Diagrams.ThreeD.Transform:aboutY from diagrams-lib-1.3.0.3

Percentage Accurate: 99.8% → 99.8%

Time: 7.4s

Alternatives: 8

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

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_] := N[(N[Sin[y], $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. Step-by-step derivation

   1. +-commutative 99.8%

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

   2. *-commutative 99.8%

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

   3. fma-def 99.8%

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

3. Applied egg-rr 99.8%

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

4. Final simplification 99.8%

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

Alternative 2: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

2. Final simplification 99.8%

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

Alternative 3: 73.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin y \cdot z\\ t_1 := x \cdot \cos y\\ \mathbf{if}\;y \leq -1.02 \cdot 10^{+239}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -6 \cdot 10^{+200}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -0.45:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{+15}:\\ \;\;\;\;x \cdot \left(1 + -0.5 \cdot \left(y \cdot y\right)\right) + y \cdot z\\ \mathbf{elif}\;y \leq 3.9 \cdot 10^{+159} \lor \neg \left(y \leq 1.35 \cdot 10^{+264}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (sin y) z)) (t_1 (* x (cos y))))
   (if (<= y -1.02e+239)
     t_0
     (if (<= y -6e+200)
       t_1
       (if (<= y -0.45)
         t_0
         (if (<= y 3.1e+15)
           (+ (* x (+ 1.0 (* -0.5 (* y y)))) (* y z))
           (if (or (<= y 3.9e+159) (not (<= y 1.35e+264))) t_1 t_0)))))))

C

double code(double x, double y, double z) {
	double t_0 = sin(y) * z;
	double t_1 = x * cos(y);
	double tmp;
	if (y <= -1.02e+239) {
		tmp = t_0;
	} else if (y <= -6e+200) {
		tmp = t_1;
	} else if (y <= -0.45) {
		tmp = t_0;
	} else if (y <= 3.1e+15) {
		tmp = (x * (1.0 + (-0.5 * (y * y)))) + (y * z);
	} else if ((y <= 3.9e+159) || !(y <= 1.35e+264)) {
		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 = sin(y) * z
    t_1 = x * cos(y)
    if (y <= (-1.02d+239)) then
        tmp = t_0
    else if (y <= (-6d+200)) then
        tmp = t_1
    else if (y <= (-0.45d0)) then
        tmp = t_0
    else if (y <= 3.1d+15) then
        tmp = (x * (1.0d0 + ((-0.5d0) * (y * y)))) + (y * z)
    else if ((y <= 3.9d+159) .or. (.not. (y <= 1.35d+264))) 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 = Math.sin(y) * z;
	double t_1 = x * Math.cos(y);
	double tmp;
	if (y <= -1.02e+239) {
		tmp = t_0;
	} else if (y <= -6e+200) {
		tmp = t_1;
	} else if (y <= -0.45) {
		tmp = t_0;
	} else if (y <= 3.1e+15) {
		tmp = (x * (1.0 + (-0.5 * (y * y)))) + (y * z);
	} else if ((y <= 3.9e+159) || !(y <= 1.35e+264)) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.sin(y) * z
	t_1 = x * math.cos(y)
	tmp = 0
	if y <= -1.02e+239:
		tmp = t_0
	elif y <= -6e+200:
		tmp = t_1
	elif y <= -0.45:
		tmp = t_0
	elif y <= 3.1e+15:
		tmp = (x * (1.0 + (-0.5 * (y * y)))) + (y * z)
	elif (y <= 3.9e+159) or not (y <= 1.35e+264):
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(sin(y) * z)
	t_1 = Float64(x * cos(y))
	tmp = 0.0
	if (y <= -1.02e+239)
		tmp = t_0;
	elseif (y <= -6e+200)
		tmp = t_1;
	elseif (y <= -0.45)
		tmp = t_0;
	elseif (y <= 3.1e+15)
		tmp = Float64(Float64(x * Float64(1.0 + Float64(-0.5 * Float64(y * y)))) + Float64(y * z));
	elseif ((y <= 3.9e+159) || !(y <= 1.35e+264))
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = sin(y) * z;
	t_1 = x * cos(y);
	tmp = 0.0;
	if (y <= -1.02e+239)
		tmp = t_0;
	elseif (y <= -6e+200)
		tmp = t_1;
	elseif (y <= -0.45)
		tmp = t_0;
	elseif (y <= 3.1e+15)
		tmp = (x * (1.0 + (-0.5 * (y * y)))) + (y * z);
	elseif ((y <= 3.9e+159) || ~((y <= 1.35e+264)))
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] * z), $MachinePrecision]}, Block[{t$95$1 = N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.02e+239], t$95$0, If[LessEqual[y, -6e+200], t$95$1, If[LessEqual[y, -0.45], t$95$0, If[LessEqual[y, 3.1e+15], N[(N[(x * N[(1.0 + N[(-0.5 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * z), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[y, 3.9e+159], N[Not[LessEqual[y, 1.35e+264]], $MachinePrecision]], t$95$1, t$95$0]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.02e239 or -5.99999999999999982e200 < y < -0.450000000000000011 or 3.9000000000000001e159 < y < 1.3500000000000001e264

