Linear.Quaternion:$csin from linear-1.19.1.3

Percentage Accurate: 100.0% → 100.0%

Time: 8.2s

Alternatives: 13

Speedup: 1.0×

• 49.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \cos x \cdot \frac{\sinh y}{y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* (cos x) (/ (sinh y) y)))

C

double code(double x, double y) {
	return cos(x) * (sinh(y) / y);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = cos(x) * (sinh(y) / y)
end function

Java

public static double code(double x, double y) {
	return Math.cos(x) * (Math.sinh(y) / y);
}

Python

def code(x, y):
	return math.cos(x) * (math.sinh(y) / y)

Julia

function code(x, y)
	return Float64(cos(x) * Float64(sinh(y) / y))
end

MATLAB

function tmp = code(x, y)
	tmp = cos(x) * (sinh(y) / y);
end

Wolfram

code[x_, y_] := N[(N[Cos[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \cos x \cdot \frac{\sinh y}{y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* (cos x) (/ (sinh y) y)))

C

double code(double x, double y) {
	return cos(x) * (sinh(y) / y);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = cos(x) * (sinh(y) / y)
end function

Java

public static double code(double x, double y) {
	return Math.cos(x) * (Math.sinh(y) / y);
}

Python

def code(x, y):
	return math.cos(x) * (math.sinh(y) / y)

Julia

function code(x, y)
	return Float64(cos(x) * Float64(sinh(y) / y))
end

MATLAB

function tmp = code(x, y)
	tmp = cos(x) * (sinh(y) / y);
end

Wolfram

code[x_, y_] := N[(N[Cos[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \cos x \cdot \frac{\sinh y}{y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* (cos x) (/ (sinh y) y)))

C

double code(double x, double y) {
	return cos(x) * (sinh(y) / y);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = cos(x) * (sinh(y) / y)
end function

Java

public static double code(double x, double y) {
	return Math.cos(x) * (Math.sinh(y) / y);
}

Python

def code(x, y):
	return math.cos(x) * (math.sinh(y) / y)

Julia

function code(x, y)
	return Float64(cos(x) * Float64(sinh(y) / y))
end

MATLAB

function tmp = code(x, y)
	tmp = cos(x) * (sinh(y) / y);
end

Wolfram

code[x_, y_] := N[(N[Cos[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\cos x \cdot \frac{\sinh y}{y} \]

2. Final simplification 100.0%

   \[\leadsto \cos x \cdot \frac{\sinh y}{y} \]

Alternative 2: 84.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y}{y}\\ t_1 := t_0 \cdot \left(1 + \left(x \cdot x\right) \cdot -0.5\right)\\ \mathbf{if}\;y \leq 90:\\ \;\;\;\;\cos x \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{+31}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 10^{+136}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+154}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\cos x \cdot \left(y \cdot 0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sinh y) y)) (t_1 (* t_0 (+ 1.0 (* (* x x) -0.5)))))
   (if (<= y 90.0)
     (* (cos x) (+ 1.0 (* 0.16666666666666666 (* y y))))
     (if (<= y 1.8e+31)
       t_1
       (if (<= y 1e+136)
         t_0
         (if (<= y 5e+154)
           t_1
           (* y (* (cos x) (* y 0.16666666666666666)))))))))

C

double code(double x, double y) {
	double t_0 = sinh(y) / y;
	double t_1 = t_0 * (1.0 + ((x * x) * -0.5));
	double tmp;
	if (y <= 90.0) {
		tmp = cos(x) * (1.0 + (0.16666666666666666 * (y * y)));
	} else if (y <= 1.8e+31) {
		tmp = t_1;
	} else if (y <= 1e+136) {
		tmp = t_0;
	} else if (y <= 5e+154) {
		tmp = t_1;
	} else {
		tmp = y * (cos(x) * (y * 0.16666666666666666));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sinh(y) / y
    t_1 = t_0 * (1.0d0 + ((x * x) * (-0.5d0)))
    if (y <= 90.0d0) then
        tmp = cos(x) * (1.0d0 + (0.16666666666666666d0 * (y * y)))
    else if (y <= 1.8d+31) then
        tmp = t_1
    else if (y <= 1d+136) then
        tmp = t_0
    else if (y <= 5d+154) then
        tmp = t_1
    else
        tmp = y * (cos(x) * (y * 0.16666666666666666d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sinh(y) / y;
	double t_1 = t_0 * (1.0 + ((x * x) * -0.5));
	double tmp;
	if (y <= 90.0) {
		tmp = Math.cos(x) * (1.0 + (0.16666666666666666 * (y * y)));
	} else if (y <= 1.8e+31) {
		tmp = t_1;
	} else if (y <= 1e+136) {
		tmp = t_0;
	} else if (y <= 5e+154) {
		tmp = t_1;
	} else {
		tmp = y * (Math.cos(x) * (y * 0.16666666666666666));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sinh(y) / y
	t_1 = t_0 * (1.0 + ((x * x) * -0.5))
	tmp = 0
	if y <= 90.0:
		tmp = math.cos(x) * (1.0 + (0.16666666666666666 * (y * y)))
	elif y <= 1.8e+31:
		tmp = t_1
	elif y <= 1e+136:
		tmp = t_0
	elif y <= 5e+154:
		tmp = t_1
	else:
		tmp = y * (math.cos(x) * (y * 0.16666666666666666))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sinh(y) / y)
	t_1 = Float64(t_0 * Float64(1.0 + Float64(Float64(x * x) * -0.5)))
	tmp = 0.0
	if (y <= 90.0)
		tmp = Float64(cos(x) * Float64(1.0 + Float64(0.16666666666666666 * Float64(y * y))));
	elseif (y <= 1.8e+31)
		tmp = t_1;
	elseif (y <= 1e+136)
		tmp = t_0;
	elseif (y <= 5e+154)
		tmp = t_1;
	else
		tmp = Float64(y * Float64(cos(x) * Float64(y * 0.16666666666666666)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sinh(y) / y;
	t_1 = t_0 * (1.0 + ((x * x) * -0.5));
	tmp = 0.0;
	if (y <= 90.0)
		tmp = cos(x) * (1.0 + (0.16666666666666666 * (y * y)));
	elseif (y <= 1.8e+31)
		tmp = t_1;
	elseif (y <= 1e+136)
		tmp = t_0;
	elseif (y <= 5e+154)
		tmp = t_1;
	else
		tmp = y * (cos(x) * (y * 0.16666666666666666));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(1.0 + N[(N[(x * x), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 90.0], N[(N[Cos[x], $MachinePrecision] * N[(1.0 + N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.8e+31], t$95$1, If[LessEqual[y, 1e+136], t$95$0, If[LessEqual[y, 5e+154], t$95$1, N[(y * N[(N[Cos[x], $MachinePrecision] * N[(y * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < 90

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]

   2. Taylor expanded in y around 0 84.5%

      \[\leadsto \cos x \cdot \color{blue}{\left(1 + 0.16666666666666666 \cdot {y}^{2}\right)} \]
   3. Step-by-step derivation

      1. unpow2 51.1%

         \[\leadsto 1 + 0.16666666666666666 \cdot \color{blue}{\left(y \cdot y\right)} \]

   4. Simplified 84.5%

      \[\leadsto \cos x \cdot \color{blue}{\left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)} \]

   if 90 < y < 1.79999999999999998e31 or 1.00000000000000006e136 < y < 5.00000000000000004e154

   1. Initial program 99.9%

      \[\cos x \cdot \frac{\sinh y}{y} \]

   2. Taylor expanded in x around 0 92.2%

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

      1. *-commutative 18.6%

         \[\leadsto 1 + \color{blue}{{x}^{2} \cdot -0.5} \]

      2. unpow2 18.6%

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

   4. Simplified 92.2%

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

   if 1.79999999999999998e31 < y < 1.00000000000000006e136

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Step-by-step derivation

      1. add-log-exp 100.0%

         \[\leadsto \color{blue}{\log \left(e^{\cos x \cdot \frac{\sinh y}{y}}\right)} \]

      2. *-un-lft-identity 100.0%

         \[\leadsto \log \color{blue}{\left(1 \cdot e^{\cos x \cdot \frac{\sinh y}{y}}\right)} \]

      3. log-prod 100.0%

         \[\leadsto \color{blue}{\log 1 + \log \left(e^{\cos x \cdot \frac{\sinh y}{y}}\right)} \]

      4. metadata-eval 100.0%

         \[\leadsto \color{blue}{0} + \log \left(e^{\cos x \cdot \frac{\sinh y}{y}}\right) \]

      5. add-log-exp 100.0%

         \[\leadsto 0 + \color{blue}{\cos x \cdot \frac{\sinh y}{y}} \]

   3. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{0 + \cos x \cdot \frac{\sinh y}{y}} \]
   4. Step-by-step derivation

