Linear.Quaternion:$ccos from linear-1.19.1.3

Percentage Accurate: 100.0% → 100.0%

Time: 7.4s

Alternatives: 11

Speedup: 1.0×

• 50.4% 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: 92.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin x \cdot \left(0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ t_1 := \frac{x \cdot \sinh y}{y}\\ \mathbf{if}\;y \leq -1.65 \cdot 10^{+202}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -9.6:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{-5}:\\ \;\;\;\;\sin x\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sin x) (* 0.16666666666666666 (* y y))))
        (t_1 (/ (* x (sinh y)) y)))
   (if (<= y -1.65e+202)
     t_0
     (if (<= y -9.6)
       t_1
       (if (<= y 1.45e-5) (sin x) (if (<= y 1.35e+154) t_1 t_0))))))

C

double code(double x, double y) {
	double t_0 = sin(x) * (0.16666666666666666 * (y * y));
	double t_1 = (x * sinh(y)) / y;
	double tmp;
	if (y <= -1.65e+202) {
		tmp = t_0;
	} else if (y <= -9.6) {
		tmp = t_1;
	} else if (y <= 1.45e-5) {
		tmp = sin(x);
	} else if (y <= 1.35e+154) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_0 = sin(x) * (0.16666666666666666d0 * (y * y))
    t_1 = (x * sinh(y)) / y
    if (y <= (-1.65d+202)) then
        tmp = t_0
    else if (y <= (-9.6d0)) then
        tmp = t_1
    else if (y <= 1.45d-5) then
        tmp = sin(x)
    else if (y <= 1.35d+154) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sin(x) * (0.16666666666666666 * (y * y));
	double t_1 = (x * Math.sinh(y)) / y;
	double tmp;
	if (y <= -1.65e+202) {
		tmp = t_0;
	} else if (y <= -9.6) {
		tmp = t_1;
	} else if (y <= 1.45e-5) {
		tmp = Math.sin(x);
	} else if (y <= 1.35e+154) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sin(x) * (0.16666666666666666 * (y * y))
	t_1 = (x * math.sinh(y)) / y
	tmp = 0
	if y <= -1.65e+202:
		tmp = t_0
	elif y <= -9.6:
		tmp = t_1
	elif y <= 1.45e-5:
		tmp = math.sin(x)
	elif y <= 1.35e+154:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sin(x) * Float64(0.16666666666666666 * Float64(y * y)))
	t_1 = Float64(Float64(x * sinh(y)) / y)
	tmp = 0.0
	if (y <= -1.65e+202)
		tmp = t_0;
	elseif (y <= -9.6)
		tmp = t_1;
	elseif (y <= 1.45e-5)
		tmp = sin(x);
	elseif (y <= 1.35e+154)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sin(x) * (0.16666666666666666 * (y * y));
	t_1 = (x * sinh(y)) / y;
	tmp = 0.0;
	if (y <= -1.65e+202)
		tmp = t_0;
	elseif (y <= -9.6)
		tmp = t_1;
	elseif (y <= 1.45e-5)
		tmp = sin(x);
	elseif (y <= 1.35e+154)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[x], $MachinePrecision] * N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[y, -1.65e+202], t$95$0, If[LessEqual[y, -9.6], t$95$1, If[LessEqual[y, 1.45e-5], N[Sin[x], $MachinePrecision], If[LessEqual[y, 1.35e+154], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.6499999999999999e202 or 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%

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

      2. associate-*r* 100.0%

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

      3. *-commutative 100.0%

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

   7. Simplified 100.0%

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

   if -1.6499999999999999e202 < y < -9.59999999999999964 or 1.45e-5 < y < 1.35000000000000003e154

   1. Initial program 100.0%

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

      1. add-log-exp 98.8%

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

      2. *-un-lft-identity 98.8%

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

      3. log-prod 98.8%

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

      4. metadata-eval 98.8%

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

      5. add-log-exp 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-*l/ 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. +-lft-identity 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*l/ 93.2%

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

      4. *-commutative 93.2%

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

   5. Simplified 93.2%

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

   6. Taylor expanded in x around 0 76.7%

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

      1. *-commutative 76.7%

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

      2. associate-*l/ 83.5%

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

   8. Applied egg-rr 83.5%

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

   if -9.59999999999999964 < y < 1.45e-5

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 99.3%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.65 \cdot 10^{+202}:\\ \;\;\;\;\sin x \cdot \left(0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \mathbf{elif}\;y \leq -9.6:\\ \;\;\;\;\frac{x \cdot \sinh y}{y}\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{-5}:\\ \;\;\;\;\sin x\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{x \cdot \sinh y}{y}\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \left(0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \end{array} \]