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0 65.1%

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

   if -1.02e239 < y < -5.99999999999999982e200 or 3.1e15 < y < 3.9000000000000001e159 or 1.3500000000000001e264 < y

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. *-commutative 99.7%

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

      3. fma-def 99.7%

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

   3. Applied egg-rr 99.7%

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

   4. Taylor expanded in z around 0 66.8%

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

   if -0.450000000000000011 < y < 3.1e15

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 99.5%

      \[\leadsto x \cdot \color{blue}{\left(1 + -0.5 \cdot {y}^{2}\right)} + z \cdot \sin y \]
   3. Step-by-step derivation

      1. unpow2 99.5%

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

   4. Simplified 99.5%

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

   5. Taylor expanded in y around 0 98.6%

      \[\leadsto x \cdot \left(1 + -0.5 \cdot \left(y \cdot y\right)\right) + \color{blue}{y \cdot z} \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.02 \cdot 10^{+239}:\\ \;\;\;\;\sin y \cdot z\\ \mathbf{elif}\;y \leq -6 \cdot 10^{+200}:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{elif}\;y \leq -0.45:\\ \;\;\;\;\sin y \cdot z\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{+15}:\\ \;\;\;\;x \cdot \left(1 + -0.5 \cdot \left(y \cdot y\right)\right) + y \cdot z\\ \mathbf{elif}\;y \leq 3.9 \cdot 10^{+159} \lor \neg \left(y \leq 1.35 \cdot 10^{+264}\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot z\\ \end{array} \]

Alternative 4: 85.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{-8} \lor \neg \left(x \leq 8.2 \cdot 10^{+92}\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x + \sin y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -5.2e-8) (not (<= x 8.2e+92)))
   (* x (cos y))
   (+ x (* (sin y) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -5.2e-8) || !(x <= 8.2e+92)) {
		tmp = x * cos(y);
	} else {
		tmp = x + (sin(y) * 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 <= (-5.2d-8)) .or. (.not. (x <= 8.2d+92))) then
        tmp = x * cos(y)
    else
        tmp = x + (sin(y) * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -5.2e-8) || !(x <= 8.2e+92)) {
		tmp = x * Math.cos(y);
	} else {
		tmp = x + (Math.sin(y) * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -5.2e-8) or not (x <= 8.2e+92):
		tmp = x * math.cos(y)
	else:
		tmp = x + (math.sin(y) * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -5.2e-8) || !(x <= 8.2e+92))
		tmp = Float64(x * cos(y));
	else
		tmp = Float64(x + Float64(sin(y) * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -5.2e-8) || ~((x <= 8.2e+92)))
		tmp = x * cos(y);
	else
		tmp = x + (sin(y) * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -5.2e-8], N[Not[LessEqual[x, 8.2e+92]], $MachinePrecision]], N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision], N[(x + N[(N[Sin[y], $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.2000000000000002e-8 or 8.20000000000000047e92 < x

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

      2. *-commutative 99.8%

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

      3. fma-def 99.8%

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

   3. Applied egg-rr 99.8%

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

   4. Taylor expanded in z around 0 89.9%

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

   if -5.2000000000000002e-8 < x < 8.20000000000000047e92

   1. Initial program 99.8%

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

   2. Taylor expanded in y around 0 89.0%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{-8} \lor \neg \left(x \leq 8.2 \cdot 10^{+92}\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x + \sin y \cdot z\\ \end{array} \]