      1. +-lft-identity 100.0%

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

      2. associate-*r/ 100.0%

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

      3. associate-*l/ 100.0%

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

      4. *-commutative 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 82.6%

      \[\leadsto \sinh y \cdot \color{blue}{\frac{1}{y}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 82.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sinh y \cdot \frac{1}{y}\right)\right)} \]

      2. expm1-udef 82.6%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sinh y \cdot \frac{1}{y}\right)} - 1} \]

      3. un-div-inv 82.6%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\sinh y}{y}}\right)} - 1 \]

   8. Applied egg-rr 82.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\sinh y}{y}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 82.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sinh y}{y}\right)\right)} \]

      2. expm1-log1p 82.6%

         \[\leadsto \color{blue}{\frac{\sinh y}{y}} \]

   10. Simplified 82.6%

       \[\leadsto \color{blue}{\frac{\sinh y}{y}} \]

   if 5.00000000000000004e154 < y

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]

   2. Taylor expanded in y around 0 100.0%

      \[\leadsto \cos x \cdot \color{blue}{\left(1 + 0.16666666666666666 \cdot {y}^{2}\right)} \]
   3. Step-by-step derivation

      1. unpow2 64.0%

         \[\leadsto 1 + 0.16666666666666666 \cdot \color{blue}{\left(y \cdot y\right)} \]

   4. Simplified 100.0%

      \[\leadsto \cos x \cdot \color{blue}{\left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)} \]

   5. Taylor expanded in y around inf 100.0%

      \[\leadsto \color{blue}{0.16666666666666666 \cdot \left(\cos x \cdot {y}^{2}\right)} \]
   6. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \color{blue}{\left(\cos x \cdot {y}^{2}\right) \cdot 0.16666666666666666} \]

      2. unpow2 100.0%

         \[\leadsto \left(\cos x \cdot \color{blue}{\left(y \cdot y\right)}\right) \cdot 0.16666666666666666 \]

      3. associate-*r* 100.0%

         \[\leadsto \color{blue}{\left(\left(\cos x \cdot y\right) \cdot y\right)} \cdot 0.16666666666666666 \]

      4. *-commutative 100.0%

         \[\leadsto \left(\color{blue}{\left(y \cdot \cos x\right)} \cdot y\right) \cdot 0.16666666666666666 \]

      5. associate-*r* 100.0%

         \[\leadsto \color{blue}{\left(y \cdot \cos x\right) \cdot \left(y \cdot 0.16666666666666666\right)} \]

      6. associate-*l* 100.0%

         \[\leadsto \color{blue}{y \cdot \left(\cos x \cdot \left(y \cdot 0.16666666666666666\right)\right)} \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{y \cdot \left(\cos x \cdot \left(y \cdot 0.16666666666666666\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 90:\\ \;\;\;\;\cos x \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{+31}:\\ \;\;\;\;\frac{\sinh y}{y} \cdot \left(1 + \left(x \cdot x\right) \cdot -0.5\right)\\ \mathbf{elif}\;y \leq 10^{+136}:\\ \;\;\;\;\frac{\sinh y}{y}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+154}:\\ \;\;\;\;\frac{\sinh y}{y} \cdot \left(1 + \left(x \cdot x\right) \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\cos x \cdot \left(y \cdot 0.16666666666666666\right)\right)\\ \end{array} \]

Alternative 3: 72.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 90:\\ \;\;\;\;\cos x\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{+154}:\\ \;\;\;\;\sinh y \cdot \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\cos x \cdot \left(y \cdot 0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 90.0)
   (cos x)
   (if (<= y 3.8e+154)
     (* (sinh y) (/ 1.0 y))
     (* y (* (cos x) (* y 0.16666666666666666))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 90.0) {
		tmp = cos(x);
	} else if (y <= 3.8e+154) {
		tmp = sinh(y) * (1.0 / y);
	} else {
		tmp = y * (cos(x) * (y * 0.16666666666666666));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 90.0d0) then
        tmp = cos(x)
    else if (y <= 3.8d+154) then
        tmp = sinh(y) * (1.0d0 / y)
    else
        tmp = y * (cos(x) * (y * 0.16666666666666666d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 90.0) {
		tmp = Math.cos(x);
	} else if (y <= 3.8e+154) {
		tmp = Math.sinh(y) * (1.0 / y);
	} else {
		tmp = y * (Math.cos(x) * (y * 0.16666666666666666));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 90.0:
		tmp = math.cos(x)
	elif y <= 3.8e+154:
		tmp = math.sinh(y) * (1.0 / y)
	else:
		tmp = y * (math.cos(x) * (y * 0.16666666666666666))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 90.0)
		tmp = cos(x);
	elseif (y <= 3.8e+154)
		tmp = Float64(sinh(y) * Float64(1.0 / y));
	else
		tmp = Float64(y * Float64(cos(x) * Float64(y * 0.16666666666666666)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 90.0)
		tmp = cos(x);
	elseif (y <= 3.8e+154)
		tmp = sinh(y) * (1.0 / y);
	else
		tmp = y * (cos(x) * (y * 0.16666666666666666));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 90.0], N[Cos[x], $MachinePrecision], If[LessEqual[y, 3.8e+154], N[(N[Sinh[y], $MachinePrecision] * N[(1.0 / y), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[Cos[x], $MachinePrecision] * N[(y * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 90

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]

   2. Taylor expanded in y around 0 59.3%

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

   if 90 < y < 3.7999999999999998e154

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Step-by-step derivation

      1. add-log-exp 97.3%

         \[\leadsto \color{blue}{\log \left(e^{\cos x \cdot \frac{\sinh y}{y}}\right)} \]

      2. *-un-lft-identity 97.3%

         \[\leadsto \log \color{blue}{\left(1 \cdot e^{\cos x \cdot \frac{\sinh y}{y}}\right)} \]

      3. log-prod 97.3%

         \[\leadsto \color{blue}{\log 1 + \log \left(e^{\cos x \cdot \frac{\sinh y}{y}}\right)} \]

      4. metadata-eval 97.3%

         \[\leadsto \color{blue}{0} + \log \left(e^{\cos x \cdot \frac{\sinh y}{y}}\right) \]

      5. add-log-exp 100.0%

         \[\leadsto 0 + \color{blue}{\cos x \cdot \frac{\sinh y}{y}} \]

   3. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{0 + \cos x \cdot \frac{\sinh y}{y}} \]
   4. Step-by-step derivation

      1. +-lft-identity 100.0%

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

      2. associate-*r/ 100.0%

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

      3. associate-*l/ 100.0%

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

      4. *-commutative 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 72.2%

      \[\leadsto \sinh y \cdot \color{blue}{\frac{1}{y}} \]

   if 3.7999999999999998e154 < y

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]

   2. Taylor expanded in y around 0 100.0%

      \[\leadsto \cos x \cdot \color{blue}{\left(1 + 0.16666666666666666 \cdot {y}^{2}\right)} \]
   3. Step-by-step derivation

      1. unpow2 64.0%

         \[\leadsto 1 + 0.16666666666666666 \cdot \color{blue}{\left(y \cdot y\right)} \]

   4. Simplified 100.0%

      \[\leadsto \cos x \cdot \color{blue}{\left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)} \]

   5. Taylor expanded in y around inf 100.0%

      \[\leadsto \color{blue}{0.16666666666666666 \cdot \left(\cos x \cdot {y}^{2}\right)} \]
   6. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \color{blue}{\left(\cos x \cdot {y}^{2}\right) \cdot 0.16666666666666666} \]

      2. unpow2 100.0%

         \[\leadsto \left(\cos x \cdot \color{blue}{\left(y \cdot y\right)}\right) \cdot 0.16666666666666666 \]

      3. associate-*r* 100.0%

         \[\leadsto \color{blue}{\left(\left(\cos x \cdot y\right) \cdot y\right)} \cdot 0.16666666666666666 \]

      4. *-commutative 100.0%

         \[\leadsto \left(\color{blue}{\left(y \cdot \cos x\right)} \cdot y\right) \cdot 0.16666666666666666 \]

      5. associate-*r* 100.0%

         \[\leadsto \color{blue}{\left(y \cdot \cos x\right) \cdot \left(y \cdot 0.16666666666666666\right)} \]

      6. associate-*l* 100.0%

         \[\leadsto \color{blue}{y \cdot \left(\cos x \cdot \left(y \cdot 0.16666666666666666\right)\right)} \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{y \cdot \left(\cos x \cdot \left(y \cdot 0.16666666666666666\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 90:\\ \;\;\;\;\cos x\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{+154}:\\ \;\;\;\;\sinh y \cdot \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\cos x \cdot \left(y \cdot 0.16666666666666666\right)\right)\\ \end{array} \]

