Linear.Quaternion:$ccos from linear-1.19.1.3

Percentage Accurate: 100.0% → 100.0%

Time: 8.4s

Alternatives: 15

Speedup: 1.0×

• 49.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Final simplification 100.0%

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

Alternative 2: 80.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 7200:\\ \;\;\;\;\sin x \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \mathbf{elif}\;y \leq 6.9 \cdot 10^{+102}:\\ \;\;\;\;{x}^{-3}\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{0.5 \cdot \left(0.3333333333333333 \cdot {y}^{3} + y \cdot 2\right)}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(\sin x \cdot \left(y \cdot y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 7200.0)
   (* (sin x) (+ 1.0 (* 0.16666666666666666 (* y y))))
   (if (<= y 6.9e+102)
     (pow x -3.0)
     (if (<= y 1.35e+154)
       (/ (* 0.5 (+ (* 0.3333333333333333 (pow y 3.0)) (* y 2.0))) (/ y x))
       (* 0.16666666666666666 (* (sin x) (* y y)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 7200.0) {
		tmp = sin(x) * (1.0 + (0.16666666666666666 * (y * y)));
	} else if (y <= 6.9e+102) {
		tmp = pow(x, -3.0);
	} else if (y <= 1.35e+154) {
		tmp = (0.5 * ((0.3333333333333333 * pow(y, 3.0)) + (y * 2.0))) / (y / x);
	} else {
		tmp = 0.16666666666666666 * (sin(x) * (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 <= 7200.0d0) then
        tmp = sin(x) * (1.0d0 + (0.16666666666666666d0 * (y * y)))
    else if (y <= 6.9d+102) then
        tmp = x ** (-3.0d0)
    else if (y <= 1.35d+154) then
        tmp = (0.5d0 * ((0.3333333333333333d0 * (y ** 3.0d0)) + (y * 2.0d0))) / (y / x)
    else
        tmp = 0.16666666666666666d0 * (sin(x) * (y * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 7200.0) {
		tmp = Math.sin(x) * (1.0 + (0.16666666666666666 * (y * y)));
	} else if (y <= 6.9e+102) {
		tmp = Math.pow(x, -3.0);
	} else if (y <= 1.35e+154) {
		tmp = (0.5 * ((0.3333333333333333 * Math.pow(y, 3.0)) + (y * 2.0))) / (y / x);
	} else {
		tmp = 0.16666666666666666 * (Math.sin(x) * (y * y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 7200.0:
		tmp = math.sin(x) * (1.0 + (0.16666666666666666 * (y * y)))
	elif y <= 6.9e+102:
		tmp = math.pow(x, -3.0)
	elif y <= 1.35e+154:
		tmp = (0.5 * ((0.3333333333333333 * math.pow(y, 3.0)) + (y * 2.0))) / (y / x)
	else:
		tmp = 0.16666666666666666 * (math.sin(x) * (y * y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 7200.0)
		tmp = Float64(sin(x) * Float64(1.0 + Float64(0.16666666666666666 * Float64(y * y))));
	elseif (y <= 6.9e+102)
		tmp = x ^ -3.0;
	elseif (y <= 1.35e+154)
		tmp = Float64(Float64(0.5 * Float64(Float64(0.3333333333333333 * (y ^ 3.0)) + Float64(y * 2.0))) / Float64(y / x));
	else
		tmp = Float64(0.16666666666666666 * Float64(sin(x) * Float64(y * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 7200.0)
		tmp = sin(x) * (1.0 + (0.16666666666666666 * (y * y)));
	elseif (y <= 6.9e+102)
		tmp = x ^ -3.0;
	elseif (y <= 1.35e+154)
		tmp = (0.5 * ((0.3333333333333333 * (y ^ 3.0)) + (y * 2.0))) / (y / x);
	else
		tmp = 0.16666666666666666 * (sin(x) * (y * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 7200.0], N[(N[Sin[x], $MachinePrecision] * N[(1.0 + N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6.9e+102], N[Power[x, -3.0], $MachinePrecision], If[LessEqual[y, 1.35e+154], N[(N[(0.5 * N[(N[(0.3333333333333333 * N[Power[y, 3.0], $MachinePrecision]), $MachinePrecision] + N[(y * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y / x), $MachinePrecision]), $MachinePrecision], N[(0.16666666666666666 * N[(N[Sin[x], $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < 7200

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 84.3%

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

      1. unpow2 84.3%

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

   4. Simplified 84.3%

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

   if 7200 < y < 6.89999999999999966e102

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 85.0%

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

      1. associate-/l* 80.0%

         \[\leadsto 0.5 \cdot \color{blue}{\frac{e^{y} - \frac{1}{e^{y}}}{\frac{y}{x}}} \]

      2. associate-*r/ 80.0%

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

      3. rec-exp 80.0%

         \[\leadsto \frac{0.5 \cdot \left(e^{y} - \color{blue}{e^{-y}}\right)}{\frac{y}{x}} \]

   4. Simplified 80.0%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \left(e^{y} - e^{-y}\right)}{\frac{y}{x}}} \]

   6. Applied egg-rr 37.0%

      \[\leadsto \color{blue}{{x}^{-3}} \]

   if 6.89999999999999966e102 < y < 1.35000000000000003e154

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 90.9%

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

      1. associate-/l* 72.7%

         \[\leadsto 0.5 \cdot \color{blue}{\frac{e^{y} - \frac{1}{e^{y}}}{\frac{y}{x}}} \]

      2. associate-*r/ 72.7%

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

      3. rec-exp 72.7%

         \[\leadsto \frac{0.5 \cdot \left(e^{y} - \color{blue}{e^{-y}}\right)}{\frac{y}{x}} \]

