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

Percentage Accurate: 99.8% → 99.8%

Time: 7.0s

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

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

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]

Derivation

1. Initial program 99.8%

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

2. Final simplification 99.8%

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

Alternative 3: 74.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \cos y\\ t_1 := x \cdot \sin y\\ \mathbf{if}\;y \leq -2 \cdot 10^{+154}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -1.7:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -0.0075:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 0.086:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \left(y \cdot y\right)\right) + \left(z + x \cdot y\right)\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{+20}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (cos y))) (t_1 (* x (sin y))))
   (if (<= y -2e+154)
     t_0
     (if (<= y -1.7)
       t_1
       (if (<= y -0.0075)
         t_0
         (if (<= y 0.086)
           (+ (* -0.5 (* z (* y y))) (+ z (* x y)))
           (if (<= y 5.8e+20) t_1 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 <= -2e+154) {
		tmp = t_0;
	} else if (y <= -1.7) {
		tmp = t_1;
	} else if (y <= -0.0075) {
		tmp = t_0;
	} else if (y <= 0.086) {
		tmp = (-0.5 * (z * (y * y))) + (z + (x * y));
	} else if (y <= 5.8e+20) {
		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 = z * cos(y)
    t_1 = x * sin(y)
    if (y <= (-2d+154)) then
        tmp = t_0
    else if (y <= (-1.7d0)) then
        tmp = t_1
    else if (y <= (-0.0075d0)) then
        tmp = t_0
    else if (y <= 0.086d0) then
        tmp = ((-0.5d0) * (z * (y * y))) + (z + (x * y))
    else if (y <= 5.8d+20) 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 = z * Math.cos(y);
	double t_1 = x * Math.sin(y);
	double tmp;
	if (y <= -2e+154) {
		tmp = t_0;
	} else if (y <= -1.7) {
		tmp = t_1;
	} else if (y <= -0.0075) {
		tmp = t_0;
	} else if (y <= 0.086) {
		tmp = (-0.5 * (z * (y * y))) + (z + (x * y));
	} else if (y <= 5.8e+20) {
		tmp = t_1;
	} 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 <= -2e+154:
		tmp = t_0
	elif y <= -1.7:
		tmp = t_1
	elif y <= -0.0075:
		tmp = t_0
	elif y <= 0.086:
		tmp = (-0.5 * (z * (y * y))) + (z + (x * y))
	elif y <= 5.8e+20:
		tmp = t_1
	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 <= -2e+154)
		tmp = t_0;
	elseif (y <= -1.7)
		tmp = t_1;
	elseif (y <= -0.0075)
		tmp = t_0;
	elseif (y <= 0.086)
		tmp = Float64(Float64(-0.5 * Float64(z * Float64(y * y))) + Float64(z + Float64(x * y)));
	elseif (y <= 5.8e+20)
		tmp = t_1;
	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 <= -2e+154)
		tmp = t_0;
	elseif (y <= -1.7)
		tmp = t_1;
	elseif (y <= -0.0075)
		tmp = t_0;
	elseif (y <= 0.086)
		tmp = (-0.5 * (z * (y * y))) + (z + (x * y));
	elseif (y <= 5.8e+20)
		tmp = t_1;
	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, -2e+154], t$95$0, If[LessEqual[y, -1.7], t$95$1, If[LessEqual[y, -0.0075], t$95$0, If[LessEqual[y, 0.086], N[(N[(-0.5 * N[(z * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(z + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.8e+20], t$95$1, t$95$0]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.00000000000000007e154 or -1.69999999999999996 < y < -0.0074999999999999997 or 5.8e20 < y

   1. Initial program 99.5%

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

   2. Taylor expanded in x around 0 99.5%

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

      1. fma-def 99.5%

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

   4. Simplified 99.5%

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

   5. Taylor expanded in z around inf 64.4%

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

   if -2.00000000000000007e154 < y < -1.69999999999999996 or 0.085999999999999993 < y < 5.8e20

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 99.8%

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

      1. fma-def 99.8%

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

   4. Simplified 99.8%

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

   5. Taylor expanded in z around 0 68.2%

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

   if -0.0074999999999999997 < y < 0.085999999999999993

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 100.0%

      \[\leadsto \color{blue}{-0.5 \cdot \left({y}^{2} \cdot z\right) + \left(y \cdot x + z\right)} \]
   3. Step-by-step derivation

      1. expm1-log1p-u 95.7%

         \[\leadsto -0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({y}^{2} \cdot z\right)\right)} + \left(y \cdot x + z\right) \]

      2. expm1-udef 95.7%

         \[\leadsto -0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({y}^{2} \cdot z\right)} - 1\right)} + \left(y \cdot x + z\right) \]

      3. unpow2 95.7%

         \[\leadsto -0.5 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\left(y \cdot y\right)} \cdot z\right)} - 1\right) + \left(y \cdot x + z\right) \]

      4. associate-*l* 95.7%

         \[\leadsto -0.5 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{y \cdot \left(y \cdot z\right)}\right)} - 1\right) + \left(y \cdot x + z\right) \]