Alternative 4: 84.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 90:\\ \;\;\;\;\cos x \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{+154}:\\ \;\;\;\;\sinh y \cdot \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\cos x \cdot \left(y \cdot 0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 90.0)
   (* (cos x) (+ 1.0 (* 0.16666666666666666 (* y y))))
   (if (<= y 3.3e+154)
     (* (sinh y) (/ 1.0 y))
     (* y (* (cos x) (* y 0.16666666666666666))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 90.0) {
		tmp = cos(x) * (1.0 + (0.16666666666666666 * (y * y)));
	} else if (y <= 3.3e+154) {
		tmp = sinh(y) * (1.0 / y);
	} else {
		tmp = y * (cos(x) * (y * 0.16666666666666666));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 90.0d0) then
        tmp = cos(x) * (1.0d0 + (0.16666666666666666d0 * (y * y)))
    else if (y <= 3.3d+154) then
        tmp = sinh(y) * (1.0d0 / y)
    else
        tmp = y * (cos(x) * (y * 0.16666666666666666d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 90.0) {
		tmp = Math.cos(x) * (1.0 + (0.16666666666666666 * (y * y)));
	} else if (y <= 3.3e+154) {
		tmp = Math.sinh(y) * (1.0 / y);
	} else {
		tmp = y * (Math.cos(x) * (y * 0.16666666666666666));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 90.0:
		tmp = math.cos(x) * (1.0 + (0.16666666666666666 * (y * y)))
	elif y <= 3.3e+154:
		tmp = math.sinh(y) * (1.0 / y)
	else:
		tmp = y * (math.cos(x) * (y * 0.16666666666666666))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 90.0)
		tmp = Float64(cos(x) * Float64(1.0 + Float64(0.16666666666666666 * Float64(y * y))));
	elseif (y <= 3.3e+154)
		tmp = Float64(sinh(y) * Float64(1.0 / y));
	else
		tmp = Float64(y * Float64(cos(x) * Float64(y * 0.16666666666666666)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 90.0)
		tmp = cos(x) * (1.0 + (0.16666666666666666 * (y * y)));
	elseif (y <= 3.3e+154)
		tmp = sinh(y) * (1.0 / y);
	else
		tmp = y * (cos(x) * (y * 0.16666666666666666));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 90.0], N[(N[Cos[x], $MachinePrecision] * N[(1.0 + N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.3e+154], N[(N[Sinh[y], $MachinePrecision] * N[(1.0 / y), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[Cos[x], $MachinePrecision] * N[(y * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 90

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]

   2. Taylor expanded in y around 0 84.5%

      \[\leadsto \cos x \cdot \color{blue}{\left(1 + 0.16666666666666666 \cdot {y}^{2}\right)} \]
   3. Step-by-step derivation

      1. unpow2 51.1%

         \[\leadsto 1 + 0.16666666666666666 \cdot \color{blue}{\left(y \cdot y\right)} \]

   4. Simplified 84.5%

      \[\leadsto \cos x \cdot \color{blue}{\left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)} \]

   if 90 < y < 3.3e154

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Step-by-step derivation

      1. add-log-exp 97.3%

         \[\leadsto \color{blue}{\log \left(e^{\cos x \cdot \frac{\sinh y}{y}}\right)} \]

      2. *-un-lft-identity 97.3%

         \[\leadsto \log \color{blue}{\left(1 \cdot e^{\cos x \cdot \frac{\sinh y}{y}}\right)} \]

      3. log-prod 97.3%

         \[\leadsto \color{blue}{\log 1 + \log \left(e^{\cos x \cdot \frac{\sinh y}{y}}\right)} \]

      4. metadata-eval 97.3%

         \[\leadsto \color{blue}{0} + \log \left(e^{\cos x \cdot \frac{\sinh y}{y}}\right) \]

      5. add-log-exp 100.0%

         \[\leadsto 0 + \color{blue}{\cos x \cdot \frac{\sinh y}{y}} \]

   3. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{0 + \cos x \cdot \frac{\sinh y}{y}} \]
   4. Step-by-step derivation

      1. +-lft-identity 100.0%

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

      2. associate-*r/ 100.0%

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

      3. associate-*l/ 100.0%

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

      4. *-commutative 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 72.2%

      \[\leadsto \sinh y \cdot \color{blue}{\frac{1}{y}} \]

   if 3.3e154 < y

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]

   2. Taylor expanded in y around 0 100.0%

      \[\leadsto \cos x \cdot \color{blue}{\left(1 + 0.16666666666666666 \cdot {y}^{2}\right)} \]
   3. Step-by-step derivation

      1. unpow2 64.0%

         \[\leadsto 1 + 0.16666666666666666 \cdot \color{blue}{\left(y \cdot y\right)} \]

   4. Simplified 100.0%

      \[\leadsto \cos x \cdot \color{blue}{\left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)} \]

   5. Taylor expanded in y around inf 100.0%

      \[\leadsto \color{blue}{0.16666666666666666 \cdot \left(\cos x \cdot {y}^{2}\right)} \]
   6. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \color{blue}{\left(\cos x \cdot {y}^{2}\right) \cdot 0.16666666666666666} \]

      2. unpow2 100.0%

         \[\leadsto \left(\cos x \cdot \color{blue}{\left(y \cdot y\right)}\right) \cdot 0.16666666666666666 \]

      3. associate-*r* 100.0%

         \[\leadsto \color{blue}{\left(\left(\cos x \cdot y\right) \cdot y\right)} \cdot 0.16666666666666666 \]

      4. *-commutative 100.0%

         \[\leadsto \left(\color{blue}{\left(y \cdot \cos x\right)} \cdot y\right) \cdot 0.16666666666666666 \]

      5. associate-*r* 100.0%

         \[\leadsto \color{blue}{\left(y \cdot \cos x\right) \cdot \left(y \cdot 0.16666666666666666\right)} \]

      6. associate-*l* 100.0%

         \[\leadsto \color{blue}{y \cdot \left(\cos x \cdot \left(y \cdot 0.16666666666666666\right)\right)} \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{y \cdot \left(\cos x \cdot \left(y \cdot 0.16666666666666666\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 90:\\ \;\;\;\;\cos x \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{+154}:\\ \;\;\;\;\sinh y \cdot \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\cos x \cdot \left(y \cdot 0.16666666666666666\right)\right)\\ \end{array} \]

Alternative 5: 69.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 90:\\ \;\;\;\;\cos x\\ \mathbf{elif}\;y \leq 4.3 \cdot 10^{+166} \lor \neg \left(y \leq 2.25 \cdot 10^{+275}\right):\\ \;\;\;\;\frac{\sinh y}{y}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot \left(1 + \left(x \cdot x\right) \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 90.0)
   (cos x)
   (if (or (<= y 4.3e+166) (not (<= y 2.25e+275)))
     (/ (sinh y) y)
     (* (+ 1.0 (* 0.16666666666666666 (* y y))) (+ 1.0 (* (* x x) -0.5))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 90.0) {
		tmp = cos(x);
	} else if ((y <= 4.3e+166) || !(y <= 2.25e+275)) {
		tmp = sinh(y) / y;
	} else {
		tmp = (1.0 + (0.16666666666666666 * (y * y))) * (1.0 + ((x * x) * -0.5));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 90.0d0) then
        tmp = cos(x)
    else if ((y <= 4.3d+166) .or. (.not. (y <= 2.25d+275))) then
        tmp = sinh(y) / y
    else
        tmp = (1.0d0 + (0.16666666666666666d0 * (y * y))) * (1.0d0 + ((x * x) * (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 90.0) {
		tmp = Math.cos(x);
	} else if ((y <= 4.3e+166) || !(y <= 2.25e+275)) {
		tmp = Math.sinh(y) / y;
	} else {
		tmp = (1.0 + (0.16666666666666666 * (y * y))) * (1.0 + ((x * x) * -0.5));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 90.0:
		tmp = math.cos(x)
	elif (y <= 4.3e+166) or not (y <= 2.25e+275):
		tmp = math.sinh(y) / y
	else:
		tmp = (1.0 + (0.16666666666666666 * (y * y))) * (1.0 + ((x * x) * -0.5))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 90.0)
		tmp = cos(x);
	elseif ((y <= 4.3e+166) || !(y <= 2.25e+275))
		tmp = Float64(sinh(y) / y);
	else
		tmp = Float64(Float64(1.0 + Float64(0.16666666666666666 * Float64(y * y))) * Float64(1.0 + Float64(Float64(x * x) * -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 90.0)
		tmp = cos(x);
	elseif ((y <= 4.3e+166) || ~((y <= 2.25e+275)))
		tmp = sinh(y) / y;
	else
		tmp = (1.0 + (0.16666666666666666 * (y * y))) * (1.0 + ((x * x) * -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 90.0], N[Cos[x], $MachinePrecision], If[Or[LessEqual[y, 4.3e+166], N[Not[LessEqual[y, 2.25e+275]], $MachinePrecision]], N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision], N[(N[(1.0 + N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(x * x), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 90

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]

   2. Taylor expanded in y around 0 59.3%

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

   if 90 < y < 4.3e166 or 2.24999999999999987e275 < y

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Step-by-step derivation

      1. add-log-exp 97.8%

         \[\leadsto \color{blue}{\log \left(e^{\cos x \cdot \frac{\sinh y}{y}}\right)} \]