   4. Simplified 72.7%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \left(e^{y} - e^{-y}\right)}{\frac{y}{x}}} \]

   5. Taylor expanded in y around 0 72.7%

      \[\leadsto \frac{0.5 \cdot \color{blue}{\left(0.3333333333333333 \cdot {y}^{3} + 2 \cdot y\right)}}{\frac{y}{x}} \]

   if 1.35000000000000003e154 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 100.0%

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

      1. unpow2 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in y around inf 100.0%

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

      1. unpow2 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7200:\\ \;\;\;\;\sin x \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \mathbf{elif}\;y \leq 6.9 \cdot 10^{+102}:\\ \;\;\;\;{x}^{-3}\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{0.5 \cdot \left(0.3333333333333333 \cdot {y}^{3} + y \cdot 2\right)}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(\sin x \cdot \left(y \cdot y\right)\right)\\ \end{array} \]

Alternative 3: 67.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 7200:\\ \;\;\;\;\sin x\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{+102}:\\ \;\;\;\;{x}^{-3}\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{x}{y} \cdot \left(0.16666666666666666 \cdot {y}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(\sin x \cdot \left(y \cdot y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 7200.0)
   (sin x)
   (if (<= y 5.5e+102)
     (pow x -3.0)
     (if (<= y 1.35e+154)
       (* (/ x y) (* 0.16666666666666666 (pow y 3.0)))
       (* 0.16666666666666666 (* (sin x) (* y y)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 7200.0) {
		tmp = sin(x);
	} else if (y <= 5.5e+102) {
		tmp = pow(x, -3.0);
	} else if (y <= 1.35e+154) {
		tmp = (x / y) * (0.16666666666666666 * pow(y, 3.0));
	} else {
		tmp = 0.16666666666666666 * (sin(x) * (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 <= 7200.0d0) then
        tmp = sin(x)
    else if (y <= 5.5d+102) then
        tmp = x ** (-3.0d0)
    else if (y <= 1.35d+154) then
        tmp = (x / y) * (0.16666666666666666d0 * (y ** 3.0d0))
    else
        tmp = 0.16666666666666666d0 * (sin(x) * (y * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 7200.0) {
		tmp = Math.sin(x);
	} else if (y <= 5.5e+102) {
		tmp = Math.pow(x, -3.0);
	} else if (y <= 1.35e+154) {
		tmp = (x / y) * (0.16666666666666666 * Math.pow(y, 3.0));
	} else {
		tmp = 0.16666666666666666 * (Math.sin(x) * (y * y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 7200.0:
		tmp = math.sin(x)
	elif y <= 5.5e+102:
		tmp = math.pow(x, -3.0)
	elif y <= 1.35e+154:
		tmp = (x / y) * (0.16666666666666666 * math.pow(y, 3.0))
	else:
		tmp = 0.16666666666666666 * (math.sin(x) * (y * y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 7200.0)
		tmp = sin(x);
	elseif (y <= 5.5e+102)
		tmp = x ^ -3.0;
	elseif (y <= 1.35e+154)
		tmp = Float64(Float64(x / y) * Float64(0.16666666666666666 * (y ^ 3.0)));
	else
		tmp = Float64(0.16666666666666666 * Float64(sin(x) * Float64(y * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 7200.0)
		tmp = sin(x);
	elseif (y <= 5.5e+102)
		tmp = x ^ -3.0;
	elseif (y <= 1.35e+154)
		tmp = (x / y) * (0.16666666666666666 * (y ^ 3.0));
	else
		tmp = 0.16666666666666666 * (sin(x) * (y * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 7200.0], N[Sin[x], $MachinePrecision], If[LessEqual[y, 5.5e+102], N[Power[x, -3.0], $MachinePrecision], If[LessEqual[y, 1.35e+154], N[(N[(x / y), $MachinePrecision] * N[(0.16666666666666666 * N[Power[y, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.16666666666666666 * N[(N[Sin[x], $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < 7200

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 68.7%

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

   if 7200 < y < 5.49999999999999981e102

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 85.0%

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

      1. associate-/l* 80.0%

         \[\leadsto 0.5 \cdot \color{blue}{\frac{e^{y} - \frac{1}{e^{y}}}{\frac{y}{x}}} \]

      2. associate-*r/ 80.0%

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

      3. rec-exp 80.0%

         \[\leadsto \frac{0.5 \cdot \left(e^{y} - \color{blue}{e^{-y}}\right)}{\frac{y}{x}} \]

   4. Simplified 80.0%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \left(e^{y} - e^{-y}\right)}{\frac{y}{x}}} \]

   6. Applied egg-rr 37.0%

      \[\leadsto \color{blue}{{x}^{-3}} \]

   if 5.49999999999999981e102 < y < 1.35000000000000003e154

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 90.9%

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

      1. associate-/l* 72.7%

         \[\leadsto 0.5 \cdot \color{blue}{\frac{e^{y} - \frac{1}{e^{y}}}{\frac{y}{x}}} \]

      2. associate-*r/ 72.7%

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

      3. rec-exp 72.7%

         \[\leadsto \frac{0.5 \cdot \left(e^{y} - \color{blue}{e^{-y}}\right)}{\frac{y}{x}} \]

   4. Simplified 72.7%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \left(e^{y} - e^{-y}\right)}{\frac{y}{x}}} \]

   5. Taylor expanded in y around 0 72.7%

      \[\leadsto \frac{0.5 \cdot \color{blue}{\left(0.3333333333333333 \cdot {y}^{3} + 2 \cdot y\right)}}{\frac{y}{x}} \]

   6. Taylor expanded in y around inf 72.7%

      \[\leadsto \frac{0.5 \cdot \color{blue}{\left(0.3333333333333333 \cdot {y}^{3}\right)}}{\frac{y}{x}} \]
   7. Step-by-step derivation

