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

Percentage Accurate: 99.8% → 99.8%

Time: 7.0s

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: 75.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \cos y\\ t_1 := \sin y \cdot z\\ \mathbf{if}\;y \leq -1 \cdot 10^{+158}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -4.9 \cdot 10^{+61}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -16:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -1.05 \cdot 10^{-7} \lor \neg \left(y \leq 4.8 \cdot 10^{-6}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;y \cdot z + \left(x + -0.5 \cdot \left(x \cdot \left(y \cdot y\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (cos y))) (t_1 (* (sin y) z)))
   (if (<= y -1e+158)
     t_0
     (if (<= y -4.9e+61)
       t_1
       (if (<= y -16.0)
         t_0
         (if (or (<= y -1.05e-7) (not (<= y 4.8e-6)))
           t_1
           (+ (* y z) (+ x (* -0.5 (* x (* y y)))))))))))

C

double code(double x, double y, double z) {
	double t_0 = x * cos(y);
	double t_1 = sin(y) * z;
	double tmp;
	if (y <= -1e+158) {
		tmp = t_0;
	} else if (y <= -4.9e+61) {
		tmp = t_1;
	} else if (y <= -16.0) {
		tmp = t_0;
	} else if ((y <= -1.05e-7) || !(y <= 4.8e-6)) {
		tmp = t_1;
	} else {
		tmp = (y * z) + (x + (-0.5 * (x * (y * 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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x * cos(y)
    t_1 = sin(y) * z
    if (y <= (-1d+158)) then
        tmp = t_0
    else if (y <= (-4.9d+61)) then
        tmp = t_1
    else if (y <= (-16.0d0)) then
        tmp = t_0
    else if ((y <= (-1.05d-7)) .or. (.not. (y <= 4.8d-6))) then
        tmp = t_1
    else
        tmp = (y * z) + (x + ((-0.5d0) * (x * (y * y))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x * Math.cos(y);
	double t_1 = Math.sin(y) * z;
	double tmp;
	if (y <= -1e+158) {
		tmp = t_0;
	} else if (y <= -4.9e+61) {
		tmp = t_1;
	} else if (y <= -16.0) {
		tmp = t_0;
	} else if ((y <= -1.05e-7) || !(y <= 4.8e-6)) {
		tmp = t_1;
	} else {
		tmp = (y * z) + (x + (-0.5 * (x * (y * y))));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * math.cos(y)
	t_1 = math.sin(y) * z
	tmp = 0
	if y <= -1e+158:
		tmp = t_0
	elif y <= -4.9e+61:
		tmp = t_1
	elif y <= -16.0:
		tmp = t_0
	elif (y <= -1.05e-7) or not (y <= 4.8e-6):
		tmp = t_1
	else:
		tmp = (y * z) + (x + (-0.5 * (x * (y * y))))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * cos(y))
	t_1 = Float64(sin(y) * z)
	tmp = 0.0
	if (y <= -1e+158)
		tmp = t_0;
	elseif (y <= -4.9e+61)
		tmp = t_1;
	elseif (y <= -16.0)
		tmp = t_0;
	elseif ((y <= -1.05e-7) || !(y <= 4.8e-6))
		tmp = t_1;
	else
		tmp = Float64(Float64(y * z) + Float64(x + Float64(-0.5 * Float64(x * Float64(y * y)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * cos(y);
	t_1 = sin(y) * z;
	tmp = 0.0;
	if (y <= -1e+158)
		tmp = t_0;
	elseif (y <= -4.9e+61)
		tmp = t_1;
	elseif (y <= -16.0)
		tmp = t_0;
	elseif ((y <= -1.05e-7) || ~((y <= 4.8e-6)))
		tmp = t_1;
	else
		tmp = (y * z) + (x + (-0.5 * (x * (y * y))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[y], $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[y, -1e+158], t$95$0, If[LessEqual[y, -4.9e+61], t$95$1, If[LessEqual[y, -16.0], t$95$0, If[Or[LessEqual[y, -1.05e-7], N[Not[LessEqual[y, 4.8e-6]], $MachinePrecision]], t$95$1, N[(N[(y * z), $MachinePrecision] + N[(x + N[(-0.5 * N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -9.99999999999999953e157 or -4.90000000000000025e61 < y < -16