      2. *-un-lft-identity 97.8%

         \[\leadsto \log \color{blue}{\left(1 \cdot e^{\cos x \cdot \frac{\sinh y}{y}}\right)} \]

      3. log-prod 97.8%

         \[\leadsto \color{blue}{\log 1 + \log \left(e^{\cos x \cdot \frac{\sinh y}{y}}\right)} \]

      4. metadata-eval 97.8%

         \[\leadsto \color{blue}{0} + \log \left(e^{\cos x \cdot \frac{\sinh y}{y}}\right) \]

      5. add-log-exp 100.0%

         \[\leadsto 0 + \color{blue}{\cos x \cdot \frac{\sinh y}{y}} \]

   3. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{0 + \cos x \cdot \frac{\sinh y}{y}} \]
   4. Step-by-step derivation

      1. +-lft-identity 100.0%

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

      2. associate-*r/ 100.0%

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

      3. associate-*l/ 100.0%

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

      4. *-commutative 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 74.4%

      \[\leadsto \sinh y \cdot \color{blue}{\frac{1}{y}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 74.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sinh y \cdot \frac{1}{y}\right)\right)} \]

      2. expm1-udef 74.2%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sinh y \cdot \frac{1}{y}\right)} - 1} \]

      3. un-div-inv 74.2%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\sinh y}{y}}\right)} - 1 \]

   8. Applied egg-rr 74.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\sinh y}{y}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 74.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sinh y}{y}\right)\right)} \]

      2. expm1-log1p 74.4%

         \[\leadsto \color{blue}{\frac{\sinh y}{y}} \]

   10. Simplified 74.4%

       \[\leadsto \color{blue}{\frac{\sinh y}{y}} \]

   if 4.3e166 < y < 2.24999999999999987e275

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]

   2. Taylor expanded in y around 0 100.0%

      \[\leadsto \cos x \cdot \color{blue}{\left(1 + 0.16666666666666666 \cdot {y}^{2}\right)} \]
   3. Step-by-step derivation

      1. unpow2 55.6%

         \[\leadsto 1 + 0.16666666666666666 \cdot \color{blue}{\left(y \cdot y\right)} \]

   4. Simplified 100.0%

      \[\leadsto \cos x \cdot \color{blue}{\left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)} \]

   5. Taylor expanded in x around 0 72.2%

      \[\leadsto \color{blue}{\left(1 + -0.5 \cdot {x}^{2}\right)} \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \]
   6. Step-by-step derivation

      1. *-commutative 19.0%

         \[\leadsto 1 + \color{blue}{{x}^{2} \cdot -0.5} \]

      2. unpow2 19.0%

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

   7. Simplified 72.2%

      \[\leadsto \color{blue}{\left(1 + \left(x \cdot x\right) \cdot -0.5\right)} \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 90:\\ \;\;\;\;\cos x\\ \mathbf{elif}\;y \leq 4.3 \cdot 10^{+166} \lor \neg \left(y \leq 2.25 \cdot 10^{+275}\right):\\ \;\;\;\;\frac{\sinh y}{y}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot \left(1 + \left(x \cdot x\right) \cdot -0.5\right)\\ \end{array} \]

Alternative 6: 69.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 90:\\ \;\;\;\;\cos x\\ \mathbf{elif}\;y \leq 10^{+165}:\\ \;\;\;\;\sinh y \cdot \frac{1}{y}\\ \mathbf{elif}\;y \leq 2.45 \cdot 10^{+275}:\\ \;\;\;\;\left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot \left(1 + \left(x \cdot x\right) \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 90.0)
   (cos x)
   (if (<= y 1e+165)
     (* (sinh y) (/ 1.0 y))
     (if (<= y 2.45e+275)
       (* (+ 1.0 (* 0.16666666666666666 (* y y))) (+ 1.0 (* (* x x) -0.5)))
       (/ (sinh y) y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 90.0) {
		tmp = cos(x);
	} else if (y <= 1e+165) {
		tmp = sinh(y) * (1.0 / y);
	} else if (y <= 2.45e+275) {
		tmp = (1.0 + (0.16666666666666666 * (y * y))) * (1.0 + ((x * x) * -0.5));
	} else {
		tmp = sinh(y) / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 90.0d0) then
        tmp = cos(x)
    else if (y <= 1d+165) then
        tmp = sinh(y) * (1.0d0 / y)
    else if (y <= 2.45d+275) then
        tmp = (1.0d0 + (0.16666666666666666d0 * (y * y))) * (1.0d0 + ((x * x) * (-0.5d0)))
    else
        tmp = sinh(y) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 90.0) {
		tmp = Math.cos(x);
	} else if (y <= 1e+165) {
		tmp = Math.sinh(y) * (1.0 / y);
	} else if (y <= 2.45e+275) {
		tmp = (1.0 + (0.16666666666666666 * (y * y))) * (1.0 + ((x * x) * -0.5));
	} else {
		tmp = Math.sinh(y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 90.0:
		tmp = math.cos(x)
	elif y <= 1e+165:
		tmp = math.sinh(y) * (1.0 / y)
	elif y <= 2.45e+275:
		tmp = (1.0 + (0.16666666666666666 * (y * y))) * (1.0 + ((x * x) * -0.5))
	else:
		tmp = math.sinh(y) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 90.0)
		tmp = cos(x);
	elseif (y <= 1e+165)
		tmp = Float64(sinh(y) * Float64(1.0 / y));
	elseif (y <= 2.45e+275)
		tmp = Float64(Float64(1.0 + Float64(0.16666666666666666 * Float64(y * y))) * Float64(1.0 + Float64(Float64(x * x) * -0.5)));
	else
		tmp = Float64(sinh(y) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 90.0)
		tmp = cos(x);
	elseif (y <= 1e+165)
		tmp = sinh(y) * (1.0 / y);
	elseif (y <= 2.45e+275)
		tmp = (1.0 + (0.16666666666666666 * (y * y))) * (1.0 + ((x * x) * -0.5));
	else
		tmp = sinh(y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 90.0], N[Cos[x], $MachinePrecision], If[LessEqual[y, 1e+165], N[(N[Sinh[y], $MachinePrecision] * N[(1.0 / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.45e+275], N[(N[(1.0 + N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(x * x), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < 90

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]

   2. Taylor expanded in y around 0 59.3%

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

   if 90 < y < 9.99999999999999899e164

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Step-by-step derivation

      1. add-log-exp 97.5%

         \[\leadsto \color{blue}{\log \left(e^{\cos x \cdot \frac{\sinh y}{y}}\right)} \]

      2. *-un-lft-identity 97.5%

         \[\leadsto \log \color{blue}{\left(1 \cdot e^{\cos x \cdot \frac{\sinh y}{y}}\right)} \]

      3. log-prod 97.5%

         \[\leadsto \color{blue}{\log 1 + \log \left(e^{\cos x \cdot \frac{\sinh y}{y}}\right)} \]

      4. metadata-eval 97.5%

         \[\leadsto \color{blue}{0} + \log \left(e^{\cos x \cdot \frac{\sinh y}{y}}\right) \]

      5. add-log-exp 100.0%

         \[\leadsto 0 + \color{blue}{\cos x \cdot \frac{\sinh y}{y}} \]

   3. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{0 + \cos x \cdot \frac{\sinh y}{y}} \]
   4. Step-by-step derivation

      1. +-lft-identity 100.0%

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

      2. associate-*r/ 100.0%

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

      3. associate-*l/ 100.0%

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

      4. *-commutative 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 73.7%

      \[\leadsto \sinh y \cdot \color{blue}{\frac{1}{y}} \]

   if 9.99999999999999899e164 < y < 2.4499999999999999e275

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]

   2. Taylor expanded in y around 0 100.0%

      \[\leadsto \cos x \cdot \color{blue}{\left(1 + 0.16666666666666666 \cdot {y}^{2}\right)} \]
   3. Step-by-step derivation

      1. unpow2 55.6%

         \[\leadsto 1 + 0.16666666666666666 \cdot \color{blue}{\left(y \cdot y\right)} \]

   4. Simplified 100.0%

      \[\leadsto \cos x \cdot \color{blue}{\left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)} \]

   5. Taylor expanded in x around 0 72.2%

      \[\leadsto \color{blue}{\left(1 + -0.5 \cdot {x}^{2}\right)} \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \]
   6. Step-by-step derivation

      1. *-commutative 19.0%

         \[\leadsto 1 + \color{blue}{{x}^{2} \cdot -0.5} \]

      2. unpow2 19.0%

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

   7. Simplified 72.2%

      \[\leadsto \color{blue}{\left(1 + \left(x \cdot x\right) \cdot -0.5\right)} \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \]

   if 2.4499999999999999e275 < y

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Step-by-step derivation

      1. add-log-exp 100.0%

         \[\leadsto \color{blue}{\log \left(e^{\cos x \cdot \frac{\sinh y}{y}}\right)} \]