Alternative 3: 92.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.16666666666666666 \cdot \left(y \cdot y\right)\\ t_1 := \sin x \cdot t_0\\ t_2 := \frac{x \cdot \sinh y}{y}\\ \mathbf{if}\;y \leq -1.65 \cdot 10^{+202}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -9.6:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 29000000:\\ \;\;\;\;\sin x \cdot \left(t_0 + 1\right)\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+153}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* 0.16666666666666666 (* y y)))
        (t_1 (* (sin x) t_0))
        (t_2 (/ (* x (sinh y)) y)))
   (if (<= y -1.65e+202)
     t_1
     (if (<= y -9.6)
       t_2
       (if (<= y 29000000.0)
         (* (sin x) (+ t_0 1.0))
         (if (<= y 5e+153) t_2 t_1))))))

C

double code(double x, double y) {
	double t_0 = 0.16666666666666666 * (y * y);
	double t_1 = sin(x) * t_0;
	double t_2 = (x * sinh(y)) / y;
	double tmp;
	if (y <= -1.65e+202) {
		tmp = t_1;
	} else if (y <= -9.6) {
		tmp = t_2;
	} else if (y <= 29000000.0) {
		tmp = sin(x) * (t_0 + 1.0);
	} else if (y <= 5e+153) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	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) :: t_2
    real(8) :: tmp
    t_0 = 0.16666666666666666d0 * (y * y)
    t_1 = sin(x) * t_0
    t_2 = (x * sinh(y)) / y
    if (y <= (-1.65d+202)) then
        tmp = t_1
    else if (y <= (-9.6d0)) then
        tmp = t_2
    else if (y <= 29000000.0d0) then
        tmp = sin(x) * (t_0 + 1.0d0)
    else if (y <= 5d+153) then
        tmp = t_2
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 0.16666666666666666 * (y * y);
	double t_1 = Math.sin(x) * t_0;
	double t_2 = (x * Math.sinh(y)) / y;
	double tmp;
	if (y <= -1.65e+202) {
		tmp = t_1;
	} else if (y <= -9.6) {
		tmp = t_2;
	} else if (y <= 29000000.0) {
		tmp = Math.sin(x) * (t_0 + 1.0);
	} else if (y <= 5e+153) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 0.16666666666666666 * (y * y)
	t_1 = math.sin(x) * t_0
	t_2 = (x * math.sinh(y)) / y
	tmp = 0
	if y <= -1.65e+202:
		tmp = t_1
	elif y <= -9.6:
		tmp = t_2
	elif y <= 29000000.0:
		tmp = math.sin(x) * (t_0 + 1.0)
	elif y <= 5e+153:
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(0.16666666666666666 * Float64(y * y))
	t_1 = Float64(sin(x) * t_0)
	t_2 = Float64(Float64(x * sinh(y)) / y)
	tmp = 0.0
	if (y <= -1.65e+202)
		tmp = t_1;
	elseif (y <= -9.6)
		tmp = t_2;
	elseif (y <= 29000000.0)
		tmp = Float64(sin(x) * Float64(t_0 + 1.0));
	elseif (y <= 5e+153)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 0.16666666666666666 * (y * y);
	t_1 = sin(x) * t_0;
	t_2 = (x * sinh(y)) / y;
	tmp = 0.0;
	if (y <= -1.65e+202)
		tmp = t_1;
	elseif (y <= -9.6)
		tmp = t_2;
	elseif (y <= 29000000.0)
		tmp = sin(x) * (t_0 + 1.0);
	elseif (y <= 5e+153)
		tmp = t_2;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[x], $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[y, -1.65e+202], t$95$1, If[LessEqual[y, -9.6], t$95$2, If[LessEqual[y, 29000000.0], N[(N[Sin[x], $MachinePrecision] * N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5e+153], t$95$2, t$95$1]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.6499999999999999e202 or 5.00000000000000018e153 < 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%

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

      2. associate-*r* 100.0%

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

      3. *-commutative 100.0%

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

   7. Simplified 100.0%

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

   if -1.6499999999999999e202 < y < -9.59999999999999964 or 2.9e7 < y < 5.00000000000000018e153

   1. Initial program 100.0%

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

      1. add-log-exp 100.0%

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

      2. *-un-lft-identity 100.0%

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

      3. log-prod 100.0%

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

      4. metadata-eval 100.0%

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

      5. add-log-exp 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-*l/ 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. +-lft-identity 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*l/ 93.0%