   4. Applied egg-rr 95.7%

      \[\leadsto -0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(y \cdot \left(y \cdot z\right)\right)} - 1\right)} + \left(y \cdot x + z\right) \]
   5. Step-by-step derivation

      1. expm1-def 95.7%

         \[\leadsto -0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(y \cdot \left(y \cdot z\right)\right)\right)} + \left(y \cdot x + z\right) \]

      2. expm1-log1p 100.0%

         \[\leadsto -0.5 \cdot \color{blue}{\left(y \cdot \left(y \cdot z\right)\right)} + \left(y \cdot x + z\right) \]

      3. *-commutative 100.0%

         \[\leadsto -0.5 \cdot \color{blue}{\left(\left(y \cdot z\right) \cdot y\right)} + \left(y \cdot x + z\right) \]

      4. *-commutative 100.0%

         \[\leadsto -0.5 \cdot \left(\color{blue}{\left(z \cdot y\right)} \cdot y\right) + \left(y \cdot x + z\right) \]

      5. associate-*r* 100.0%

         \[\leadsto -0.5 \cdot \color{blue}{\left(z \cdot \left(y \cdot y\right)\right)} + \left(y \cdot x + z\right) \]

   6. Simplified 100.0%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{+154}:\\ \;\;\;\;z \cdot \cos y\\ \mathbf{elif}\;y \leq -1.7:\\ \;\;\;\;x \cdot \sin y\\ \mathbf{elif}\;y \leq -0.0075:\\ \;\;\;\;z \cdot \cos y\\ \mathbf{elif}\;y \leq 0.086:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \left(y \cdot y\right)\right) + \left(z + x \cdot y\right)\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{+20}:\\ \;\;\;\;x \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \cos y\\ \end{array} \]

Alternative 4: 85.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{+126} \lor \neg \left(z \leq 2.1 \cdot 10^{+36}\right):\\ \;\;\;\;z \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;z + x \cdot \sin y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1e+126) (not (<= z 2.1e+36)))
   (* z (cos y))
   (+ z (* x (sin y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1e+126) || !(z <= 2.1e+36)) {
		tmp = z * cos(y);
	} else {
		tmp = z + (x * sin(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 ((z <= (-1d+126)) .or. (.not. (z <= 2.1d+36))) then
        tmp = z * cos(y)
    else
        tmp = z + (x * sin(y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1e+126) || !(z <= 2.1e+36)) {
		tmp = z * Math.cos(y);
	} else {
		tmp = z + (x * Math.sin(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1e+126) or not (z <= 2.1e+36):
		tmp = z * math.cos(y)
	else:
		tmp = z + (x * math.sin(y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1e+126) || !(z <= 2.1e+36))
		tmp = Float64(z * cos(y));
	else
		tmp = Float64(z + Float64(x * sin(y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1e+126) || ~((z <= 2.1e+36)))
		tmp = z * cos(y);
	else
		tmp = z + (x * sin(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1e+126], N[Not[LessEqual[z, 2.1e+36]], $MachinePrecision]], N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision], N[(z + N[(x * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -9.99999999999999925e125 or 2.10000000000000004e36 < z

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 99.8%

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

      1. fma-def 99.8%

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

   4. Simplified 99.8%

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

   5. Taylor expanded in z around inf 91.1%

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

   if -9.99999999999999925e125 < z < 2.10000000000000004e36

   1. Initial program 99.8%

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

   2. Taylor expanded in y around 0 90.4%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{+126} \lor \neg \left(z \leq 2.1 \cdot 10^{+36}\right):\\ \;\;\;\;z \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;z + x \cdot \sin y\\ \end{array} \]

Alternative 5: 74.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.0071 \lor \neg \left(y \leq 4.1 \cdot 10^{+17}\right):\\ \;\;\;\;z \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \left(y \cdot y\right)\right) + \left(z + x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -0.0071) (not (<= y 4.1e+17)))
   (* z (cos y))
   (+ (* -0.5 (* z (* y y))) (+ z (* x y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.0071) || !(y <= 4.1e+17)) {
		tmp = z * cos(y);
	} else {
		tmp = (-0.5 * (z * (y * y))) + (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 <= (-0.0071d0)) .or. (.not. (y <= 4.1d+17))) then
        tmp = z * cos(y)
    else
        tmp = ((-0.5d0) * (z * (y * y))) + (z + (x * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.0071) || !(y <= 4.1e+17)) {
		tmp = z * Math.cos(y);
	} else {
		tmp = (-0.5 * (z * (y * y))) + (z + (x * y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -0.0071) or not (y <= 4.1e+17):
		tmp = z * math.cos(y)
	else:
		tmp = (-0.5 * (z * (y * y))) + (z + (x * y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -0.0071) || !(y <= 4.1e+17))
		tmp = Float64(z * cos(y));
	else
		tmp = Float64(Float64(-0.5 * Float64(z * Float64(y * y))) + Float64(z + Float64(x * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -0.0071) || ~((y <= 4.1e+17)))
		tmp = z * cos(y);
	else
		tmp = (-0.5 * (z * (y * y))) + (z + (x * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -0.0071], N[Not[LessEqual[y, 4.1e+17]], $MachinePrecision]], N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * N[(z * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(z + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -0.0071000000000000004 or 4.1e17 < y