      2. *-un-lft-identity 100.0%

         \[\leadsto \log \color{blue}{\left(1 \cdot e^{\cos x \cdot \frac{\sinh y}{y}}\right)} \]

      3. log-prod 100.0%

         \[\leadsto \color{blue}{\log 1 + \log \left(e^{\cos x \cdot \frac{\sinh y}{y}}\right)} \]

      4. metadata-eval 100.0%

         \[\leadsto \color{blue}{0} + \log \left(e^{\cos x \cdot \frac{\sinh y}{y}}\right) \]

      5. add-log-exp 100.0%

         \[\leadsto 0 + \color{blue}{\cos x \cdot \frac{\sinh y}{y}} \]

   3. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{0 + \cos x \cdot \frac{\sinh y}{y}} \]
   4. Step-by-step derivation

      1. +-lft-identity 100.0%

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

      2. associate-*r/ 100.0%

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

      3. associate-*l/ 100.0%

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

      4. *-commutative 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 80.0%

      \[\leadsto \sinh y \cdot \color{blue}{\frac{1}{y}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 80.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sinh y \cdot \frac{1}{y}\right)\right)} \]

      2. expm1-udef 80.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sinh y \cdot \frac{1}{y}\right)} - 1} \]

      3. un-div-inv 80.0%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\sinh y}{y}}\right)} - 1 \]

   8. Applied egg-rr 80.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\sinh y}{y}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 80.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sinh y}{y}\right)\right)} \]

      2. expm1-log1p 80.0%

         \[\leadsto \color{blue}{\frac{\sinh y}{y}} \]

   10. Simplified 80.0%

       \[\leadsto \color{blue}{\frac{\sinh y}{y}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 90:\\ \;\;\;\;\cos x\\ \mathbf{elif}\;y \leq 10^{+165}:\\ \;\;\;\;\sinh y \cdot \frac{1}{y}\\ \mathbf{elif}\;y \leq 2.45 \cdot 10^{+275}:\\ \;\;\;\;\left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot \left(1 + \left(x \cdot x\right) \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y}{y}\\ \end{array} \]

Alternative 7: 62.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\\ \mathbf{if}\;y \leq 115:\\ \;\;\;\;\cos x\\ \mathbf{elif}\;y \leq 10^{+275}:\\ \;\;\;\;t_0 \cdot \left(1 + \left(x \cdot x\right) \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* 0.16666666666666666 (* y y)))))
   (if (<= y 115.0)
     (cos x)
     (if (<= y 1e+275) (* t_0 (+ 1.0 (* (* x x) -0.5))) t_0))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (0.16666666666666666 * (y * y));
	double tmp;
	if (y <= 115.0) {
		tmp = cos(x);
	} else if (y <= 1e+275) {
		tmp = t_0 * (1.0 + ((x * x) * -0.5));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (0.16666666666666666d0 * (y * y))
    if (y <= 115.0d0) then
        tmp = cos(x)
    else if (y <= 1d+275) then
        tmp = t_0 * (1.0d0 + ((x * x) * (-0.5d0)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + (0.16666666666666666 * (y * y));
	double tmp;
	if (y <= 115.0) {
		tmp = Math.cos(x);
	} else if (y <= 1e+275) {
		tmp = t_0 * (1.0 + ((x * x) * -0.5));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (0.16666666666666666 * (y * y))
	tmp = 0
	if y <= 115.0:
		tmp = math.cos(x)
	elif y <= 1e+275:
		tmp = t_0 * (1.0 + ((x * x) * -0.5))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(0.16666666666666666 * Float64(y * y)))
	tmp = 0.0
	if (y <= 115.0)
		tmp = cos(x);
	elseif (y <= 1e+275)
		tmp = Float64(t_0 * Float64(1.0 + Float64(Float64(x * x) * -0.5)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + (0.16666666666666666 * (y * y));
	tmp = 0.0;
	if (y <= 115.0)
		tmp = cos(x);
	elseif (y <= 1e+275)
		tmp = t_0 * (1.0 + ((x * x) * -0.5));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 115.0], N[Cos[x], $MachinePrecision], If[LessEqual[y, 1e+275], N[(t$95$0 * N[(1.0 + N[(N[(x * x), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 3 regimes

2. if y < 115

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]

   2. Taylor expanded in y around 0 59.3%

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

   if 115 < y < 9.9999999999999996e274

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]

   2. Taylor expanded in y around 0 39.2%

      \[\leadsto \cos x \cdot \color{blue}{\left(1 + 0.16666666666666666 \cdot {y}^{2}\right)} \]
   3. Step-by-step derivation

      1. unpow2 24.1%

         \[\leadsto 1 + 0.16666666666666666 \cdot \color{blue}{\left(y \cdot y\right)} \]

   4. Simplified 39.2%

      \[\leadsto \cos x \cdot \color{blue}{\left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)} \]

   5. Taylor expanded in x around 0 38.1%

      \[\leadsto \color{blue}{\left(1 + -0.5 \cdot {x}^{2}\right)} \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \]
   6. Step-by-step derivation

      1. *-commutative 14.9%

         \[\leadsto 1 + \color{blue}{{x}^{2} \cdot -0.5} \]

      2. unpow2 14.9%

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

   7. Simplified 38.1%

      \[\leadsto \color{blue}{\left(1 + \left(x \cdot x\right) \cdot -0.5\right)} \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \]

   if 9.9999999999999996e274 < y

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Step-by-step derivation

      1. add-log-exp 100.0%

         \[\leadsto \color{blue}{\log \left(e^{\cos x \cdot \frac{\sinh y}{y}}\right)} \]

      2. *-un-lft-identity 100.0%

         \[\leadsto \log \color{blue}{\left(1 \cdot e^{\cos x \cdot \frac{\sinh y}{y}}\right)} \]

      3. log-prod 100.0%

         \[\leadsto \color{blue}{\log 1 + \log \left(e^{\cos x \cdot \frac{\sinh y}{y}}\right)} \]

      4. metadata-eval 100.0%

         \[\leadsto \color{blue}{0} + \log \left(e^{\cos x \cdot \frac{\sinh y}{y}}\right) \]

      5. add-log-exp 100.0%

         \[\leadsto 0 + \color{blue}{\cos x \cdot \frac{\sinh y}{y}} \]

   3. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{0 + \cos x \cdot \frac{\sinh y}{y}} \]
   4. Step-by-step derivation

      1. +-lft-identity 100.0%

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

      2. associate-*r/ 100.0%

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

      3. associate-*l/ 100.0%

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

      4. *-commutative 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 80.0%

      \[\leadsto \sinh y \cdot \color{blue}{\frac{1}{y}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 80.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sinh y \cdot \frac{1}{y}\right)\right)} \]

      2. expm1-udef 80.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sinh y \cdot \frac{1}{y}\right)} - 1} \]

      3. un-div-inv 80.0%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\sinh y}{y}}\right)} - 1 \]

   8. Applied egg-rr 80.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\sinh y}{y}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 80.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sinh y}{y}\right)\right)} \]

      2. expm1-log1p 80.0%

         \[\leadsto \color{blue}{\frac{\sinh y}{y}} \]

   10. Simplified 80.0%

       \[\leadsto \color{blue}{\frac{\sinh y}{y}} \]

   11. Taylor expanded in y around 0 80.0%

       \[\leadsto \color{blue}{1 + 0.16666666666666666 \cdot {y}^{2}} \]
   12. Step-by-step derivation

       1. unpow2 80.0%

          \[\leadsto 1 + 0.16666666666666666 \cdot \color{blue}{\left(y \cdot y\right)} \]

   13. Simplified 80.0%

       \[\leadsto \color{blue}{1 + 0.16666666666666666 \cdot \left(y \cdot y\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 55.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 115:\\ \;\;\;\;\cos x\\ \mathbf{elif}\;y \leq 10^{+275}:\\ \;\;\;\;\left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot \left(1 + \left(x \cdot x\right) \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\\ \end{array} \]

Alternative 8: 49.8% accurate, 10.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\\ \mathbf{if}\;y \leq 1.25 \cdot 10^{+275}:\\ \;\;\;\;t_0 \cdot \left(\left(2 + x \cdot \left(x \cdot -0.5\right)\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* 0.16666666666666666 (* y y)))))
   (if (<= y 1.25e+275) (* t_0 (+ (+ 2.0 (* x (* x -0.5))) -1.0)) t_0)))