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

      4. *-commutative 93.0%

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

   5. Simplified 93.0%

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

   6. Taylor expanded in x around 0 77.5%

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

      1. *-commutative 77.5%

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

      2. associate-*l/ 84.5%

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

   8. Applied egg-rr 84.5%

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

   if -9.59999999999999964 < y < 2.9e7

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 98.6%

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

      1. unpow2 98.6%

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

   4. Simplified 98.6%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.65 \cdot 10^{+202}:\\ \;\;\;\;\sin x \cdot \left(0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \mathbf{elif}\;y \leq -9.6:\\ \;\;\;\;\frac{x \cdot \sinh y}{y}\\ \mathbf{elif}\;y \leq 29000000:\\ \;\;\;\;\sin x \cdot \left(0.16666666666666666 \cdot \left(y \cdot y\right) + 1\right)\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+153}:\\ \;\;\;\;\frac{x \cdot \sinh y}{y}\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \left(0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \end{array} \]

Alternative 4: 84.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \sinh y}{y}\\ \mathbf{if}\;y \leq -9.6:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.42 \cdot 10^{-5}:\\ \;\;\;\;\sin x\\ \mathbf{elif}\;y \leq 2.05 \cdot 10^{+201} \lor \neg \left(y \leq 3 \cdot 10^{+237}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-0.16666666666666666 \cdot {x}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (sinh y)) y)))
   (if (<= y -9.6)
     t_0
     (if (<= y 1.42e-5)
       (sin x)
       (if (or (<= y 2.05e+201) (not (<= y 3e+237)))
         t_0
         (* -0.16666666666666666 (pow x 3.0)))))))

C

double code(double x, double y) {
	double t_0 = (x * sinh(y)) / y;
	double tmp;
	if (y <= -9.6) {
		tmp = t_0;
	} else if (y <= 1.42e-5) {
		tmp = sin(x);
	} else if ((y <= 2.05e+201) || !(y <= 3e+237)) {
		tmp = t_0;
	} else {
		tmp = -0.16666666666666666 * pow(x, 3.0);
	}
	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 = (x * sinh(y)) / y
    if (y <= (-9.6d0)) then
        tmp = t_0
    else if (y <= 1.42d-5) then
        tmp = sin(x)
    else if ((y <= 2.05d+201) .or. (.not. (y <= 3d+237))) then
        tmp = t_0
    else
        tmp = (-0.16666666666666666d0) * (x ** 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x * Math.sinh(y)) / y;
	double tmp;
	if (y <= -9.6) {
		tmp = t_0;
	} else if (y <= 1.42e-5) {
		tmp = Math.sin(x);
	} else if ((y <= 2.05e+201) || !(y <= 3e+237)) {
		tmp = t_0;
	} else {
		tmp = -0.16666666666666666 * Math.pow(x, 3.0);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x * math.sinh(y)) / y
	tmp = 0
	if y <= -9.6:
		tmp = t_0
	elif y <= 1.42e-5:
		tmp = math.sin(x)
	elif (y <= 2.05e+201) or not (y <= 3e+237):
		tmp = t_0
	else:
		tmp = -0.16666666666666666 * math.pow(x, 3.0)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x * sinh(y)) / y)
	tmp = 0.0
	if (y <= -9.6)
		tmp = t_0;
	elseif (y <= 1.42e-5)
		tmp = sin(x);
	elseif ((y <= 2.05e+201) || !(y <= 3e+237))
		tmp = t_0;
	else
		tmp = Float64(-0.16666666666666666 * (x ^ 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x * sinh(y)) / y;
	tmp = 0.0;
	if (y <= -9.6)
		tmp = t_0;
	elseif (y <= 1.42e-5)
		tmp = sin(x);
	elseif ((y <= 2.05e+201) || ~((y <= 3e+237)))
		tmp = t_0;
	else
		tmp = -0.16666666666666666 * (x ^ 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[y, -9.6], t$95$0, If[LessEqual[y, 1.42e-5], N[Sin[x], $MachinePrecision], If[Or[LessEqual[y, 2.05e+201], N[Not[LessEqual[y, 3e+237]], $MachinePrecision]], t$95$0, N[(-0.16666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -9.59999999999999964 or 1.42e-5 < y < 2.0500000000000001e201 or 3e237 < y