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0 99.6%

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

      1. fma-def 99.6%

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

   4. Simplified 99.6%

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

   5. Taylor expanded in z around inf 59.3%

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

   if -0.0071000000000000004 < y < 4.1e17

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 96.7%

      \[\leadsto \color{blue}{-0.5 \cdot \left({y}^{2} \cdot z\right) + \left(y \cdot x + z\right)} \]
   3. Step-by-step derivation

      1. expm1-log1p-u 92.6%

         \[\leadsto -0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({y}^{2} \cdot z\right)\right)} + \left(y \cdot x + z\right) \]

      2. expm1-udef 92.6%

         \[\leadsto -0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({y}^{2} \cdot z\right)} - 1\right)} + \left(y \cdot x + z\right) \]

      3. unpow2 92.6%

         \[\leadsto -0.5 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\left(y \cdot y\right)} \cdot z\right)} - 1\right) + \left(y \cdot x + z\right) \]

      4. associate-*l* 92.6%

         \[\leadsto -0.5 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{y \cdot \left(y \cdot z\right)}\right)} - 1\right) + \left(y \cdot x + z\right) \]

   4. Applied egg-rr 92.6%

      \[\leadsto -0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(y \cdot \left(y \cdot z\right)\right)} - 1\right)} + \left(y \cdot x + z\right) \]
   5. Step-by-step derivation

      1. expm1-def 92.6%

         \[\leadsto -0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(y \cdot \left(y \cdot z\right)\right)\right)} + \left(y \cdot x + z\right) \]

      2. expm1-log1p 96.7%

         \[\leadsto -0.5 \cdot \color{blue}{\left(y \cdot \left(y \cdot z\right)\right)} + \left(y \cdot x + z\right) \]

      3. *-commutative 96.7%

         \[\leadsto -0.5 \cdot \color{blue}{\left(\left(y \cdot z\right) \cdot y\right)} + \left(y \cdot x + z\right) \]

      4. *-commutative 96.7%

         \[\leadsto -0.5 \cdot \left(\color{blue}{\left(z \cdot y\right)} \cdot y\right) + \left(y \cdot x + z\right) \]

      5. associate-*r* 96.7%

         \[\leadsto -0.5 \cdot \color{blue}{\left(z \cdot \left(y \cdot y\right)\right)} + \left(y \cdot x + z\right) \]

   6. Simplified 96.7%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.0071 \lor \neg \left(y \leq 4.1 \cdot 10^{+17}\right):\\ \;\;\;\;z \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \left(y \cdot y\right)\right) + \left(z + x \cdot y\right)\\ \end{array} \]

Alternative 6: 40.8% accurate, 29.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.8 \cdot 10^{-53}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq 1.04 \cdot 10^{-100}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -4.8e-53) z (if (<= z 1.04e-100) (* x y) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -4.8e-53) {
		tmp = z;
	} else if (z <= 1.04e-100) {
		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 (z <= (-4.8d-53)) then
        tmp = z
    else if (z <= 1.04d-100) 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 (z <= -4.8e-53) {
		tmp = z;
	} else if (z <= 1.04e-100) {
		tmp = x * y;
	} else {
		tmp = z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -4.8e-53:
		tmp = z
	elif z <= 1.04e-100:
		tmp = x * y
	else:
		tmp = z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -4.8e-53)
		tmp = z;
	elseif (z <= 1.04e-100)
		tmp = Float64(x * y);
	else
		tmp = z;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -4.8e-53)
		tmp = z;
	elseif (z <= 1.04e-100)
		tmp = x * y;
	else
		tmp = z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -4.8e-53], z, If[LessEqual[z, 1.04e-100], N[(x * y), $MachinePrecision], z]]

Derivation

1. Split input into 2 regimes

2. if z < -4.80000000000000015e-53 or 1.04e-100 < z

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 99.7%

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

      1. fma-def 99.8%

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

   4. Simplified 99.8%

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

   5. Taylor expanded in y around 0 47.3%

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

   if -4.80000000000000015e-53 < z < 1.04e-100

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 99.9%

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

      1. fma-def 99.9%

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

   4. Simplified 99.9%

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

   5. Taylor expanded in z around 0 72.5%

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

   6. Taylor expanded in y around 0 42.6%

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

4. Final simplification 45.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.8 \cdot 10^{-53}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq 1.04 \cdot 10^{-100}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]

Alternative 7: 52.7% 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 57.0%

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

3. Final simplification 57.0%

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

Alternative 8: 38.7% 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. Taylor expanded in x around 0 99.8%

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

   1. fma-def 99.8%

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

4. Simplified 99.8%

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

5. Taylor expanded in y around 0 39.1%

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

6. Final simplification 39.1%

   \[\leadsto z \]

Reproduce

herbie shell --seed 2023224 
(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))))