C

double code(double x, double y) {
	double t_0 = 1.0 + (0.16666666666666666 * (y * y));
	double tmp;
	if (y <= 1.25e+275) {
		tmp = t_0 * ((2.0 + (x * (x * -0.5))) + -1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (0.16666666666666666d0 * (y * y))
    if (y <= 1.25d+275) then
        tmp = t_0 * ((2.0d0 + (x * (x * (-0.5d0)))) + (-1.0d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + (0.16666666666666666 * (y * y));
	double tmp;
	if (y <= 1.25e+275) {
		tmp = t_0 * ((2.0 + (x * (x * -0.5))) + -1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (0.16666666666666666 * (y * y))
	tmp = 0
	if y <= 1.25e+275:
		tmp = t_0 * ((2.0 + (x * (x * -0.5))) + -1.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(0.16666666666666666 * Float64(y * y)))
	tmp = 0.0
	if (y <= 1.25e+275)
		tmp = Float64(t_0 * Float64(Float64(2.0 + Float64(x * Float64(x * -0.5))) + -1.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + (0.16666666666666666 * (y * y));
	tmp = 0.0;
	if (y <= 1.25e+275)
		tmp = t_0 * ((2.0 + (x * (x * -0.5))) + -1.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 1.25e+275], N[(t$95$0 * N[(N[(2.0 + N[(x * N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if y < 1.2500000000000001e275

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]

   2. Taylor expanded in y around 0 74.4%

      \[\leadsto \cos x \cdot \color{blue}{\left(1 + 0.16666666666666666 \cdot {y}^{2}\right)} \]
   3. Step-by-step derivation

      1. unpow2 45.1%

         \[\leadsto 1 + 0.16666666666666666 \cdot \color{blue}{\left(y \cdot y\right)} \]

   4. Simplified 74.4%

      \[\leadsto \cos x \cdot \color{blue}{\left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)} \]
   5. Step-by-step derivation

      1. expm1-log1p-u 74.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos x\right)\right)} \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \]

      2. expm1-udef 74.3%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\cos x\right)} - 1\right)} \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \]

      3. log1p-udef 74.3%

         \[\leadsto \left(e^{\color{blue}{\log \left(1 + \cos x\right)}} - 1\right) \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \]

      4. rem-exp-log 74.3%

         \[\leadsto \left(\color{blue}{\left(1 + \cos x\right)} - 1\right) \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \]

   6. Applied egg-rr 74.3%

      \[\leadsto \color{blue}{\left(\left(1 + \cos x\right) - 1\right)} \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \]

   7. Taylor expanded in x around 0 47.0%

      \[\leadsto \left(\color{blue}{\left(2 + -0.5 \cdot {x}^{2}\right)} - 1\right) \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \]
   8. Step-by-step derivation

      1. *-commutative 47.0%

         \[\leadsto \left(\left(2 + \color{blue}{{x}^{2} \cdot -0.5}\right) - 1\right) \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \]

      2. unpow2 47.0%

         \[\leadsto \left(\left(2 + \color{blue}{\left(x \cdot x\right)} \cdot -0.5\right) - 1\right) \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \]

      3. associate-*r* 47.0%

         \[\leadsto \left(\left(2 + \color{blue}{x \cdot \left(x \cdot -0.5\right)}\right) - 1\right) \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \]

      4. *-commutative 47.0%

         \[\leadsto \left(\left(2 + x \cdot \color{blue}{\left(-0.5 \cdot x\right)}\right) - 1\right) \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \]

   9. Simplified 47.0%

      \[\leadsto \left(\color{blue}{\left(2 + x \cdot \left(-0.5 \cdot x\right)\right)} - 1\right) \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \]

   if 1.2500000000000001e275 < y

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Step-by-step derivation

      1. add-log-exp 100.0%

         \[\leadsto \color{blue}{\log \left(e^{\cos x \cdot \frac{\sinh y}{y}}\right)} \]

      2. *-un-lft-identity 100.0%

         \[\leadsto \log \color{blue}{\left(1 \cdot e^{\cos x \cdot \frac{\sinh y}{y}}\right)} \]

      3. log-prod 100.0%

         \[\leadsto \color{blue}{\log 1 + \log \left(e^{\cos x \cdot \frac{\sinh y}{y}}\right)} \]

      4. metadata-eval 100.0%

         \[\leadsto \color{blue}{0} + \log \left(e^{\cos x \cdot \frac{\sinh y}{y}}\right) \]

      5. add-log-exp 100.0%

         \[\leadsto 0 + \color{blue}{\cos x \cdot \frac{\sinh y}{y}} \]

   3. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{0 + \cos x \cdot \frac{\sinh y}{y}} \]
   4. Step-by-step derivation

      1. +-lft-identity 100.0%

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

      2. associate-*r/ 100.0%

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

      3. associate-*l/ 100.0%

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

      4. *-commutative 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 80.0%

      \[\leadsto \sinh y \cdot \color{blue}{\frac{1}{y}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 80.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sinh y \cdot \frac{1}{y}\right)\right)} \]

      2. expm1-udef 80.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sinh y \cdot \frac{1}{y}\right)} - 1} \]

      3. un-div-inv 80.0%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\sinh y}{y}}\right)} - 1 \]

   8. Applied egg-rr 80.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\sinh y}{y}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 80.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sinh y}{y}\right)\right)} \]

      2. expm1-log1p 80.0%

         \[\leadsto \color{blue}{\frac{\sinh y}{y}} \]

   10. Simplified 80.0%

       \[\leadsto \color{blue}{\frac{\sinh y}{y}} \]

   11. Taylor expanded in y around 0 80.0%

       \[\leadsto \color{blue}{1 + 0.16666666666666666 \cdot {y}^{2}} \]
   12. Step-by-step derivation

       1. unpow2 80.0%

          \[\leadsto 1 + 0.16666666666666666 \cdot \color{blue}{\left(y \cdot y\right)} \]

   13. Simplified 80.0%

       \[\leadsto \color{blue}{1 + 0.16666666666666666 \cdot \left(y \cdot y\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.25 \cdot 10^{+275}:\\ \;\;\;\;\left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot \left(\left(2 + x \cdot \left(x \cdot -0.5\right)\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\\ \end{array} \]

Alternative 9: 49.8% accurate, 12.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\\ \mathbf{if}\;y \leq 1.1 \cdot 10^{+275}:\\ \;\;\;\;t_0 \cdot \left(1 + \left(x \cdot x\right) \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* 0.16666666666666666 (* y y)))))
   (if (<= y 1.1e+275) (* t_0 (+ 1.0 (* (* x x) -0.5))) t_0)))

C

double code(double x, double y) {
	double t_0 = 1.0 + (0.16666666666666666 * (y * y));
	double tmp;
	if (y <= 1.1e+275) {
		tmp = t_0 * (1.0 + ((x * x) * -0.5));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (0.16666666666666666d0 * (y * y))
    if (y <= 1.1d+275) then
        tmp = t_0 * (1.0d0 + ((x * x) * (-0.5d0)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + (0.16666666666666666 * (y * y));
	double tmp;
	if (y <= 1.1e+275) {
		tmp = t_0 * (1.0 + ((x * x) * -0.5));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (0.16666666666666666 * (y * y))
	tmp = 0
	if y <= 1.1e+275:
		tmp = t_0 * (1.0 + ((x * x) * -0.5))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(0.16666666666666666 * Float64(y * y)))
	tmp = 0.0
	if (y <= 1.1e+275)
		tmp = Float64(t_0 * Float64(1.0 + Float64(Float64(x * x) * -0.5)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + (0.16666666666666666 * (y * y));
	tmp = 0.0;
	if (y <= 1.1e+275)
		tmp = t_0 * (1.0 + ((x * x) * -0.5));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 1.1e+275], N[(t$95$0 * N[(1.0 + N[(N[(x * x), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if y < 1.1e275

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]

   2. Taylor expanded in y around 0 74.4%

      \[\leadsto \cos x \cdot \color{blue}{\left(1 + 0.16666666666666666 \cdot {y}^{2}\right)} \]
   3. Step-by-step derivation

      1. unpow2 45.1%

         \[\leadsto 1 + 0.16666666666666666 \cdot \color{blue}{\left(y \cdot y\right)} \]

   4. Simplified 74.4%

      \[\leadsto \cos x \cdot \color{blue}{\left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)} \]

   5. Taylor expanded in x around 0 47.0%

      \[\leadsto \color{blue}{\left(1 + -0.5 \cdot {x}^{2}\right)} \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \]
   6. Step-by-step derivation

      1. *-commutative 31.6%

         \[\leadsto 1 + \color{blue}{{x}^{2} \cdot -0.5} \]

      2. unpow2 31.6%

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

   7. Simplified 47.0%

      \[\leadsto \color{blue}{\left(1 + \left(x \cdot x\right) \cdot -0.5\right)} \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \]

   if 1.1e275 < y

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Step-by-step derivation

      1. add-log-exp 100.0%

         \[\leadsto \color{blue}{\log \left(e^{\cos x \cdot \frac{\sinh y}{y}}\right)} \]

      2. *-un-lft-identity 100.0%

         \[\leadsto \log \color{blue}{\left(1 \cdot e^{\cos x \cdot \frac{\sinh y}{y}}\right)} \]

      3. log-prod 100.0%

         \[\leadsto \color{blue}{\log 1 + \log \left(e^{\cos x \cdot \frac{\sinh y}{y}}\right)} \]