      1. clear-num 72.7%

         \[\leadsto \color{blue}{\frac{1}{\frac{\frac{y}{x}}{0.5 \cdot \left(0.3333333333333333 \cdot {y}^{3}\right)}}} \]

      2. associate-/r/ 72.7%

         \[\leadsto \color{blue}{\frac{1}{\frac{y}{x}} \cdot \left(0.5 \cdot \left(0.3333333333333333 \cdot {y}^{3}\right)\right)} \]

      3. clear-num 72.7%

         \[\leadsto \color{blue}{\frac{x}{y}} \cdot \left(0.5 \cdot \left(0.3333333333333333 \cdot {y}^{3}\right)\right) \]

      4. associate-*r* 72.7%

         \[\leadsto \frac{x}{y} \cdot \color{blue}{\left(\left(0.5 \cdot 0.3333333333333333\right) \cdot {y}^{3}\right)} \]

      5. metadata-eval 72.7%

         \[\leadsto \frac{x}{y} \cdot \left(\color{blue}{0.16666666666666666} \cdot {y}^{3}\right) \]

   8. Applied egg-rr 72.7%

      \[\leadsto \color{blue}{\frac{x}{y} \cdot \left(0.16666666666666666 \cdot {y}^{3}\right)} \]

   if 1.35000000000000003e154 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 100.0%

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

      1. unpow2 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in y around inf 100.0%

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

      1. unpow2 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7200:\\ \;\;\;\;\sin x\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{+102}:\\ \;\;\;\;{x}^{-3}\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{x}{y} \cdot \left(0.16666666666666666 \cdot {y}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(\sin x \cdot \left(y \cdot y\right)\right)\\ \end{array} \]

Alternative 4: 80.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 7200:\\ \;\;\;\;\sin x \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \mathbf{elif}\;y \leq 6.9 \cdot 10^{+102}:\\ \;\;\;\;{x}^{-3}\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{x}{y} \cdot \left(0.16666666666666666 \cdot {y}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(\sin x \cdot \left(y \cdot y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 7200.0)
   (* (sin x) (+ 1.0 (* 0.16666666666666666 (* y y))))
   (if (<= y 6.9e+102)
     (pow x -3.0)
     (if (<= y 1.35e+154)
       (* (/ x y) (* 0.16666666666666666 (pow y 3.0)))
       (* 0.16666666666666666 (* (sin x) (* y y)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 7200.0) {
		tmp = sin(x) * (1.0 + (0.16666666666666666 * (y * y)));
	} else if (y <= 6.9e+102) {
		tmp = pow(x, -3.0);
	} else if (y <= 1.35e+154) {
		tmp = (x / y) * (0.16666666666666666 * pow(y, 3.0));
	} else {
		tmp = 0.16666666666666666 * (sin(x) * (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 <= 7200.0d0) then
        tmp = sin(x) * (1.0d0 + (0.16666666666666666d0 * (y * y)))
    else if (y <= 6.9d+102) then
        tmp = x ** (-3.0d0)
    else if (y <= 1.35d+154) then
        tmp = (x / y) * (0.16666666666666666d0 * (y ** 3.0d0))
    else
        tmp = 0.16666666666666666d0 * (sin(x) * (y * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 7200.0) {
		tmp = Math.sin(x) * (1.0 + (0.16666666666666666 * (y * y)));
	} else if (y <= 6.9e+102) {
		tmp = Math.pow(x, -3.0);
	} else if (y <= 1.35e+154) {
		tmp = (x / y) * (0.16666666666666666 * Math.pow(y, 3.0));
	} else {
		tmp = 0.16666666666666666 * (Math.sin(x) * (y * y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 7200.0:
		tmp = math.sin(x) * (1.0 + (0.16666666666666666 * (y * y)))
	elif y <= 6.9e+102:
		tmp = math.pow(x, -3.0)
	elif y <= 1.35e+154:
		tmp = (x / y) * (0.16666666666666666 * math.pow(y, 3.0))
	else:
		tmp = 0.16666666666666666 * (math.sin(x) * (y * y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 7200.0)
		tmp = Float64(sin(x) * Float64(1.0 + Float64(0.16666666666666666 * Float64(y * y))));
	elseif (y <= 6.9e+102)
		tmp = x ^ -3.0;
	elseif (y <= 1.35e+154)
		tmp = Float64(Float64(x / y) * Float64(0.16666666666666666 * (y ^ 3.0)));
	else
		tmp = Float64(0.16666666666666666 * Float64(sin(x) * Float64(y * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 7200.0)
		tmp = sin(x) * (1.0 + (0.16666666666666666 * (y * y)));
	elseif (y <= 6.9e+102)
		tmp = x ^ -3.0;
	elseif (y <= 1.35e+154)
		tmp = (x / y) * (0.16666666666666666 * (y ^ 3.0));
	else
		tmp = 0.16666666666666666 * (sin(x) * (y * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 7200.0], N[(N[Sin[x], $MachinePrecision] * N[(1.0 + N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6.9e+102], N[Power[x, -3.0], $MachinePrecision], If[LessEqual[y, 1.35e+154], N[(N[(x / y), $MachinePrecision] * N[(0.16666666666666666 * N[Power[y, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.16666666666666666 * N[(N[Sin[x], $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < 7200

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 84.3%

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

      1. unpow2 84.3%

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

   4. Simplified 84.3%

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

   if 7200 < y < 6.89999999999999966e102

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 85.0%

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

      1. associate-/l* 80.0%

         \[\leadsto 0.5 \cdot \color{blue}{\frac{e^{y} - \frac{1}{e^{y}}}{\frac{y}{x}}} \]

      2. associate-*r/ 80.0%

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

      3. rec-exp 80.0%

         \[\leadsto \frac{0.5 \cdot \left(e^{y} - \color{blue}{e^{-y}}\right)}{\frac{y}{x}} \]