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      2. *-commutative 99.5%

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

      3. fma-def 99.6%

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

   3. Applied egg-rr 99.6%

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

   4. Taylor expanded in z around 0 67.6%

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

   if -9.99999999999999953e157 < y < -4.90000000000000025e61 or -16 < y < -1.05e-7 or 4.7999999999999998e-6 < y

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 65.5%

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

   if -1.05e-7 < y < 4.7999999999999998e-6

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 100.0%

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

      1. expm1-log1p-u 97.7%

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

      2. expm1-udef 97.7%

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

      3. unpow2 97.7%

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

      4. associate-*l* 97.7%

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

   4. Applied egg-rr 97.7%

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

      1. expm1-def 97.7%

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

      2. expm1-log1p 100.0%

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

      3. *-commutative 100.0%

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

      4. *-commutative 100.0%

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

      5. associate-*r* 100.0%

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

   6. Simplified 100.0%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{+158}:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{elif}\;y \leq -4.9 \cdot 10^{+61}:\\ \;\;\;\;\sin y \cdot z\\ \mathbf{elif}\;y \leq -16:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{elif}\;y \leq -1.05 \cdot 10^{-7} \lor \neg \left(y \leq 4.8 \cdot 10^{-6}\right):\\ \;\;\;\;\sin y \cdot z\\ \mathbf{else}:\\ \;\;\;\;y \cdot z + \left(x + -0.5 \cdot \left(x \cdot \left(y \cdot y\right)\right)\right)\\ \end{array} \]

Alternative 4: 85.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{+125} \lor \neg \left(x \leq 1.8 \cdot 10^{+70}\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x + \sin y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -4e+125) (not (<= x 1.8e+70)))
   (* x (cos y))
   (+ x (* (sin y) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -4e+125) || !(x <= 1.8e+70)) {
		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 <= (-4d+125)) .or. (.not. (x <= 1.8d+70))) 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 <= -4e+125) || !(x <= 1.8e+70)) {
		tmp = x * Math.cos(y);
	} else {
		tmp = x + (Math.sin(y) * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -4e+125) or not (x <= 1.8e+70):
		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 <= -4e+125) || !(x <= 1.8e+70))
		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 <= -4e+125) || ~((x <= 1.8e+70)))
		tmp = x * cos(y);
	else
		tmp = x + (sin(y) * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -4e+125], N[Not[LessEqual[x, 1.8e+70]], $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 < -3.9999999999999997e125 or 1.8e70 < 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 91.5%

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

   if -3.9999999999999997e125 < x < 1.8e70

   1. Initial program 99.8%

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

   2. Taylor expanded in y around 0 87.5%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{+125} \lor \neg \left(x \leq 1.8 \cdot 10^{+70}\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x + \sin y \cdot z\\ \end{array} \]

Alternative 5: 74.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{-7} \lor \neg \left(y \leq 4.8 \cdot 10^{-6}\right):\\ \;\;\;\;\sin y \cdot z\\ \mathbf{else}:\\ \;\;\;\;y \cdot z + \left(x + -0.5 \cdot \left(x \cdot \left(y \cdot y\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.05e-7) (not (<= y 4.8e-6)))
   (* (sin y) z)
   (+ (* y z) (+ x (* -0.5 (* x (* y y)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.05e-7) || !(y <= 4.8e-6)) {
		tmp = sin(y) * z;
	} else {
		tmp = (y * z) + (x + (-0.5 * (x * (y * 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 <= (-1.05d-7)) .or. (.not. (y <= 4.8d-6))) then
        tmp = sin(y) * z
    else
        tmp = (y * z) + (x + ((-0.5d0) * (x * (y * y))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.05e-7) || !(y <= 4.8e-6)) {
		tmp = Math.sin(y) * z;
	} else {
		tmp = (y * z) + (x + (-0.5 * (x * (y * y))));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -1.05e-7) or not (y <= 4.8e-6):
		tmp = math.sin(y) * z
	else:
		tmp = (y * z) + (x + (-0.5 * (x * (y * y))))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1.05e-7) || !(y <= 4.8e-6))
		tmp = Float64(sin(y) * z);
	else
		tmp = Float64(Float64(y * z) + Float64(x + Float64(-0.5 * Float64(x * Float64(y * y)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -1.05e-7) || ~((y <= 4.8e-6)))
		tmp = sin(y) * z;
	else
		tmp = (y * z) + (x + (-0.5 * (x * (y * y))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1.05e-7], N[Not[LessEqual[y, 4.8e-6]], $MachinePrecision]], N[(N[Sin[y], $MachinePrecision] * z), $MachinePrecision], N[(N[(y * z), $MachinePrecision] + N[(x + N[(-0.5 * N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.05e-7 or 4.7999999999999998e-6 < y