      4. metadata-eval 100.0%

         \[\leadsto \color{blue}{0} + \log \left(e^{\cos x \cdot \frac{\sinh y}{y}}\right) \]

      5. add-log-exp 100.0%

         \[\leadsto 0 + \color{blue}{\cos x \cdot \frac{\sinh y}{y}} \]

   3. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{0 + \cos x \cdot \frac{\sinh y}{y}} \]
   4. Step-by-step derivation

      1. +-lft-identity 100.0%

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

      2. associate-*r/ 100.0%

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

      3. associate-*l/ 100.0%

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

      4. *-commutative 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 80.0%

      \[\leadsto \sinh y \cdot \color{blue}{\frac{1}{y}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 80.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sinh y \cdot \frac{1}{y}\right)\right)} \]

      2. expm1-udef 80.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sinh y \cdot \frac{1}{y}\right)} - 1} \]

      3. un-div-inv 80.0%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\sinh y}{y}}\right)} - 1 \]

   8. Applied egg-rr 80.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\sinh y}{y}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 80.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sinh y}{y}\right)\right)} \]

      2. expm1-log1p 80.0%

         \[\leadsto \color{blue}{\frac{\sinh y}{y}} \]

   10. Simplified 80.0%

       \[\leadsto \color{blue}{\frac{\sinh y}{y}} \]

   11. Taylor expanded in y around 0 80.0%

       \[\leadsto \color{blue}{1 + 0.16666666666666666 \cdot {y}^{2}} \]
   12. Step-by-step derivation

       1. unpow2 80.0%

          \[\leadsto 1 + 0.16666666666666666 \cdot \color{blue}{\left(y \cdot y\right)} \]

   13. Simplified 80.0%

       \[\leadsto \color{blue}{1 + 0.16666666666666666 \cdot \left(y \cdot y\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.1 \cdot 10^{+275}:\\ \;\;\;\;\left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot \left(1 + \left(x \cdot x\right) \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\\ \end{array} \]

Alternative 10: 48.8% accurate, 13.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\\ \mathbf{if}\;x \leq 1.4 \cdot 10^{+81}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(\left(x \cdot x\right) \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* 0.16666666666666666 (* y y)))))
   (if (<= x 1.4e+81) t_0 (* t_0 (* (* x x) -0.5)))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (0.16666666666666666 * (y * y));
	double tmp;
	if (x <= 1.4e+81) {
		tmp = t_0;
	} else {
		tmp = t_0 * ((x * x) * -0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (0.16666666666666666d0 * (y * y))
    if (x <= 1.4d+81) then
        tmp = t_0
    else
        tmp = t_0 * ((x * x) * (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + (0.16666666666666666 * (y * y));
	double tmp;
	if (x <= 1.4e+81) {
		tmp = t_0;
	} else {
		tmp = t_0 * ((x * x) * -0.5);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (0.16666666666666666 * (y * y))
	tmp = 0
	if x <= 1.4e+81:
		tmp = t_0
	else:
		tmp = t_0 * ((x * x) * -0.5)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(0.16666666666666666 * Float64(y * y)))
	tmp = 0.0
	if (x <= 1.4e+81)
		tmp = t_0;
	else
		tmp = Float64(t_0 * Float64(Float64(x * x) * -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + (0.16666666666666666 * (y * y));
	tmp = 0.0;
	if (x <= 1.4e+81)
		tmp = t_0;
	else
		tmp = t_0 * ((x * x) * -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1.4e+81], t$95$0, N[(t$95$0 * N[(N[(x * x), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.39999999999999997e81

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Step-by-step derivation

      1. add-log-exp 98.5%

         \[\leadsto \color{blue}{\log \left(e^{\cos x \cdot \frac{\sinh y}{y}}\right)} \]

      2. *-un-lft-identity 98.5%

         \[\leadsto \log \color{blue}{\left(1 \cdot e^{\cos x \cdot \frac{\sinh y}{y}}\right)} \]

      3. log-prod 98.5%

         \[\leadsto \color{blue}{\log 1 + \log \left(e^{\cos x \cdot \frac{\sinh y}{y}}\right)} \]

      4. metadata-eval 98.5%

         \[\leadsto \color{blue}{0} + \log \left(e^{\cos x \cdot \frac{\sinh y}{y}}\right) \]

      5. add-log-exp 100.0%

         \[\leadsto 0 + \color{blue}{\cos x \cdot \frac{\sinh y}{y}} \]

   3. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{0 + \cos x \cdot \frac{\sinh y}{y}} \]
   4. Step-by-step derivation

      1. +-lft-identity 100.0%

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

      2. associate-*r/ 99.9%

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

      3. associate-*l/ 99.9%

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

      4. *-commutative 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in x around 0 72.3%

      \[\leadsto \sinh y \cdot \color{blue}{\frac{1}{y}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 72.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sinh y \cdot \frac{1}{y}\right)\right)} \]

      2. expm1-udef 72.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sinh y \cdot \frac{1}{y}\right)} - 1} \]

      3. un-div-inv 72.3%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\sinh y}{y}}\right)} - 1 \]

   8. Applied egg-rr 72.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\sinh y}{y}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 72.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sinh y}{y}\right)\right)} \]

      2. expm1-log1p 72.3%

         \[\leadsto \color{blue}{\frac{\sinh y}{y}} \]

   10. Simplified 72.3%

       \[\leadsto \color{blue}{\frac{\sinh y}{y}} \]

   11. Taylor expanded in y around 0 51.6%

       \[\leadsto \color{blue}{1 + 0.16666666666666666 \cdot {y}^{2}} \]
   12. Step-by-step derivation

       1. unpow2 51.6%

          \[\leadsto 1 + 0.16666666666666666 \cdot \color{blue}{\left(y \cdot y\right)} \]

   13. Simplified 51.6%

       \[\leadsto \color{blue}{1 + 0.16666666666666666 \cdot \left(y \cdot y\right)} \]

   if 1.39999999999999997e81 < x

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]

   2. Taylor expanded in y around 0 79.4%

      \[\leadsto \cos x \cdot \color{blue}{\left(1 + 0.16666666666666666 \cdot {y}^{2}\right)} \]
   3. Step-by-step derivation

      1. unpow2 19.0%

         \[\leadsto 1 + 0.16666666666666666 \cdot \color{blue}{\left(y \cdot y\right)} \]

   4. Simplified 79.4%

      \[\leadsto \cos x \cdot \color{blue}{\left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)} \]
   5. Step-by-step derivation

      1. expm1-log1p-u 79.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos x\right)\right)} \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \]

      2. expm1-udef 79.1%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\cos x\right)} - 1\right)} \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \]

      3. log1p-udef 79.1%

         \[\leadsto \left(e^{\color{blue}{\log \left(1 + \cos x\right)}} - 1\right) \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \]

      4. rem-exp-log 79.1%

         \[\leadsto \left(\color{blue}{\left(1 + \cos x\right)} - 1\right) \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \]

   6. Applied egg-rr 79.1%

      \[\leadsto \color{blue}{\left(\left(1 + \cos x\right) - 1\right)} \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \]

   7. Taylor expanded in x around 0 29.5%

      \[\leadsto \left(\color{blue}{\left(2 + -0.5 \cdot {x}^{2}\right)} - 1\right) \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \]
   8. Step-by-step derivation

      1. *-commutative 29.5%

         \[\leadsto \left(\left(2 + \color{blue}{{x}^{2} \cdot -0.5}\right) - 1\right) \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \]

      2. unpow2 29.5%

         \[\leadsto \left(\left(2 + \color{blue}{\left(x \cdot x\right)} \cdot -0.5\right) - 1\right) \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \]

      3. associate-*r* 29.5%

         \[\leadsto \left(\left(2 + \color{blue}{x \cdot \left(x \cdot -0.5\right)}\right) - 1\right) \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \]

      4. *-commutative 29.5%

         \[\leadsto \left(\left(2 + x \cdot \color{blue}{\left(-0.5 \cdot x\right)}\right) - 1\right) \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \]

   9. Simplified 29.5%

      \[\leadsto \left(\color{blue}{\left(2 + x \cdot \left(-0.5 \cdot x\right)\right)} - 1\right) \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \]

   10. Taylor expanded in x around inf 29.5%

       \[\leadsto \color{blue}{\left(-0.5 \cdot {x}^{2}\right)} \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \]
   11. Step-by-step derivation

       1. unpow2 29.5%

          \[\leadsto \left(-0.5 \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \]

   12. Simplified 29.5%

       \[\leadsto \color{blue}{\left(-0.5 \cdot \left(x \cdot x\right)\right)} \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.4 \cdot 10^{+81}:\\ \;\;\;\;1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot \left(\left(x \cdot x\right) \cdot -0.5\right)\\ \end{array} \]