   4. Simplified 80.0%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \left(e^{y} - e^{-y}\right)}{\frac{y}{x}}} \]

   6. Applied egg-rr 37.0%

      \[\leadsto \color{blue}{{x}^{-3}} \]

   if 6.89999999999999966e102 < y < 1.35000000000000003e154

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 90.9%

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

      1. associate-/l* 72.7%

         \[\leadsto 0.5 \cdot \color{blue}{\frac{e^{y} - \frac{1}{e^{y}}}{\frac{y}{x}}} \]

      2. associate-*r/ 72.7%

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

      3. rec-exp 72.7%

         \[\leadsto \frac{0.5 \cdot \left(e^{y} - \color{blue}{e^{-y}}\right)}{\frac{y}{x}} \]

   4. Simplified 72.7%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \left(e^{y} - e^{-y}\right)}{\frac{y}{x}}} \]

   5. Taylor expanded in y around 0 72.7%

      \[\leadsto \frac{0.5 \cdot \color{blue}{\left(0.3333333333333333 \cdot {y}^{3} + 2 \cdot y\right)}}{\frac{y}{x}} \]

   6. Taylor expanded in y around inf 72.7%

      \[\leadsto \frac{0.5 \cdot \color{blue}{\left(0.3333333333333333 \cdot {y}^{3}\right)}}{\frac{y}{x}} \]
   7. Step-by-step derivation

      1. clear-num 72.7%

         \[\leadsto \color{blue}{\frac{1}{\frac{\frac{y}{x}}{0.5 \cdot \left(0.3333333333333333 \cdot {y}^{3}\right)}}} \]

      2. associate-/r/ 72.7%

         \[\leadsto \color{blue}{\frac{1}{\frac{y}{x}} \cdot \left(0.5 \cdot \left(0.3333333333333333 \cdot {y}^{3}\right)\right)} \]

      3. clear-num 72.7%

         \[\leadsto \color{blue}{\frac{x}{y}} \cdot \left(0.5 \cdot \left(0.3333333333333333 \cdot {y}^{3}\right)\right) \]

      4. associate-*r* 72.7%

         \[\leadsto \frac{x}{y} \cdot \color{blue}{\left(\left(0.5 \cdot 0.3333333333333333\right) \cdot {y}^{3}\right)} \]

      5. metadata-eval 72.7%

         \[\leadsto \frac{x}{y} \cdot \left(\color{blue}{0.16666666666666666} \cdot {y}^{3}\right) \]

   8. Applied egg-rr 72.7%

      \[\leadsto \color{blue}{\frac{x}{y} \cdot \left(0.16666666666666666 \cdot {y}^{3}\right)} \]

   if 1.35000000000000003e154 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 100.0%

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

      1. unpow2 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in y around inf 100.0%

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

      1. unpow2 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7200:\\ \;\;\;\;\sin x \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \mathbf{elif}\;y \leq 6.9 \cdot 10^{+102}:\\ \;\;\;\;{x}^{-3}\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{x}{y} \cdot \left(0.16666666666666666 \cdot {y}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(\sin x \cdot \left(y \cdot y\right)\right)\\ \end{array} \]

Alternative 5: 67.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 7200:\\ \;\;\;\;\sin x\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;{x}^{-3}\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(\sin x \cdot \left(y \cdot y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 7200.0)
   (sin x)
   (if (<= y 1.35e+154)
     (pow x -3.0)
     (* 0.16666666666666666 (* (sin x) (* y y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 7200.0) {
		tmp = sin(x);
	} else if (y <= 1.35e+154) {
		tmp = pow(x, -3.0);
	} else {
		tmp = 0.16666666666666666 * (sin(x) * (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 <= 7200.0d0) then
        tmp = sin(x)
    else if (y <= 1.35d+154) then
        tmp = x ** (-3.0d0)
    else
        tmp = 0.16666666666666666d0 * (sin(x) * (y * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 7200.0) {
		tmp = Math.sin(x);
	} else if (y <= 1.35e+154) {
		tmp = Math.pow(x, -3.0);
	} else {
		tmp = 0.16666666666666666 * (Math.sin(x) * (y * y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 7200.0:
		tmp = math.sin(x)
	elif y <= 1.35e+154:
		tmp = math.pow(x, -3.0)
	else:
		tmp = 0.16666666666666666 * (math.sin(x) * (y * y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 7200.0)
		tmp = sin(x);
	elseif (y <= 1.35e+154)
		tmp = x ^ -3.0;
	else
		tmp = Float64(0.16666666666666666 * Float64(sin(x) * Float64(y * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 7200.0)
		tmp = sin(x);
	elseif (y <= 1.35e+154)
		tmp = x ^ -3.0;
	else
		tmp = 0.16666666666666666 * (sin(x) * (y * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 7200.0], N[Sin[x], $MachinePrecision], If[LessEqual[y, 1.35e+154], N[Power[x, -3.0], $MachinePrecision], N[(0.16666666666666666 * N[(N[Sin[x], $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 7200

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 68.7%

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

   if 7200 < y < 1.35000000000000003e154

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 87.1%

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

      1. associate-/l* 77.4%

         \[\leadsto 0.5 \cdot \color{blue}{\frac{e^{y} - \frac{1}{e^{y}}}{\frac{y}{x}}} \]

      2. associate-*r/ 77.4%

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

      3. rec-exp 77.4%

         \[\leadsto \frac{0.5 \cdot \left(e^{y} - \color{blue}{e^{-y}}\right)}{\frac{y}{x}} \]