   1. Initial program 100.0%

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

      1. add-log-exp 99.3%

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

      2. *-un-lft-identity 99.3%

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

      3. log-prod 99.3%

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

      4. metadata-eval 99.3%

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

      5. add-log-exp 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-*l/ 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. +-lft-identity 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*l/ 81.5%

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

      4. *-commutative 81.5%

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

   5. Simplified 81.5%

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

   6. Taylor expanded in x around 0 57.3%

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

      1. *-commutative 57.3%

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

      2. associate-*l/ 75.8%

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

   8. Applied egg-rr 75.8%

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

   if -9.59999999999999964 < y < 1.42e-5

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 99.3%

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

   if 2.0500000000000001e201 < y < 3e237

   1. Initial program 100.0%

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

      1. add-log-exp 100.0%

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

      2. *-un-lft-identity 100.0%

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

      3. log-prod 100.0%

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

      4. metadata-eval 100.0%

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

      5. add-log-exp 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-*l/ 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. +-lft-identity 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*l/ 100.0%

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

      4. *-commutative 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around 0 3.1%

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

   7. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot {x}^{3} + x} \]

   8. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot {x}^{3}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.6:\\ \;\;\;\;\frac{x \cdot \sinh y}{y}\\ \mathbf{elif}\;y \leq 1.42 \cdot 10^{-5}:\\ \;\;\;\;\sin x\\ \mathbf{elif}\;y \leq 2.05 \cdot 10^{+201} \lor \neg \left(y \leq 3 \cdot 10^{+237}\right):\\ \;\;\;\;\frac{x \cdot \sinh y}{y}\\ \mathbf{else}:\\ \;\;\;\;-0.16666666666666666 \cdot {x}^{3}\\ \end{array} \]

Alternative 5: 83.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.16666666666666666 \cdot \left(x \cdot \left(y \cdot y\right)\right)\\ t_1 := \sinh y \cdot \frac{x}{y}\\ \mathbf{if}\;y \leq -1.42 \cdot 10^{+190}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -9.6:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{-5}:\\ \;\;\;\;\sin x\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* 0.16666666666666666 (* x (* y y)))) (t_1 (* (sinh y) (/ x y))))
   (if (<= y -1.42e+190)
     t_0
     (if (<= y -9.6)
       t_1
       (if (<= y 1.45e-5) (sin x) (if (<= y 1.35e+154) t_1 t_0))))))

C

double code(double x, double y) {
	double t_0 = 0.16666666666666666 * (x * (y * y));
	double t_1 = sinh(y) * (x / y);
	double tmp;
	if (y <= -1.42e+190) {
		tmp = t_0;
	} else if (y <= -9.6) {
		tmp = t_1;
	} else if (y <= 1.45e-5) {
		tmp = sin(x);
	} else if (y <= 1.35e+154) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_0 = 0.16666666666666666d0 * (x * (y * y))
    t_1 = sinh(y) * (x / y)
    if (y <= (-1.42d+190)) then
        tmp = t_0
    else if (y <= (-9.6d0)) then
        tmp = t_1
    else if (y <= 1.45d-5) then
        tmp = sin(x)
    else if (y <= 1.35d+154) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 0.16666666666666666 * (x * (y * y));
	double t_1 = Math.sinh(y) * (x / y);
	double tmp;
	if (y <= -1.42e+190) {
		tmp = t_0;
	} else if (y <= -9.6) {
		tmp = t_1;
	} else if (y <= 1.45e-5) {
		tmp = Math.sin(x);
	} else if (y <= 1.35e+154) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 0.16666666666666666 * (x * (y * y))
	t_1 = math.sinh(y) * (x / y)
	tmp = 0
	if y <= -1.42e+190:
		tmp = t_0
	elif y <= -9.6:
		tmp = t_1
	elif y <= 1.45e-5:
		tmp = math.sin(x)
	elif y <= 1.35e+154:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(0.16666666666666666 * Float64(x * Float64(y * y)))
	t_1 = Float64(sinh(y) * Float64(x / y))
	tmp = 0.0
	if (y <= -1.42e+190)
		tmp = t_0;
	elseif (y <= -9.6)
		tmp = t_1;
	elseif (y <= 1.45e-5)
		tmp = sin(x);
	elseif (y <= 1.35e+154)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 0.16666666666666666 * (x * (y * y));
	t_1 = sinh(y) * (x / y);
	tmp = 0.0;
	if (y <= -1.42e+190)
		tmp = t_0;
	elseif (y <= -9.6)
		tmp = t_1;
	elseif (y <= 1.45e-5)
		tmp = sin(x);
	elseif (y <= 1.35e+154)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(0.16666666666666666 * N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sinh[y], $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.42e+190], t$95$0, If[LessEqual[y, -9.6], t$95$1, If[LessEqual[y, 1.45e-5], N[Sin[x], $MachinePrecision], If[LessEqual[y, 1.35e+154], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.42e190 or 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%