Alternative 11: 47.6% accurate, 18.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.7 \cdot 10^{+149} \lor \neg \left(x \leq 8.6 \cdot 10^{+202}\right):\\ \;\;\;\;1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x 3.7e+149) (not (<= x 8.6e+202)))
   (+ 1.0 (* 0.16666666666666666 (* y y)))
   (+ 1.0 (* (* x x) -0.5))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= 3.7e+149) || !(x <= 8.6e+202)) {
		tmp = 1.0 + (0.16666666666666666 * (y * y));
	} else {
		tmp = 1.0 + ((x * x) * -0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= 3.7d+149) .or. (.not. (x <= 8.6d+202))) then
        tmp = 1.0d0 + (0.16666666666666666d0 * (y * y))
    else
        tmp = 1.0d0 + ((x * x) * (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= 3.7e+149) || !(x <= 8.6e+202)) {
		tmp = 1.0 + (0.16666666666666666 * (y * y));
	} else {
		tmp = 1.0 + ((x * x) * -0.5);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= 3.7e+149) or not (x <= 8.6e+202):
		tmp = 1.0 + (0.16666666666666666 * (y * y))
	else:
		tmp = 1.0 + ((x * x) * -0.5)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= 3.7e+149) || !(x <= 8.6e+202))
		tmp = Float64(1.0 + Float64(0.16666666666666666 * Float64(y * y)));
	else
		tmp = Float64(1.0 + Float64(Float64(x * x) * -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= 3.7e+149) || ~((x <= 8.6e+202)))
		tmp = 1.0 + (0.16666666666666666 * (y * y));
	else
		tmp = 1.0 + ((x * x) * -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, 3.7e+149], N[Not[LessEqual[x, 8.6e+202]], $MachinePrecision]], N[(1.0 + N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(x * x), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.69999999999999978e149 or 8.6000000000000005e202 < x

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Step-by-step derivation

      1. add-log-exp 98.6%

         \[\leadsto \color{blue}{\log \left(e^{\cos x \cdot \frac{\sinh y}{y}}\right)} \]

      2. *-un-lft-identity 98.6%

         \[\leadsto \log \color{blue}{\left(1 \cdot e^{\cos x \cdot \frac{\sinh y}{y}}\right)} \]

      3. log-prod 98.6%

         \[\leadsto \color{blue}{\log 1 + \log \left(e^{\cos x \cdot \frac{\sinh y}{y}}\right)} \]

      4. metadata-eval 98.6%

         \[\leadsto \color{blue}{0} + \log \left(e^{\cos x \cdot \frac{\sinh y}{y}}\right) \]

      5. add-log-exp 100.0%

         \[\leadsto 0 + \color{blue}{\cos x \cdot \frac{\sinh y}{y}} \]

   3. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{0 + \cos x \cdot \frac{\sinh y}{y}} \]
   4. Step-by-step derivation

      1. +-lft-identity 100.0%

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

      2. associate-*r/ 99.9%

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

      3. associate-*l/ 99.9%

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

      4. *-commutative 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in x around 0 67.0%

      \[\leadsto \sinh y \cdot \color{blue}{\frac{1}{y}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 67.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sinh y \cdot \frac{1}{y}\right)\right)} \]

      2. expm1-udef 67.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sinh y \cdot \frac{1}{y}\right)} - 1} \]

      3. un-div-inv 67.0%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\sinh y}{y}}\right)} - 1 \]

   8. Applied egg-rr 67.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\sinh y}{y}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 67.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sinh y}{y}\right)\right)} \]

      2. expm1-log1p 67.1%

         \[\leadsto \color{blue}{\frac{\sinh y}{y}} \]

   10. Simplified 67.1%

       \[\leadsto \color{blue}{\frac{\sinh y}{y}} \]

   11. Taylor expanded in y around 0 47.5%

       \[\leadsto \color{blue}{1 + 0.16666666666666666 \cdot {y}^{2}} \]
   12. Step-by-step derivation

       1. unpow2 47.5%

          \[\leadsto 1 + 0.16666666666666666 \cdot \color{blue}{\left(y \cdot y\right)} \]

   13. Simplified 47.5%

       \[\leadsto \color{blue}{1 + 0.16666666666666666 \cdot \left(y \cdot y\right)} \]

   if 3.69999999999999978e149 < x < 8.6000000000000005e202

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]

   2. Taylor expanded in y around 0 47.1%

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

   3. Taylor expanded in x around 0 35.0%

      \[\leadsto \color{blue}{1 + -0.5 \cdot {x}^{2}} \]
   4. Step-by-step derivation

      1. *-commutative 35.0%

         \[\leadsto 1 + \color{blue}{{x}^{2} \cdot -0.5} \]

      2. unpow2 35.0%

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

   5. Simplified 35.0%

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

4. Final simplification 46.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.7 \cdot 10^{+149} \lor \neg \left(x \leq 8.6 \cdot 10^{+202}\right):\\ \;\;\;\;1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot -0.5\\ \end{array} \]

Alternative 12: 47.5% accurate, 29.3× speedup

Math

\[\begin{array}{l} \\ 1 + 0.16666666666666666 \cdot \left(y \cdot y\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ 1.0 (* 0.16666666666666666 (* y y))))

C

double code(double x, double y) {
	return 1.0 + (0.16666666666666666 * (y * y));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 + (0.16666666666666666d0 * (y * y))
end function

Java

public static double code(double x, double y) {
	return 1.0 + (0.16666666666666666 * (y * y));
}

Python

def code(x, y):
	return 1.0 + (0.16666666666666666 * (y * y))

Julia

function code(x, y)
	return Float64(1.0 + Float64(0.16666666666666666 * Float64(y * y)))
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 + (0.16666666666666666 * (y * y));
end

Wolfram

code[x_, y_] := N[(1.0 + N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\cos x \cdot \frac{\sinh y}{y} \]
2. Step-by-step derivation

   1. add-log-exp 98.7%

      \[\leadsto \color{blue}{\log \left(e^{\cos x \cdot \frac{\sinh y}{y}}\right)} \]

   2. *-un-lft-identity 98.7%

      \[\leadsto \log \color{blue}{\left(1 \cdot e^{\cos x \cdot \frac{\sinh y}{y}}\right)} \]

   3. log-prod 98.7%

      \[\leadsto \color{blue}{\log 1 + \log \left(e^{\cos x \cdot \frac{\sinh y}{y}}\right)} \]

   4. metadata-eval 98.7%

      \[\leadsto \color{blue}{0} + \log \left(e^{\cos x \cdot \frac{\sinh y}{y}}\right) \]

   5. add-log-exp 100.0%

      \[\leadsto 0 + \color{blue}{\cos x \cdot \frac{\sinh y}{y}} \]

3. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{0 + \cos x \cdot \frac{\sinh y}{y}} \]
4. Step-by-step derivation

   1. +-lft-identity 100.0%

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

   2. associate-*r/ 99.9%

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

   3. associate-*l/ 99.9%

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

   4. *-commutative 99.9%

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

5. Simplified 99.9%

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

6. Taylor expanded in x around 0 64.2%

   \[\leadsto \sinh y \cdot \color{blue}{\frac{1}{y}} \]
7. Step-by-step derivation

   1. expm1-log1p-u 64.2%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sinh y \cdot \frac{1}{y}\right)\right)} \]

   2. expm1-udef 64.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sinh y \cdot \frac{1}{y}\right)} - 1} \]

   3. un-div-inv 64.2%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\sinh y}{y}}\right)} - 1 \]

8. Applied egg-rr 64.2%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\sinh y}{y}\right)} - 1} \]
9. Step-by-step derivation

   1. expm1-def 64.2%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sinh y}{y}\right)\right)} \]

   2. expm1-log1p 64.2%

      \[\leadsto \color{blue}{\frac{\sinh y}{y}} \]

10. Simplified 64.2%

    \[\leadsto \color{blue}{\frac{\sinh y}{y}} \]

11. Taylor expanded in y around 0 45.8%

    \[\leadsto \color{blue}{1 + 0.16666666666666666 \cdot {y}^{2}} \]
12. Step-by-step derivation

    1. unpow2 45.8%

       \[\leadsto 1 + 0.16666666666666666 \cdot \color{blue}{\left(y \cdot y\right)} \]

13. Simplified 45.8%

    \[\leadsto \color{blue}{1 + 0.16666666666666666 \cdot \left(y \cdot y\right)} \]

14. Final simplification 45.8%

    \[\leadsto 1 + 0.16666666666666666 \cdot \left(y \cdot y\right) \]

Alternative 13: 29.2% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 1.0)

C

double code(double x, double y) {
	return 1.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0
end function

Java

public static double code(double x, double y) {
	return 1.0;
}

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0;
end

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 100.0%

   \[\cos x \cdot \frac{\sinh y}{y} \]

2. Taylor expanded in y around 0 45.9%

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

3. Taylor expanded in x around 0 26.0%

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

4. Final simplification 26.0%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023207 
(FPCore (x y)
  :name "Linear.Quaternion:$csin from linear-1.19.1.3"
  :precision binary64
  (* (cos x) (/ (sinh y) y)))