   4. Simplified 77.4%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \left(e^{y} - e^{-y}\right)}{\frac{y}{x}}} \]

   6. Applied egg-rr 34.3%

      \[\leadsto \color{blue}{{x}^{-3}} \]

   if 1.35000000000000003e154 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 100.0%

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

      1. unpow2 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in y around inf 100.0%

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

      1. unpow2 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7200:\\ \;\;\;\;\sin x\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;{x}^{-3}\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(\sin x \cdot \left(y \cdot y\right)\right)\\ \end{array} \]

Alternative 6: 64.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 7200:\\ \;\;\;\;\sin x\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{+154}:\\ \;\;\;\;{x}^{-3}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + y \cdot \left(y \cdot 0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 7200.0)
   (sin x)
   (if (<= y 3.3e+154)
     (pow x -3.0)
     (* x (+ 1.0 (* y (* y 0.16666666666666666)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 7200.0) {
		tmp = sin(x);
	} else if (y <= 3.3e+154) {
		tmp = pow(x, -3.0);
	} else {
		tmp = x * (1.0 + (y * (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 <= 7200.0d0) then
        tmp = sin(x)
    else if (y <= 3.3d+154) then
        tmp = x ** (-3.0d0)
    else
        tmp = x * (1.0d0 + (y * (y * 0.16666666666666666d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 7200.0) {
		tmp = Math.sin(x);
	} else if (y <= 3.3e+154) {
		tmp = Math.pow(x, -3.0);
	} else {
		tmp = x * (1.0 + (y * (y * 0.16666666666666666)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 7200.0:
		tmp = math.sin(x)
	elif y <= 3.3e+154:
		tmp = math.pow(x, -3.0)
	else:
		tmp = x * (1.0 + (y * (y * 0.16666666666666666)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 7200.0)
		tmp = sin(x);
	elseif (y <= 3.3e+154)
		tmp = x ^ -3.0;
	else
		tmp = Float64(x * Float64(1.0 + Float64(y * Float64(y * 0.16666666666666666))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 7200.0)
		tmp = sin(x);
	elseif (y <= 3.3e+154)
		tmp = x ^ -3.0;
	else
		tmp = x * (1.0 + (y * (y * 0.16666666666666666)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 7200.0], N[Sin[x], $MachinePrecision], If[LessEqual[y, 3.3e+154], N[Power[x, -3.0], $MachinePrecision], N[(x * N[(1.0 + N[(y * N[(y * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 7200

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 68.7%

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

   if 7200 < y < 3.3e154

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 87.1%

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

      1. associate-/l* 77.4%

         \[\leadsto 0.5 \cdot \color{blue}{\frac{e^{y} - \frac{1}{e^{y}}}{\frac{y}{x}}} \]

      2. associate-*r/ 77.4%

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

      3. rec-exp 77.4%

         \[\leadsto \frac{0.5 \cdot \left(e^{y} - \color{blue}{e^{-y}}\right)}{\frac{y}{x}} \]

   4. Simplified 77.4%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \left(e^{y} - e^{-y}\right)}{\frac{y}{x}}} \]

   6. Applied egg-rr 34.3%

      \[\leadsto \color{blue}{{x}^{-3}} \]

   if 3.3e154 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 66.7%

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

      1. associate-/l* 30.6%

         \[\leadsto 0.5 \cdot \color{blue}{\frac{e^{y} - \frac{1}{e^{y}}}{\frac{y}{x}}} \]

      2. associate-*r/ 30.6%

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

      3. rec-exp 30.6%

         \[\leadsto \frac{0.5 \cdot \left(e^{y} - \color{blue}{e^{-y}}\right)}{\frac{y}{x}} \]

   4. Simplified 30.6%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \left(e^{y} - e^{-y}\right)}{\frac{y}{x}}} \]

   5. Taylor expanded in y around 0 66.7%

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

      1. +-commutative 66.7%

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

      2. *-un-lft-identity 66.7%

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

      3. associate-*r* 66.7%

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

      4. distribute-rgt-out 66.7%

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

      5. unpow2 66.7%

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

      6. associate-*r* 66.7%

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

   7. Applied egg-rr 66.7%

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

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7200:\\ \;\;\;\;\sin x\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{+154}:\\ \;\;\;\;{x}^{-3}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + y \cdot \left(y \cdot 0.16666666666666666\right)\right)\\ \end{array} \]

Alternative 7: 62.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 550:\\ \;\;\;\;\sin x\\ \mathbf{elif}\;y \leq 6.3 \cdot 10^{+75}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;x + 0.16666666666666666 \cdot \left(x \cdot \left(y \cdot y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 550.0)
   (sin x)
   (if (<= y 6.3e+75) (* x x) (+ x (* 0.16666666666666666 (* x (* y y)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 550.0) {
		tmp = sin(x);
	} else if (y <= 6.3e+75) {
		tmp = x * x;
	} else {
		tmp = x + (0.16666666666666666 * (x * (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 <= 550.0d0) then
        tmp = sin(x)
    else if (y <= 6.3d+75) then
        tmp = x * x
    else
        tmp = x + (0.16666666666666666d0 * (x * (y * y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 550.0) {
		tmp = Math.sin(x);
	} else if (y <= 6.3e+75) {
		tmp = x * x;
	} else {
		tmp = x + (0.16666666666666666 * (x * (y * y)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 550.0:
		tmp = math.sin(x)
	elif y <= 6.3e+75:
		tmp = x * x
	else:
		tmp = x + (0.16666666666666666 * (x * (y * y)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 550.0)
		tmp = sin(x);
	elseif (y <= 6.3e+75)
		tmp = Float64(x * x);
	else
		tmp = Float64(x + Float64(0.16666666666666666 * Float64(x * Float64(y * y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 550.0)
		tmp = sin(x);
	elseif (y <= 6.3e+75)
		tmp = x * x;
	else
		tmp = x + (0.16666666666666666 * (x * (y * y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 550.0], N[Sin[x], $MachinePrecision], If[LessEqual[y, 6.3e+75], N[(x * x), $MachinePrecision], N[(x + N[(0.16666666666666666 * N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 550

