Linear.Quaternion:$ccosh from linear-1.19.1.3

Percentage Accurate: 88.7% → 99.8%

Time: 9.5s

Alternatives: 7

Speedup: 1.0×

• 48.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.1%

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

   1. associate-/l* 99.9%

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

3. Simplified 99.9%

   \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
4. Add Preprocessing

5. Final simplification 99.9%

   \[\leadsto \sin x \cdot \frac{\sinh y}{x} \]
6. Add Preprocessing

Alternative 2: 70.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sinh y \leq 0.005:\\ \;\;\;\;y \cdot \frac{\sin x}{x}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sinh y) 0.005) (* y (/ (sin x) x)) (sinh y)))

C

double code(double x, double y) {
	double tmp;
	if (sinh(y) <= 0.005) {
		tmp = y * (sin(x) / x);
	} else {
		tmp = sinh(y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (sinh(y) <= 0.005d0) then
        tmp = y * (sin(x) / x)
    else
        tmp = sinh(y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (Math.sinh(y) <= 0.005) {
		tmp = y * (Math.sin(x) / x);
	} else {
		tmp = Math.sinh(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.sinh(y) <= 0.005:
		tmp = y * (math.sin(x) / x)
	else:
		tmp = math.sinh(y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (sinh(y) <= 0.005)
		tmp = Float64(y * Float64(sin(x) / x));
	else
		tmp = sinh(y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (sinh(y) <= 0.005)
		tmp = y * (sin(x) / x);
	else
		tmp = sinh(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sinh[y], $MachinePrecision], 0.005], N[(y * N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[Sinh[y], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sinh.f64 y) < 0.0050000000000000001

   1. Initial program 81.8%

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

      1. associate-/l* 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 46.1%

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

      1. associate-/l* 64.2%

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

   7. Simplified 64.2%

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

   if 0.0050000000000000001 < (sinh.f64 y)

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. add-sqr-sqrt 52.5%

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

      3. times-frac 52.5%

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

   4. Applied egg-rr 52.5%

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

      1. frac-times 52.5%

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

      2. *-commutative 52.5%

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

      3. add-sqr-sqrt 100.0%

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

      4. associate-*l/ 100.0%

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

      5. associate-/r/ 100.0%

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

      6. div-inv 100.0%

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

      7. associate-/r* 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0 75.4%

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

      1. remove-double-div 75.4%

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

      2. expm1-log1p-u 75.4%

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

      3. expm1-undefine 75.4%

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

   9. Applied egg-rr 75.4%

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

       1. sub-neg 75.4%

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

       2. metadata-eval 75.4%

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

       3. +-commutative 75.4%

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

       4. log1p-undefine 75.4%

          \[\leadsto -1 + e^{\color{blue}{\log \left(1 + \sinh y\right)}} \]

       5. rem-exp-log 75.4%

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

       6. associate-+r+ 75.4%

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

       7. metadata-eval 75.4%

          \[\leadsto \color{blue}{0} + \sinh y \]

       8. +-lft-identity 75.4%

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

   11. Simplified 75.4%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sinh y \leq 0.005:\\ \;\;\;\;y \cdot \frac{\sin x}{x}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 63.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sinh y \leq 10^{-88}:\\ \;\;\;\;x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sinh y) 1e-88) (* x (/ y x)) (sinh y)))

C

double code(double x, double y) {
	double tmp;
	if (sinh(y) <= 1e-88) {
		tmp = x * (y / x);
	} else {
		tmp = sinh(y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (sinh(y) <= 1d-88) then
        tmp = x * (y / x)
    else
        tmp = sinh(y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (Math.sinh(y) <= 1e-88) {
		tmp = x * (y / x);
	} else {
		tmp = Math.sinh(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.sinh(y) <= 1e-88:
		tmp = x * (y / x)
	else:
		tmp = math.sinh(y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (sinh(y) <= 1e-88)
		tmp = Float64(x * Float64(y / x));
	else
		tmp = sinh(y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (sinh(y) <= 1e-88)
		tmp = x * (y / x);
	else
		tmp = sinh(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sinh[y], $MachinePrecision], 1e-88], N[(x * N[(y / x), $MachinePrecision]), $MachinePrecision], N[Sinh[y], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sinh.f64 y) < 9.99999999999999934e-89

   1. Initial program 81.2%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 43.2%

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

   4. Taylor expanded in x around 0 23.4%

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

      1. associate-/l* 58.4%

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

      2. *-commutative 58.4%

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

   6. Applied egg-rr 58.4%

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

   if 9.99999999999999934e-89 < (sinh.f64 y)

   1. Initial program 98.6%

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

      1. *-commutative 98.6%

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

      2. add-sqr-sqrt 53.3%

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

      3. times-frac 53.3%

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

   4. Applied egg-rr 53.3%

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

      1. frac-times 53.3%

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

      2. *-commutative 53.3%

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

      3. add-sqr-sqrt 98.6%

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

      4. associate-*l/ 100.0%

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

      5. associate-/r/ 99.9%

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

      6. div-inv 99.9%

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

      7. associate-/r* 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in x around 0 74.0%