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

      2. associate-*r* 100.0%

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

      3. *-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around 0 59.6%

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

      1. *-commutative 59.6%

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

      2. unpow2 59.6%

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

   10. Simplified 59.6%

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

   if -1.42e190 < y < -9.59999999999999964 or 1.45e-5 < y < 1.35000000000000003e154

   1. Initial program 100.0%

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

      1. add-log-exp 98.8%

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

      2. *-un-lft-identity 98.8%

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

      3. log-prod 98.8%

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

      4. metadata-eval 98.8%

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

      5. add-log-exp 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-*l/ 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. +-lft-identity 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*l/ 94.4%

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

      4. *-commutative 94.4%

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

   5. Simplified 94.4%

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

   6. Taylor expanded in x around 0 77.8%

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

   if -9.59999999999999964 < y < 1.45e-5

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 99.3%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.42 \cdot 10^{+190}:\\ \;\;\;\;0.16666666666666666 \cdot \left(x \cdot \left(y \cdot y\right)\right)\\ \mathbf{elif}\;y \leq -9.6:\\ \;\;\;\;\sinh y \cdot \frac{x}{y}\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{-5}:\\ \;\;\;\;\sin x\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\sinh y \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(x \cdot \left(y \cdot y\right)\right)\\ \end{array} \]

Alternative 6: 67.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.16666666666666666 \cdot \left(x \cdot \left(y \cdot y\right)\right)\\ \mathbf{if}\;y \leq -3000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{+31}:\\ \;\;\;\;\sin x\\ \mathbf{elif}\;y \leq 3 \cdot 10^{+237}:\\ \;\;\;\;-0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* 0.16666666666666666 (* x (* y y)))))
   (if (<= y -3000.0)
     t_0
     (if (<= y 1.25e+31)
       (sin x)
       (if (<= y 3e+237) (* -0.16666666666666666 (pow x 3.0)) t_0)))))

C

double code(double x, double y) {
	double t_0 = 0.16666666666666666 * (x * (y * y));
	double tmp;
	if (y <= -3000.0) {
		tmp = t_0;
	} else if (y <= 1.25e+31) {
		tmp = sin(x);
	} else if (y <= 3e+237) {
		tmp = -0.16666666666666666 * pow(x, 3.0);
	} 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 = 0.16666666666666666d0 * (x * (y * y))
    if (y <= (-3000.0d0)) then
        tmp = t_0
    else if (y <= 1.25d+31) then
        tmp = sin(x)
    else if (y <= 3d+237) then
        tmp = (-0.16666666666666666d0) * (x ** 3.0d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 0.16666666666666666 * (x * (y * y));
	double tmp;
	if (y <= -3000.0) {
		tmp = t_0;
	} else if (y <= 1.25e+31) {
		tmp = Math.sin(x);
	} else if (y <= 3e+237) {
		tmp = -0.16666666666666666 * Math.pow(x, 3.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 0.16666666666666666 * (x * (y * y))
	tmp = 0
	if y <= -3000.0:
		tmp = t_0
	elif y <= 1.25e+31:
		tmp = math.sin(x)
	elif y <= 3e+237:
		tmp = -0.16666666666666666 * math.pow(x, 3.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(0.16666666666666666 * Float64(x * Float64(y * y)))
	tmp = 0.0
	if (y <= -3000.0)
		tmp = t_0;
	elseif (y <= 1.25e+31)
		tmp = sin(x);
	elseif (y <= 3e+237)
		tmp = Float64(-0.16666666666666666 * (x ^ 3.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 0.16666666666666666 * (x * (y * y));
	tmp = 0.0;
	if (y <= -3000.0)
		tmp = t_0;
	elseif (y <= 1.25e+31)
		tmp = sin(x);
	elseif (y <= 3e+237)
		tmp = -0.16666666666666666 * (x ^ 3.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(0.16666666666666666 * N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -3000.0], t$95$0, If[LessEqual[y, 1.25e+31], N[Sin[x], $MachinePrecision], If[LessEqual[y, 3e+237], N[(-0.16666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3e3 or 3e237 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 62.8%