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 69.4%

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

   if 550 < y < 6.30000000000000036e75

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 81.3%

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

      1. associate-/l* 75.0%

         \[\leadsto 0.5 \cdot \color{blue}{\frac{e^{y} - \frac{1}{e^{y}}}{\frac{y}{x}}} \]

      2. associate-*r/ 75.0%

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

      3. rec-exp 75.0%

         \[\leadsto \frac{0.5 \cdot \left(e^{y} - \color{blue}{e^{-y}}\right)}{\frac{y}{x}} \]

   4. Simplified 75.0%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \left(e^{y} - e^{-y}\right)}{\frac{y}{x}}} \]

   6. Applied egg-rr 14.3%

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

   if 6.30000000000000036e75 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 73.6%

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

      1. associate-/l* 45.3%

         \[\leadsto 0.5 \cdot \color{blue}{\frac{e^{y} - \frac{1}{e^{y}}}{\frac{y}{x}}} \]

      2. associate-*r/ 45.3%

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

      3. rec-exp 45.3%

         \[\leadsto \frac{0.5 \cdot \left(e^{y} - \color{blue}{e^{-y}}\right)}{\frac{y}{x}} \]

   4. Simplified 45.3%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \left(e^{y} - e^{-y}\right)}{\frac{y}{x}}} \]

   5. Taylor expanded in y around 0 53.8%

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

      1. expm1-log1p-u 32.7%

         \[\leadsto 0.16666666666666666 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({y}^{2} \cdot x\right)\right)} + x \]

      2. expm1-udef 32.7%

         \[\leadsto 0.16666666666666666 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({y}^{2} \cdot x\right)} - 1\right)} + x \]

      3. *-commutative 32.7%

         \[\leadsto 0.16666666666666666 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{x \cdot {y}^{2}}\right)} - 1\right) + x \]

      4. unpow2 32.7%

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

      5. associate-*r* 29.2%

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

   7. Applied egg-rr 29.2%

      \[\leadsto 0.16666666666666666 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\left(x \cdot y\right) \cdot y\right)} - 1\right)} + x \]
   8. Step-by-step derivation

      1. expm1-def 29.2%

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

      2. expm1-log1p 41.5%

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

      3. associate-*l* 53.8%

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

   9. Simplified 53.8%

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

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 550:\\ \;\;\;\;\sin x\\ \mathbf{elif}\;y \leq 6.3 \cdot 10^{+75}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;x + 0.16666666666666666 \cdot \left(x \cdot \left(y \cdot y\right)\right)\\ \end{array} \]

Alternative 8: 37.3% accurate, 18.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 490:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 9 \cdot 10^{+75}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(x \cdot \left(y \cdot y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 490.0)
   x
   (if (<= y 9e+75) (* x x) (* 0.16666666666666666 (* x (* y y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 490.0) {
		tmp = x;
	} else if (y <= 9e+75) {
		tmp = x * x;
	} else {
		tmp = 0.16666666666666666 * (x * (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 <= 490.0d0) then
        tmp = x
    else if (y <= 9d+75) then
        tmp = x * x
    else
        tmp = 0.16666666666666666d0 * (x * (y * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 490.0) {
		tmp = x;
	} else if (y <= 9e+75) {
		tmp = x * x;
	} else {
		tmp = 0.16666666666666666 * (x * (y * y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 490.0:
		tmp = x
	elif y <= 9e+75:
		tmp = x * x
	else:
		tmp = 0.16666666666666666 * (x * (y * y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 490.0)
		tmp = x;
	elseif (y <= 9e+75)
		tmp = Float64(x * x);
	else
		tmp = Float64(0.16666666666666666 * Float64(x * Float64(y * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 490.0)
		tmp = x;
	elseif (y <= 9e+75)
		tmp = x * x;
	else
		tmp = 0.16666666666666666 * (x * (y * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 490.0], x, If[LessEqual[y, 9e+75], N[(x * x), $MachinePrecision], N[(0.16666666666666666 * N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 490

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 26.1%

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

      1. associate-/l* 16.0%

         \[\leadsto 0.5 \cdot \color{blue}{\frac{e^{y} - \frac{1}{e^{y}}}{\frac{y}{x}}} \]

      2. associate-*r/ 16.0%

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

      3. rec-exp 16.0%

         \[\leadsto \frac{0.5 \cdot \left(e^{y} - \color{blue}{e^{-y}}\right)}{\frac{y}{x}} \]

   4. Simplified 16.0%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \left(e^{y} - e^{-y}\right)}{\frac{y}{x}}} \]

   5. Taylor expanded in y around 0 36.0%

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

   if 490 < y < 9.0000000000000007e75

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 81.3%

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

      1. associate-/l* 75.0%

         \[\leadsto 0.5 \cdot \color{blue}{\frac{e^{y} - \frac{1}{e^{y}}}{\frac{y}{x}}} \]

      2. associate-*r/ 75.0%

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

      3. rec-exp 75.0%

         \[\leadsto \frac{0.5 \cdot \left(e^{y} - \color{blue}{e^{-y}}\right)}{\frac{y}{x}} \]

   4. Simplified 75.0%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \left(e^{y} - e^{-y}\right)}{\frac{y}{x}}} \]

   6. Applied egg-rr 14.3%

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

   if 9.0000000000000007e75 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 73.6%

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

      1. associate-/l* 45.3%

         \[\leadsto 0.5 \cdot \color{blue}{\frac{e^{y} - \frac{1}{e^{y}}}{\frac{y}{x}}} \]

