin y + z \cdot \cos y}\right) \cdot \sqrt[3]{x \cdot \sin y + z \cdot \cos y}} \]

      2. pow3 98.4%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{x \cdot \sin y + z \cdot \cos y}\right)}^{3}} \]

      3. fma-def 98.4%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{\mathsf{fma}\left(x, \sin y, z \cdot \cos y\right)}}\right)}^{3} \]

   3. Applied egg-rr 98.4%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(x, \sin y, z \cdot \cos y\right)}\right)}^{3}} \]

   4. Taylor expanded in z around 0 83.6%

      \[\leadsto \color{blue}{{1}^{0.3333333333333333} \cdot \left(\sin y \cdot x\right)} \]
   5. Step-by-step derivation

      1. pow-base-1 83.6%

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

      2. *-lft-identity 83.6%

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

   6. Simplified 83.6%

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

   if -0.010999999999999999 < y < 2.39999999999999979e-7

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 100.0%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{+206}:\\ \;\;\;\;z \cdot \cos y\\ \mathbf{elif}\;y \leq -3 \cdot 10^{+162}:\\ \;\;\;\;x \cdot \sin y\\ \mathbf{elif}\;y \leq -7.2 \cdot 10^{+28}:\\ \;\;\;\;z \cdot \cos y\\ \mathbf{elif}\;y \leq -0.011:\\ \;\;\;\;x \cdot \sin y\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{-7}:\\ \;\;\;\;z + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \cos y\\ \end{array} \]

Alternative 4: 85.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -10500000000 \lor \neg \left(x \leq 175000000\right):\\ \;\;\;\;z + x \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \cos y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -10500000000.0) (not (<= x 175000000.0)))
   (+ z (* x (sin y)))
   (* z (cos y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -10500000000.0) || !(x <= 175000000.0)) {
		tmp = z + (x * sin(y));
	} else {
		tmp = z * cos(y);
	}
	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 <= (-10500000000.0d0)) .or. (.not. (x <= 175000000.0d0))) then
        tmp = z + (x * sin(y))
    else
        tmp = z * cos(y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -10500000000.0) || !(x <= 175000000.0)) {
		tmp = z + (x * Math.sin(y));
	} else {
		tmp = z * Math.cos(y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -10500000000.0) or not (x <= 175000000.0):
		tmp = z + (x * math.sin(y))
	else:
		tmp = z * math.cos(y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -10500000000.0) || !(x <= 175000000.0))
		tmp = Float64(z + Float64(x * sin(y)));
	else
		tmp = Float64(z * cos(y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -10500000000.0) || ~((x <= 175000000.0)))
		tmp = z + (x * sin(y));
	else
		tmp = z * cos(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -10500000000.0], N[Not[LessEqual[x, 175000000.0]], $MachinePrecision]], N[(z + N[(x * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.05e10 or 1.75e8 < x

   1. Initial program 99.8%

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

   2. Taylor expanded in y around 0 91.1%

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

   if -1.05e10 < x < 1.75e8

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

      2. fma-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 88.5%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -10500000000 \lor \neg \left(x \leq 175000000\right):\\ \;\;\;\;z + x \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \cos y\\ \end{array} \]

Alternative 5: 74.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.0038 \lor \neg \left(y \leq 2.4 \cdot 10^{-7}\right):\\ \;\;\;\;z \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;z + x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -0.0038) (not (<= y 2.4e-7))) (* z (cos y)) (+ z (* x y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.0038) || !(y <= 2.4e-7)) {
		tmp = z * cos(y);
	} else {
		tmp = z + (x * y);
	}
	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 ((y <= (-0.0038d0)) .or. (.not. (y <= 2.4d-7))) then
        tmp = z * cos(y)
    else
        tmp = z + (x * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.0038) || !(y <= 2.4e-7)) {
		tmp = z * Math.cos(y);
	} else {
		tmp = z + (x * y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -0.0038) or not (y <= 2.4e-7):
		tmp = z * math.cos(y)
	else:
		tmp = z + (x * y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -0.0038) || !(y <= 2.4e-7))
		tmp = Float64(z * cos(y));
	else
		tmp = Float64(z + Float64(x * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -0.0038) || ~((y <= 2.4e-7)))
		tmp = z * cos(y);
	else
		tmp = z + (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -0.0038], N[Not[LessEqual[y, 2.4e-7]], $MachinePrecision]], N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision], N[(z + N[(x * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -0.00379999999999999999 or 2.39999999999999979e-7 < y

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. fma-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around 0 57.5%

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

   if -0.00379999999999999999 < y < 2.39999999999999979e-7

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 100.0%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.0038 \lor \neg \left(y \leq 2.4 \cdot 10^{-7}\right):\\ \;\;\;\;z \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;z + x \cdot y\\ \end{array} \]

Alternative 6: 39.8% accurate, 40.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.6 \cdot 10^{+211}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= x -6.6e+211) (* x y) z))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -6.6e+211) {
		tmp = x * y;
	} 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 (x <= (-6.6d+211)) then
        tmp = x * y
    else
        tmp = z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -6.6e+211) {
		tmp = x * y;
	} else {
		tmp = z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -6.6e+211:
		tmp = x * y
	else:
		tmp = z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -6.6e+211)
		tmp = Float64(x * y);
	else
		tmp = z;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -6.6e+211)
		tmp = x * y;
	else
		tmp = z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -6.6e+211], N[(x * y), $MachinePrecision], z]