Alternative 5: 74.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.16 \lor \neg \left(y \leq 0.011\right):\\ \;\;\;\;\sin y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + -0.5 \cdot \left(y \cdot y\right)\right) + y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -0.16) (not (<= y 0.011)))
   (* (sin y) z)
   (+ (* x (+ 1.0 (* -0.5 (* y y)))) (* y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.16) || !(y <= 0.011)) {
		tmp = sin(y) * z;
	} else {
		tmp = (x * (1.0 + (-0.5 * (y * y)))) + (y * 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 ((y <= (-0.16d0)) .or. (.not. (y <= 0.011d0))) then
        tmp = sin(y) * z
    else
        tmp = (x * (1.0d0 + ((-0.5d0) * (y * y)))) + (y * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.16) || !(y <= 0.011)) {
		tmp = Math.sin(y) * z;
	} else {
		tmp = (x * (1.0 + (-0.5 * (y * y)))) + (y * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -0.16) or not (y <= 0.011):
		tmp = math.sin(y) * z
	else:
		tmp = (x * (1.0 + (-0.5 * (y * y)))) + (y * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -0.16) || !(y <= 0.011))
		tmp = Float64(sin(y) * z);
	else
		tmp = Float64(Float64(x * Float64(1.0 + Float64(-0.5 * Float64(y * y)))) + Float64(y * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -0.16) || ~((y <= 0.011)))
		tmp = sin(y) * z;
	else
		tmp = (x * (1.0 + (-0.5 * (y * y)))) + (y * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -0.16], N[Not[LessEqual[y, 0.011]], $MachinePrecision]], N[(N[Sin[y], $MachinePrecision] * z), $MachinePrecision], N[(N[(x * N[(1.0 + N[(-0.5 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -0.160000000000000003 or 0.010999999999999999 < y

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0 51.9%

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

   if -0.160000000000000003 < y < 0.010999999999999999

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 99.6%

      \[\leadsto x \cdot \color{blue}{\left(1 + -0.5 \cdot {y}^{2}\right)} + z \cdot \sin y \]
   3. Step-by-step derivation

      1. unpow2 99.6%

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

   4. Simplified 99.6%

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

   5. Taylor expanded in y around 0 99.3%

      \[\leadsto x \cdot \left(1 + -0.5 \cdot \left(y \cdot y\right)\right) + \color{blue}{y \cdot z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.16 \lor \neg \left(y \leq 0.011\right):\\ \;\;\;\;\sin y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + -0.5 \cdot \left(y \cdot y\right)\right) + y \cdot z\\ \end{array} \]

Alternative 6: 42.0% accurate, 29.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3 \cdot 10^{+143}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{+93}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -3e+143) (* y z) (if (<= z 4.2e+93) x (* y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -3e+143) {
		tmp = y * z;
	} else if (z <= 4.2e+93) {
		tmp = x;
	} else {
		tmp = y * 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 <= (-3d+143)) then
        tmp = y * z
    else if (z <= 4.2d+93) then
        tmp = x
    else
        tmp = y * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -3e+143) {
		tmp = y * z;
	} else if (z <= 4.2e+93) {
		tmp = x;
	} else {
		tmp = y * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -3e+143:
		tmp = y * z
	elif z <= 4.2e+93:
		tmp = x
	else:
		tmp = y * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -3e+143)
		tmp = Float64(y * z);
	elseif (z <= 4.2e+93)
		tmp = x;
	else
		tmp = Float64(y * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -3e+143)
		tmp = y * z;
	elseif (z <= 4.2e+93)
		tmp = x;
	else
		tmp = y * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -3e+143], N[(y * z), $MachinePrecision], If[LessEqual[z, 4.2e+93], x, N[(y * z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.0000000000000001e143 or 4.1999999999999996e93 < z

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 63.8%

      \[\leadsto x \cdot \color{blue}{\left(1 + -0.5 \cdot {y}^{2}\right)} + z \cdot \sin y \]
   3. Step-by-step derivation

      1. unpow2 63.8%

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

   4. Simplified 63.8%

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

   5. Taylor expanded in y around 0 48.0%

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

   6. Taylor expanded in x around 0 34.2%

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

   if -3.0000000000000001e143 < z < 4.1999999999999996e93

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

      2. *-commutative 99.8%

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

      3. fma-def 99.8%

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

   3. Applied egg-rr 99.8%

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

   4. Taylor expanded in y around 0 50.4%

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

4. Final simplification 46.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3 \cdot 10^{+143}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{+93}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \]

Alternative 7: 52.2% 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. Taylor expanded in y around 0 53.2%

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

3. Final simplification 53.2%

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

Alternative 8: 38.5% 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. Step-by-step derivation

   1. +-commutative 99.8%

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

   2. *-commutative 99.8%

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

   3. fma-def 99.8%

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

3. Applied egg-rr 99.8%

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

4. Taylor expanded in y around 0 41.5%

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

5. Final simplification 41.5%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023213 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Transform:aboutY from diagrams-lib-1.3.0.3"
  :precision binary64
  (+ (* x (cos y)) (* z (sin y))))