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

      1. unpow2 62.8%

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

   4. Simplified 62.8%

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

   5. Taylor expanded in y around inf 62.8%

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

      1. unpow2 62.8%

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

      2. associate-*r* 62.8%

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

      3. *-commutative 62.8%

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

   7. Simplified 62.8%

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

   8. Taylor expanded in x around 0 47.1%

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

      1. *-commutative 47.1%

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

      2. unpow2 47.1%

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

   10. Simplified 47.1%

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

   if -3e3 < y < 1.25000000000000007e31

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 92.1%

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

   if 1.25000000000000007e31 < y < 3e237

   1. Initial program 100.0%

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

      1. add-log-exp 100.0%

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

      2. *-un-lft-identity 100.0%

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

      3. log-prod 100.0%

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

      4. metadata-eval 100.0%

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

      5. add-log-exp 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-*l/ 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. +-lft-identity 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*l/ 87.9%

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

      4. *-commutative 87.9%

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

   5. Simplified 87.9%

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

   6. Taylor expanded in y around 0 2.7%

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

   7. Taylor expanded in x around 0 31.7%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot {x}^{3} + x} \]

   8. Taylor expanded in x around inf 31.3%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot {x}^{3}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3000:\\ \;\;\;\;0.16666666666666666 \cdot \left(x \cdot \left(y \cdot y\right)\right)\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{+31}:\\ \;\;\;\;\sin x\\ \mathbf{elif}\;y \leq 3 \cdot 10^{+237}:\\ \;\;\;\;-0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(x \cdot \left(y \cdot y\right)\right)\\ \end{array} \]

Alternative 7: 71.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3100 \lor \neg \left(y \leq 17200000000000\right):\\ \;\;\;\;0.16666666666666666 \cdot \left(x \cdot \left(y \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -3100.0) (not (<= y 17200000000000.0)))
   (* 0.16666666666666666 (* x (* y y)))
   (sin x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -3100.0) || !(y <= 17200000000000.0)) {
		tmp = 0.16666666666666666 * (x * (y * y));
	} else {
		tmp = sin(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 <= (-3100.0d0)) .or. (.not. (y <= 17200000000000.0d0))) then
        tmp = 0.16666666666666666d0 * (x * (y * y))
    else
        tmp = sin(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -3100.0) || !(y <= 17200000000000.0)) {
		tmp = 0.16666666666666666 * (x * (y * y));
	} else {
		tmp = Math.sin(x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -3100.0) or not (y <= 17200000000000.0):
		tmp = 0.16666666666666666 * (x * (y * y))
	else:
		tmp = math.sin(x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -3100.0) || !(y <= 17200000000000.0))
		tmp = Float64(0.16666666666666666 * Float64(x * Float64(y * y)));
	else
		tmp = sin(x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -3100.0) || ~((y <= 17200000000000.0)))
		tmp = 0.16666666666666666 * (x * (y * y));
	else
		tmp = sin(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -3100.0], N[Not[LessEqual[y, 17200000000000.0]], $MachinePrecision]], N[(0.16666666666666666 * N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[x], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3100 or 1.72e13 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 50.9%

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

      1. unpow2 50.9%

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

   4. Simplified 50.9%

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

   5. Taylor expanded in y around inf 50.9%

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

      1. unpow2 50.9%

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

      2. associate-*r* 50.9%

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

      3. *-commutative 50.9%

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

   7. Simplified 50.9%

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

   8. Taylor expanded in x around 0 36.2%

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

      1. *-commutative 36.2%

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

      2. unpow2 36.2%

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

   10. Simplified 36.2%

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

   if -3100 < y < 1.72e13

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 98.4%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3100 \lor \neg \left(y \leq 17200000000000\right):\\ \;\;\;\;0.16666666666666666 \cdot \left(x \cdot \left(y \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin x\\ \end{array} \]