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

      1. remove-double-div 74.0%

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

      2. expm1-log1p-u 74.0%

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

      3. expm1-undefine 64.9%

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

   9. Applied egg-rr 64.9%

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

       1. sub-neg 64.9%

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

       2. metadata-eval 64.9%

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

       3. +-commutative 64.9%

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

       4. log1p-undefine 64.9%

          \[\leadsto -1 + e^{\color{blue}{\log \left(1 + \sinh y\right)}} \]

       5. rem-exp-log 64.9%

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

       6. associate-+r+ 74.0%

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

       7. metadata-eval 74.0%

          \[\leadsto \color{blue}{0} + \sinh y \]

       8. +-lft-identity 74.0%

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

   11. Simplified 74.0%

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

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sinh y \leq 10^{-88}:\\ \;\;\;\;x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 50.9% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{y}{x}\\ t_1 := -0.16666666666666666 \cdot \left(x \cdot x\right)\\ \mathbf{if}\;x \leq 2.3 \cdot 10^{+102}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{+158}:\\ \;\;\;\;y \cdot \left(1 + t\_1\right)\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{+226} \lor \neg \left(x \leq 1.5 \cdot 10^{+246}\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;y \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (/ y x))) (t_1 (* -0.16666666666666666 (* x x))))
   (if (<= x 2.3e+102)
     t_0
     (if (<= x 1.8e+158)
       (* y (+ 1.0 t_1))
       (if (or (<= x 1.75e+226) (not (<= x 1.5e+246))) t_0 (* y t_1))))))

C

double code(double x, double y) {
	double t_0 = x * (y / x);
	double t_1 = -0.16666666666666666 * (x * x);
	double tmp;
	if (x <= 2.3e+102) {
		tmp = t_0;
	} else if (x <= 1.8e+158) {
		tmp = y * (1.0 + t_1);
	} else if ((x <= 1.75e+226) || !(x <= 1.5e+246)) {
		tmp = t_0;
	} else {
		tmp = y * 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) :: tmp
    t_0 = x * (y / x)
    t_1 = (-0.16666666666666666d0) * (x * x)
    if (x <= 2.3d+102) then
        tmp = t_0
    else if (x <= 1.8d+158) then
        tmp = y * (1.0d0 + t_1)
    else if ((x <= 1.75d+226) .or. (.not. (x <= 1.5d+246))) then
        tmp = t_0
    else
        tmp = y * t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x * (y / x);
	double t_1 = -0.16666666666666666 * (x * x);
	double tmp;
	if (x <= 2.3e+102) {
		tmp = t_0;
	} else if (x <= 1.8e+158) {
		tmp = y * (1.0 + t_1);
	} else if ((x <= 1.75e+226) || !(x <= 1.5e+246)) {
		tmp = t_0;
	} else {
		tmp = y * t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x * (y / x)
	t_1 = -0.16666666666666666 * (x * x)
	tmp = 0
	if x <= 2.3e+102:
		tmp = t_0
	elif x <= 1.8e+158:
		tmp = y * (1.0 + t_1)
	elif (x <= 1.75e+226) or not (x <= 1.5e+246):
		tmp = t_0
	else:
		tmp = y * t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x * Float64(y / x))
	t_1 = Float64(-0.16666666666666666 * Float64(x * x))
	tmp = 0.0
	if (x <= 2.3e+102)
		tmp = t_0;
	elseif (x <= 1.8e+158)
		tmp = Float64(y * Float64(1.0 + t_1));
	elseif ((x <= 1.75e+226) || !(x <= 1.5e+246))
		tmp = t_0;
	else
		tmp = Float64(y * t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x * (y / x);
	t_1 = -0.16666666666666666 * (x * x);
	tmp = 0.0;
	if (x <= 2.3e+102)
		tmp = t_0;
	elseif (x <= 1.8e+158)
		tmp = y * (1.0 + t_1);
	elseif ((x <= 1.75e+226) || ~((x <= 1.5e+246)))
		tmp = t_0;
	else
		tmp = y * t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x * N[(y / x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 2.3e+102], t$95$0, If[LessEqual[x, 1.8e+158], N[(y * N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[x, 1.75e+226], N[Not[LessEqual[x, 1.5e+246]], $MachinePrecision]], t$95$0, N[(y * t$95$1), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 2.2999999999999999e102 or 1.79999999999999994e158 < x < 1.7499999999999999e226 or 1.5e246 < x