      2. associate-*r/ 45.3%

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

      3. rec-exp 45.3%

         \[\leadsto \frac{0.5 \cdot \left(e^{y} - \color{blue}{e^{-y}}\right)}{\frac{y}{x}} \]

   4. Simplified 45.3%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \left(e^{y} - e^{-y}\right)}{\frac{y}{x}}} \]

   5. Taylor expanded in y around 0 53.8%

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

   6. Taylor expanded in y around inf 53.8%

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

      1. unpow2 53.8%

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

      2. *-commutative 53.8%

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

   8. Simplified 53.8%

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

4. Final simplification 38.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 490:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 9 \cdot 10^{+75}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(x \cdot \left(y \cdot y\right)\right)\\ \end{array} \]

Alternative 9: 47.3% accurate, 18.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{+174}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + y \cdot \left(y \cdot 0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.9e+174) (* x x) (* x (+ 1.0 (* y (* y 0.16666666666666666))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.9e+174) {
		tmp = x * x;
	} else {
		tmp = x * (1.0 + (y * (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 (x <= (-2.9d+174)) then
        tmp = x * x
    else
        tmp = x * (1.0d0 + (y * (y * 0.16666666666666666d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -2.9e+174) {
		tmp = x * x;
	} else {
		tmp = x * (1.0 + (y * (y * 0.16666666666666666)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.9e+174:
		tmp = x * x
	else:
		tmp = x * (1.0 + (y * (y * 0.16666666666666666)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.9e+174)
		tmp = Float64(x * x);
	else
		tmp = Float64(x * Float64(1.0 + Float64(y * Float64(y * 0.16666666666666666))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.9e+174)
		tmp = x * x;
	else
		tmp = x * (1.0 + (y * (y * 0.16666666666666666)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.9e+174], N[(x * x), $MachinePrecision], N[(x * N[(1.0 + N[(y * N[(y * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.9e174

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 15.1%

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

      1. associate-/l* 14.1%

         \[\leadsto 0.5 \cdot \color{blue}{\frac{e^{y} - \frac{1}{e^{y}}}{\frac{y}{x}}} \]

      2. associate-*r/ 14.1%

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

      3. rec-exp 14.1%

         \[\leadsto \frac{0.5 \cdot \left(e^{y} - \color{blue}{e^{-y}}\right)}{\frac{y}{x}} \]

   4. Simplified 14.1%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \left(e^{y} - e^{-y}\right)}{\frac{y}{x}}} \]

   6. Applied egg-rr 42.4%

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

   if -2.9e174 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 42.5%

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

      1. associate-/l* 27.2%

         \[\leadsto 0.5 \cdot \color{blue}{\frac{e^{y} - \frac{1}{e^{y}}}{\frac{y}{x}}} \]

      2. associate-*r/ 27.2%

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

      3. rec-exp 27.2%

         \[\leadsto \frac{0.5 \cdot \left(e^{y} - \color{blue}{e^{-y}}\right)}{\frac{y}{x}} \]

   4. Simplified 27.2%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \left(e^{y} - e^{-y}\right)}{\frac{y}{x}}} \]

   5. Taylor expanded in y around 0 51.5%

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

      1. +-commutative 51.5%

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

      2. *-un-lft-identity 51.5%

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

      3. associate-*r* 51.5%

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

      4. distribute-rgt-out 51.5%

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

      5. unpow2 51.5%

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

      6. associate-*r* 51.5%

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

   7. Applied egg-rr 51.5%

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

4. Final simplification 50.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{+174}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + y \cdot \left(y \cdot 0.16666666666666666\right)\right)\\ \end{array} \]

Alternative 10: 47.4% accurate, 18.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{+174}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;x + 0.16666666666666666 \cdot \left(x \cdot \left(y \cdot y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.9e+174) (* x x) (+ x (* 0.16666666666666666 (* x (* y y))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.9e+174) {
		tmp = x * x;
	} else {
		tmp = x + (0.16666666666666666 * (x * (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 (x <= (-2.9d+174)) then
        tmp = x * x
    else
        tmp = x + (0.16666666666666666d0 * (x * (y * y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -2.9e+174) {
		tmp = x * x;
	} else {
		tmp = x + (0.16666666666666666 * (x * (y * y)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.9e+174:
		tmp = x * x
	else:
		tmp = x + (0.16666666666666666 * (x * (y * y)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.9e+174)
		tmp = Float64(x * x);
	else
		tmp = Float64(x + Float64(0.16666666666666666 * Float64(x * Float64(y * y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.9e+174)
		tmp = x * x;
	else
		tmp = x + (0.16666666666666666 * (x * (y * y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.9e+174], N[(x * x), $MachinePrecision], N[(x + N[(0.16666666666666666 * N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.9e174

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 15.1%

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

      1. associate-/l* 14.1%

         \[\leadsto 0.5 \cdot \color{blue}{\frac{e^{y} - \frac{1}{e^{y}}}{\frac{y}{x}}} \]

      2. associate-*r/ 14.1%

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

      3. rec-exp 14.1%

         \[\leadsto \frac{0.5 \cdot \left(e^{y} - \color{blue}{e^{-y}}\right)}{\frac{y}{x}} \]

   4. Simplified 14.1%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \left(e^{y} - e^{-y}\right)}{\frac{y}{x}}} \]

   6. Applied egg-rr 42.4%

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

   if -2.9e174 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 42.5%

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

      1. associate-/l* 27.2%

         \[\leadsto 0.5 \cdot \color{blue}{\frac{e^{y} - \frac{1}{e^{y}}}{\frac{y}{x}}} \]

      2. associate-*r/ 27.2%

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

      3. rec-exp 27.2%

         \[\leadsto \frac{0.5 \cdot \left(e^{y} - \color{blue}{e^{-y}}\right)}{\frac{y}{x}} \]