Alternative 8: 48.1% accurate, 13.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -54 \lor \neg \left(y \leq 85000000\right):\\ \;\;\;\;0.16666666666666666 \cdot \left(x \cdot \left(y \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{y}{x} + 0.16666666666666666 \cdot \left(x \cdot y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -54.0) (not (<= y 85000000.0)))
   (* 0.16666666666666666 (* x (* y y)))
   (/ y (+ (/ y x) (* 0.16666666666666666 (* x y))))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -54.0) || !(y <= 85000000.0)) {
		tmp = 0.16666666666666666 * (x * (y * y));
	} else {
		tmp = y / ((y / x) + (0.16666666666666666 * (x * 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 <= (-54.0d0)) .or. (.not. (y <= 85000000.0d0))) then
        tmp = 0.16666666666666666d0 * (x * (y * y))
    else
        tmp = y / ((y / x) + (0.16666666666666666d0 * (x * y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -54.0) || !(y <= 85000000.0)) {
		tmp = 0.16666666666666666 * (x * (y * y));
	} else {
		tmp = y / ((y / x) + (0.16666666666666666 * (x * y)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -54.0) or not (y <= 85000000.0):
		tmp = 0.16666666666666666 * (x * (y * y))
	else:
		tmp = y / ((y / x) + (0.16666666666666666 * (x * y)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -54.0) || !(y <= 85000000.0))
		tmp = Float64(0.16666666666666666 * Float64(x * Float64(y * y)));
	else
		tmp = Float64(y / Float64(Float64(y / x) + Float64(0.16666666666666666 * Float64(x * y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -54.0) || ~((y <= 85000000.0)))
		tmp = 0.16666666666666666 * (x * (y * y));
	else
		tmp = y / ((y / x) + (0.16666666666666666 * (x * y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -54.0], N[Not[LessEqual[y, 85000000.0]], $MachinePrecision]], N[(0.16666666666666666 * N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y / N[(N[(y / x), $MachinePrecision] + N[(0.16666666666666666 * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -54 or 8.5e7 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 50.9%

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

      1. unpow2 50.9%

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

   4. Simplified 50.9%

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

   5. Taylor expanded in y around inf 50.9%

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

      1. unpow2 50.9%

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

      2. associate-*r* 50.9%

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

      3. *-commutative 50.9%

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

   7. Simplified 50.9%

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

   8. Taylor expanded in x around 0 36.2%

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

      1. *-commutative 36.2%

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

      2. unpow2 36.2%

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

   10. Simplified 36.2%

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

   if -54 < y < 8.5e7

   1. Initial program 100.0%

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

      1. add-log-exp 58.7%

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

      2. *-un-lft-identity 58.7%

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

      3. log-prod 58.7%

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

      4. metadata-eval 58.7%

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

      5. add-log-exp 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-*l/ 85.8%

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

   3. Applied egg-rr 85.8%

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

      1. +-lft-identity 85.8%

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

      2. *-commutative 85.8%

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

      3. associate-*l/ 99.8%

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

      4. *-commutative 99.8%

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

   5. Simplified 99.8%

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

   6. Taylor expanded in y around 0 98.2%

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

      1. clear-num 97.9%

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

      2. un-div-inv 98.3%

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

   8. Applied egg-rr 98.3%

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

   9. Taylor expanded in x around 0 47.5%

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

4. Final simplification 41.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -54 \lor \neg \left(y \leq 85000000\right):\\ \;\;\;\;0.16666666666666666 \cdot \left(x \cdot \left(y \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{y}{x} + 0.16666666666666666 \cdot \left(x \cdot y\right)}\\ \end{array} \]

Alternative 9: 47.7% accurate, 18.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.12 \cdot 10^{-8} \lor \neg \left(y \leq 29000000\right):\\ \;\;\;\;0.16666666666666666 \cdot \left(x \cdot \left(y \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.12e-8) (not (<= y 29000000.0)))
   (* 0.16666666666666666 (* x (* y y)))
   x))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.12e-8) || !(y <= 29000000.0)) {
		tmp = 0.16666666666666666 * (x * (y * y));
	} else {
		tmp = 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 <= (-1.12d-8)) .or. (.not. (y <= 29000000.0d0))) then
        tmp = 0.16666666666666666d0 * (x * (y * y))
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.12e-8) || !(y <= 29000000.0)) {
		tmp = 0.16666666666666666 * (x * (y * y));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.12e-8) or not (y <= 29000000.0):
		tmp = 0.16666666666666666 * (x * (y * y))
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.12e-8) || !(y <= 29000000.0))
		tmp = Float64(0.16666666666666666 * Float64(x * Float64(y * y)));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.12e-8) || ~((y <= 29000000.0)))
		tmp = 0.16666666666666666 * (x * (y * y));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.12e-8], N[Not[LessEqual[y, 29000000.0]], $MachinePrecision]], N[(0.16666666666666666 * N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if y < -1.11999999999999994e-8 or 2.9e7 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 51.0%