   1. Initial program 85.5%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 36.9%

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

   4. Taylor expanded in x around 0 23.2%

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

      1. associate-/l* 55.5%

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

      2. *-commutative 55.5%

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

   6. Applied egg-rr 55.5%

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

   if 2.2999999999999999e102 < x < 1.79999999999999994e158

   1. Initial program 99.8%

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

      1. associate-/l* 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 17.2%

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

      1. associate-/l* 17.4%

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

   7. Simplified 17.4%

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

   8. Taylor expanded in x around 0 59.0%

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

      1. unpow2 59.0%

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

   10. Applied egg-rr 59.0%

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

   if 1.7499999999999999e226 < x < 1.5e246

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 26.9%

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

      1. associate-/l* 26.9%

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

   7. Simplified 26.9%

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

   8. Taylor expanded in x around 0 25.4%

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

   9. Taylor expanded in x around inf 25.4%

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

       1. associate-*r* 25.4%

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

       2. *-commutative 25.4%

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

       3. *-commutative 25.4%

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

   11. Simplified 25.4%

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

       1. unpow2 25.4%

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

   13. Applied egg-rr 25.4%

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

4. Final simplification 55.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.3 \cdot 10^{+102}:\\ \;\;\;\;x \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{+158}:\\ \;\;\;\;y \cdot \left(1 + -0.16666666666666666 \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{+226} \lor \neg \left(x \leq 1.5 \cdot 10^{+246}\right):\\ \;\;\;\;x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-0.16666666666666666 \cdot \left(x \cdot x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 51.3% accurate, 12.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.3 \cdot 10^{+102} \lor \neg \left(x \leq 1.65 \cdot 10^{+158}\right):\\ \;\;\;\;x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-0.16666666666666666 \cdot \left(x \cdot x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x 2.3e+102) (not (<= x 1.65e+158)))
   (* x (/ y x))
   (* y (* -0.16666666666666666 (* x x)))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= 2.3e+102) || !(x <= 1.65e+158)) {
		tmp = x * (y / x);
	} else {
		tmp = y * (-0.16666666666666666 * (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 ((x <= 2.3d+102) .or. (.not. (x <= 1.65d+158))) then
        tmp = x * (y / x)
    else
        tmp = y * ((-0.16666666666666666d0) * (x * x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= 2.3e+102) || !(x <= 1.65e+158)) {
		tmp = x * (y / x);
	} else {
		tmp = y * (-0.16666666666666666 * (x * x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= 2.3e+102) or not (x <= 1.65e+158):
		tmp = x * (y / x)
	else:
		tmp = y * (-0.16666666666666666 * (x * x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= 2.3e+102) || !(x <= 1.65e+158))
		tmp = Float64(x * Float64(y / x));
	else
		tmp = Float64(y * Float64(-0.16666666666666666 * Float64(x * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= 2.3e+102) || ~((x <= 1.65e+158)))
		tmp = x * (y / x);
	else
		tmp = y * (-0.16666666666666666 * (x * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, 2.3e+102], N[Not[LessEqual[x, 1.65e+158]], $MachinePrecision]], N[(x * N[(y / x), $MachinePrecision]), $MachinePrecision], N[(y * N[(-0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.2999999999999999e102 or 1.65000000000000009e158 < x

   1. Initial program 85.8%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 36.7%

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

   4. Taylor expanded in x around 0 23.2%

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

      1. associate-/l* 54.6%

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

      2. *-commutative 54.6%

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

   6. Applied egg-rr 54.6%

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

   if 2.2999999999999999e102 < x < 1.65000000000000009e158

   1. Initial program 99.8%

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

      1. associate-/l* 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 17.2%

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

      1. associate-/l* 17.4%

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

   7. Simplified 17.4%

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

   8. Taylor expanded in x around 0 59.0%

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

   9. Taylor expanded in x around inf 59.0%

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

       1. associate-*r* 59.0%

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

       2. *-commutative 59.0%

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

       3. *-commutative 59.0%

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

   11. Simplified 59.0%

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

       1. unpow2 59.0%

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

   13. Applied egg-rr 59.0%

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

4. Final simplification 54.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.3 \cdot 10^{+102} \lor \neg \left(x \leq 1.65 \cdot 10^{+158}\right):\\ \;\;\;\;x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-0.16666666666666666 \cdot \left(x \cdot x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 51.2% accurate, 41.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return x * (y / x)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.1%

   \[\frac{\sin x \cdot \sinh y}{x} \]
2. Add Preprocessing

3. Taylor expanded in y around 0 36.2%

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

4. Taylor expanded in x around 0 23.0%

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

   1. associate-/l* 53.2%

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

   2. *-commutative 53.2%

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

6. Applied egg-rr 53.2%

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

7. Final simplification 53.2%

   \[\leadsto x \cdot \frac{y}{x} \]
8. Add Preprocessing

Alternative 7: 28.8% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 y)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return y

Julia

function code(x, y)
	return y
end

MATLAB

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

Wolfram

code[x_, y_] := y

Derivation

1. Initial program 86.1%

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

   1. associate-/l* 99.9%

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

3. Simplified 99.9%

   \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
4. Add Preprocessing

5. Taylor expanded in y around 0 36.2%

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

   1. associate-/l* 50.0%

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

7. Simplified 50.0%

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

8. Taylor expanded in x around 0 31.0%

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

9. Final simplification 31.0%

   \[\leadsto y \]
10. Add Preprocessing

Developer target: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024076 
(FPCore (x y)
  :name "Linear.Quaternion:$ccosh from linear-1.19.1.3"
  :precision binary64

  :alt
  (* (sin x) (/ (sinh y) x))

  (/ (* (sin x) (sinh y)) x))