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0 54.0%

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

   if -1.05e-7 < y < 4.7999999999999998e-6

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 100.0%

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

      1. expm1-log1p-u 97.7%

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

      2. expm1-udef 97.7%

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

      3. unpow2 97.7%

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

      4. associate-*l* 97.7%

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

   4. Applied egg-rr 97.7%

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

      1. expm1-def 97.7%

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

      2. expm1-log1p 100.0%

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

      3. *-commutative 100.0%

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

      4. *-commutative 100.0%

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

      5. associate-*r* 100.0%

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

   6. Simplified 100.0%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{-7} \lor \neg \left(y \leq 4.8 \cdot 10^{-6}\right):\\ \;\;\;\;\sin y \cdot z\\ \mathbf{else}:\\ \;\;\;\;y \cdot z + \left(x + -0.5 \cdot \left(x \cdot \left(y \cdot y\right)\right)\right)\\ \end{array} \]

Alternative 6: 39.5% accurate, 22.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 5.8 \cdot 10^{+67}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{+147}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{+203}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 5.8e+67)
   x
   (if (<= z 2.9e+147) (* y z) (if (<= z 1.05e+203) x (* y z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 5.8e+67) {
		tmp = x;
	} else if (z <= 2.9e+147) {
		tmp = y * z;
	} else if (z <= 1.05e+203) {
		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 <= 5.8d+67) then
        tmp = x
    else if (z <= 2.9d+147) then
        tmp = y * z
    else if (z <= 1.05d+203) 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 <= 5.8e+67) {
		tmp = x;
	} else if (z <= 2.9e+147) {
		tmp = y * z;
	} else if (z <= 1.05e+203) {
		tmp = x;
	} else {
		tmp = y * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 5.8e+67:
		tmp = x
	elif z <= 2.9e+147:
		tmp = y * z
	elif z <= 1.05e+203:
		tmp = x
	else:
		tmp = y * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 5.8e+67)
		tmp = x;
	elseif (z <= 2.9e+147)
		tmp = Float64(y * z);
	elseif (z <= 1.05e+203)
		tmp = x;
	else
		tmp = Float64(y * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 5.8e+67)
		tmp = x;
	elseif (z <= 2.9e+147)
		tmp = y * z;
	elseif (z <= 1.05e+203)
		tmp = x;
	else
		tmp = y * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 5.8e+67], x, If[LessEqual[z, 2.9e+147], N[(y * z), $MachinePrecision], If[LessEqual[z, 1.05e+203], x, N[(y * z), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if z < 5.80000000000000047e67 or 2.8999999999999998e147 < z < 1.04999999999999992e203

   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 46.3%

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

   if 5.80000000000000047e67 < z < 2.8999999999999998e147 or 1.04999999999999992e203 < z

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 55.8%

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

   3. Taylor expanded in y around inf 50.2%

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

4. Final simplification 46.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 5.8 \cdot 10^{+67}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{+147}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{+203}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \]

Alternative 7: 52.7% 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 54.8%

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

3. Final simplification 54.8%

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

Alternative 8: 38.9% 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 41.4%

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

5. Final simplification 41.4%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023256 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Transform:aboutY from diagrams-lib-1.3.0.3"
  :precision binary64
  (+ (* x (cos y)) (* z (sin y))))