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

      1. unpow2 51.0%

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

   4. Simplified 51.0%

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

   5. Taylor expanded in y around inf 50.3%

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

      1. unpow2 50.3%

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

      2. associate-*r* 50.3%

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

      3. *-commutative 50.3%

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

   7. Simplified 50.3%

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

   8. Taylor expanded in x around 0 35.7%

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

      1. *-commutative 35.7%

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

      2. unpow2 35.7%

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

   10. Simplified 35.7%

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

   if -1.11999999999999994e-8 < y < 2.9e7

   1. Initial program 100.0%

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

      1. add-log-exp 58.1%

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

      2. *-un-lft-identity 58.1%

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

      3. log-prod 58.1%

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

      4. metadata-eval 58.1%

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

      5. add-log-exp 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-*l/ 85.5%

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

   3. Applied egg-rr 85.5%

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

      1. +-lft-identity 85.5%

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

      2. *-commutative 85.5%

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

      3. associate-*l/ 99.8%

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

      4. *-commutative 99.8%

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

   5. Simplified 99.8%

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

   6. Taylor expanded in y around 0 98.8%

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

   7. Taylor expanded in x around 0 47.0%

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

4. Final simplification 41.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.12 \cdot 10^{-8} \lor \neg \left(y \leq 29000000\right):\\ \;\;\;\;0.16666666666666666 \cdot \left(x \cdot \left(y \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 10: 28.9% accurate, 29.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{+61}:\\ \;\;\;\;\frac{x \cdot y}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x -7.2e+61) (/ (* x y) y) x))

C

double code(double x, double y) {
	double tmp;
	if (x <= -7.2e+61) {
		tmp = (x * y) / y;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-7.2d+61)) then
        tmp = (x * y) / y
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -7.2e+61) {
		tmp = (x * y) / y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -7.2e+61:
		tmp = (x * y) / y
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -7.2e+61)
		tmp = Float64(Float64(x * y) / y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -7.2e+61)
		tmp = (x * y) / y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -7.2e+61], N[(N[(x * y), $MachinePrecision] / y), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if x < -7.20000000000000021e61

   1. Initial program 100.0%

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

      1. add-log-exp 99.4%

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

      2. *-un-lft-identity 99.4%

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

      3. log-prod 99.4%

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

      4. metadata-eval 99.4%

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

      5. add-log-exp 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-*l/ 99.9%

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

   3. Applied egg-rr 99.9%

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

      1. +-lft-identity 99.9%

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

      2. *-commutative 99.9%

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

      3. associate-*l/ 99.9%

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

      4. *-commutative 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in y around 0 56.9%

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

   7. Taylor expanded in x around 0 2.8%

      \[\leadsto y \cdot \color{blue}{\frac{x}{y}} \]
   8. Step-by-step derivation

      1. associate-*r/ 14.6%

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

      2. *-commutative 14.6%

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

   9. Applied egg-rr 14.6%

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

   if -7.20000000000000021e61 < x

   1. Initial program 100.0%

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

      1. add-log-exp 72.4%

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

      2. *-un-lft-identity 72.4%

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

      3. log-prod 72.4%

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

      4. metadata-eval 72.4%

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

      5. add-log-exp 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-*l/ 90.4%

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

   3. Applied egg-rr 90.4%

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

      1. +-lft-identity 90.4%

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

      2. *-commutative 90.4%

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

      3. associate-*l/ 87.8%

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

      4. *-commutative 87.8%

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

   5. Simplified 87.8%

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

   6. Taylor expanded in y around 0 48.7%

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

   7. Taylor expanded in x around 0 31.8%

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

4. Final simplification 27.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{+61}:\\ \;\;\;\;\frac{x \cdot y}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 11: 26.1% 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. Step-by-step derivation

   1. add-log-exp 79.2%

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

   2. *-un-lft-identity 79.2%

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

   3. log-prod 79.2%

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

   4. metadata-eval 79.2%

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

   5. add-log-exp 100.0%

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

   6. *-commutative 100.0%

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

   7. associate-*l/ 92.8%

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

3. Applied egg-rr 92.8%

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

   1. +-lft-identity 92.8%

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

   2. *-commutative 92.8%

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

   3. associate-*l/ 90.9%

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

   4. *-commutative 90.9%

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

5. Simplified 90.9%

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

6. Taylor expanded in y around 0 50.8%

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

7. Taylor expanded in x around 0 24.5%

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

8. Final simplification 24.5%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023192 
(FPCore (x y)
  :name "Linear.Quaternion:$ccos from linear-1.19.1.3"
  :precision binary64
  (* (sin x) (/ (sinh y) y)))