Derivation

1. Split input into 2 regimes

2. if x < -6.59999999999999966e211

   1. Initial program 99.8%

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

      1. add-cube-cbrt 97.9%

         \[\leadsto \color{blue}{\left(\sqrt[3]{x \cdot \sin y + z \cdot \cos y} \cdot \sqrt[3]{x \cdot \sin y + z \cdot \cos y}\right) \cdot \sqrt[3]{x \cdot \sin y + z \cdot \cos y}} \]

      2. pow3 97.9%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{x \cdot \sin y + z \cdot \cos y}\right)}^{3}} \]

      3. fma-def 97.9%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{\mathsf{fma}\left(x, \sin y, z \cdot \cos y\right)}}\right)}^{3} \]

   3. Applied egg-rr 97.9%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(x, \sin y, z \cdot \cos y\right)}\right)}^{3}} \]

   4. Taylor expanded in z around 0 83.1%

      \[\leadsto \color{blue}{{1}^{0.3333333333333333} \cdot \left(\sin y \cdot x\right)} \]
   5. Step-by-step derivation

      1. pow-base-1 83.1%

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

      2. *-lft-identity 83.1%

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

   6. Simplified 83.1%

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

   7. Taylor expanded in y around 0 47.5%

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

   if -6.59999999999999966e211 < x

   1. Initial program 99.8%

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

      1. add-cube-cbrt 98.0%

         \[\leadsto \color{blue}{\left(\sqrt[3]{x \cdot \sin y + z \cdot \cos y} \cdot \sqrt[3]{x \cdot \sin y + z \cdot \cos y}\right) \cdot \sqrt[3]{x \cdot \sin y + z \cdot \cos y}} \]

      2. pow3 98.0%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{x \cdot \sin y + z \cdot \cos y}\right)}^{3}} \]

      3. fma-def 98.0%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{\mathsf{fma}\left(x, \sin y, z \cdot \cos y\right)}}\right)}^{3} \]

   3. Applied egg-rr 98.0%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(x, \sin y, z \cdot \cos y\right)}\right)}^{3}} \]
   4. Step-by-step derivation

      1. fma-def 98.0%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{x \cdot \sin y + z \cdot \cos y}}\right)}^{3} \]

      2. +-commutative 98.0%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{z \cdot \cos y + x \cdot \sin y}}\right)}^{3} \]

   5. Applied egg-rr 98.0%

      \[\leadsto {\left(\sqrt[3]{\color{blue}{z \cdot \cos y + x \cdot \sin y}}\right)}^{3} \]

   6. Taylor expanded in y around 0 46.0%

      \[\leadsto \color{blue}{{1}^{0.3333333333333333} \cdot z} \]
   7. Step-by-step derivation

      1. pow-base-1 46.0%

         \[\leadsto \color{blue}{1} \cdot z \]

      2. *-lft-identity 46.0%

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

   8. Simplified 46.0%

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

4. Final simplification 46.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.6 \cdot 10^{+211}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]

Alternative 7: 52.4% accurate, 41.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in y around 0 55.0%

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

3. Final simplification 55.0%

   \[\leadsto z + x \cdot y \]

Alternative 8: 38.3% 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. Step-by-step derivation

   1. add-cube-cbrt 97.9%

      \[\leadsto \color{blue}{\left(\sqrt[3]{x \cdot \sin y + z \cdot \cos y} \cdot \sqrt[3]{x \cdot \sin y + z \cdot \cos y}\right) \cdot \sqrt[3]{x \cdot \sin y + z \cdot \cos y}} \]

   2. pow3 98.0%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{x \cdot \sin y + z \cdot \cos y}\right)}^{3}} \]

   3. fma-def 98.0%

      \[\leadsto {\left(\sqrt[3]{\color{blue}{\mathsf{fma}\left(x, \sin y, z \cdot \cos y\right)}}\right)}^{3} \]

3. Applied egg-rr 98.0%

   \[\leadsto \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(x, \sin y, z \cdot \cos y\right)}\right)}^{3}} \]
4. Step-by-step derivation

   1. fma-def 98.0%

      \[\leadsto {\left(\sqrt[3]{\color{blue}{x \cdot \sin y + z \cdot \cos y}}\right)}^{3} \]

   2. +-commutative 98.0%

      \[\leadsto {\left(\sqrt[3]{\color{blue}{z \cdot \cos y + x \cdot \sin y}}\right)}^{3} \]

5. Applied egg-rr 98.0%

   \[\leadsto {\left(\sqrt[3]{\color{blue}{z \cdot \cos y + x \cdot \sin y}}\right)}^{3} \]

6. Taylor expanded in y around 0 43.5%

   \[\leadsto \color{blue}{{1}^{0.3333333333333333} \cdot z} \]
7. Step-by-step derivation

   1. pow-base-1 43.5%

      \[\leadsto \color{blue}{1} \cdot z \]

   2. *-lft-identity 43.5%

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

8. Simplified 43.5%

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

9. Final simplification 43.5%

   \[\leadsto z \]

Reproduce

herbie shell --seed 2023274 
(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))))