   4. Simplified 27.2%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \left(e^{y} - e^{-y}\right)}{\frac{y}{x}}} \]

   5. Taylor expanded in y around 0 51.5%

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

      1. expm1-log1p-u 41.5%

         \[\leadsto 0.16666666666666666 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({y}^{2} \cdot x\right)\right)} + x \]

      2. expm1-udef 41.3%

         \[\leadsto 0.16666666666666666 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({y}^{2} \cdot x\right)} - 1\right)} + x \]

      3. *-commutative 41.3%

         \[\leadsto 0.16666666666666666 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{x \cdot {y}^{2}}\right)} - 1\right) + x \]

      4. unpow2 41.3%

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

      5. associate-*r* 38.4%

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

   7. Applied egg-rr 38.4%

      \[\leadsto 0.16666666666666666 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\left(x \cdot y\right) \cdot y\right)} - 1\right)} + x \]
   8. Step-by-step derivation

      1. expm1-def 38.6%

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

      2. expm1-log1p 44.9%

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

      3. associate-*l* 51.5%

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

   9. Simplified 51.5%

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

4. Final simplification 50.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{+174}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;x + 0.16666666666666666 \cdot \left(x \cdot \left(y \cdot y\right)\right)\\ \end{array} \]

Alternative 11: 28.7% accurate, 40.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 660:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 660.0) x (* x x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 660.0) {
		tmp = x;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 660.0d0) then
        tmp = x
    else
        tmp = x * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 660.0) {
		tmp = x;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 660.0:
		tmp = x
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 660.0)
		tmp = x;
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 660.0)
		tmp = x;
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 660.0], x, N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 660

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 26.1%

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

      1. associate-/l* 16.0%

         \[\leadsto 0.5 \cdot \color{blue}{\frac{e^{y} - \frac{1}{e^{y}}}{\frac{y}{x}}} \]

      2. associate-*r/ 16.0%

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

      3. rec-exp 16.0%

         \[\leadsto \frac{0.5 \cdot \left(e^{y} - \color{blue}{e^{-y}}\right)}{\frac{y}{x}} \]

   4. Simplified 16.0%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \left(e^{y} - e^{-y}\right)}{\frac{y}{x}}} \]

   5. Taylor expanded in y around 0 36.0%

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

   if 660 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 75.4%

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

      1. associate-/l* 52.2%

         \[\leadsto 0.5 \cdot \color{blue}{\frac{e^{y} - \frac{1}{e^{y}}}{\frac{y}{x}}} \]

      2. associate-*r/ 52.2%

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

      3. rec-exp 52.2%

         \[\leadsto \frac{0.5 \cdot \left(e^{y} - \color{blue}{e^{-y}}\right)}{\frac{y}{x}} \]

   4. Simplified 52.2%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \left(e^{y} - e^{-y}\right)}{\frac{y}{x}}} \]

   6. Applied egg-rr 19.0%

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

4. Final simplification 31.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 660:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 12: 4.2% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ -3 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 -3.0)

C

double code(double x, double y) {
	return -3.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = -3.0d0
end function

Java

public static double code(double x, double y) {
	return -3.0;
}

Python

def code(x, y):
	return -3.0

Julia

function code(x, y)
	return -3.0
end

MATLAB

function tmp = code(x, y)
	tmp = -3.0;
end

Wolfram

code[x_, y_] := -3.0

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in y around 0 76.8%

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

   1. unpow2 76.8%

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

4. Simplified 76.8%

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

6. Applied egg-rr 4.2%

   \[\leadsto \color{blue}{-3} \]

7. Final simplification 4.2%

   \[\leadsto -3 \]

Alternative 13: 4.7% 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%

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

2. Taylor expanded in y around 0 76.8%

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

   1. unpow2 76.8%

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

4. Simplified 76.8%

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

6. Applied egg-rr 4.6%

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

7. Final simplification 4.6%

   \[\leadsto -1 \]

Alternative 14: 4.6% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ -0.5 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 -0.5)

C

double code(double x, double y) {
	return -0.5;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = -0.5d0
end function

Java

public static double code(double x, double y) {
	return -0.5;
}

Python

def code(x, y):
	return -0.5

Julia

function code(x, y)
	return -0.5
end

MATLAB

function tmp = code(x, y)
	tmp = -0.5;
end

Wolfram

code[x_, y_] := -0.5

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in y around 0 76.8%

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

   1. unpow2 76.8%

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

4. Simplified 76.8%

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

6. Applied egg-rr 4.6%

   \[\leadsto \color{blue}{-0.5} \]

7. Final simplification 4.6%

   \[\leadsto -0.5 \]

Alternative 15: 26.3% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 x)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return x

Julia

function code(x, y)
	return x
end

MATLAB

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

Wolfram

code[x_, y_] := x

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in x around 0 39.4%

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

   1. associate-/l* 25.8%

      \[\leadsto 0.5 \cdot \color{blue}{\frac{e^{y} - \frac{1}{e^{y}}}{\frac{y}{x}}} \]

   2. associate-*r/ 25.8%

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

   3. rec-exp 25.8%

      \[\leadsto \frac{0.5 \cdot \left(e^{y} - \color{blue}{e^{-y}}\right)}{\frac{y}{x}} \]

4. Simplified 25.8%

   \[\leadsto \color{blue}{\frac{0.5 \cdot \left(e^{y} - e^{-y}\right)}{\frac{y}{x}}} \]

5. Taylor expanded in y around 0 27.0%

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

6. Final simplification 27.0%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023240 
(FPCore (x y)
  :name "Linear.Quaternion:$ccos from linear-1.19.1.3"
  :precision binary64
  (* (sin x) (/ (sinh y) y)))