Linear.Quaternion:$csinh from linear-1.19.1.3

Percentage Accurate: 99.9% → 99.9%

Time: 7.5s

Alternatives: 8

Speedup: 1.0×

• 49.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

2. Final simplification 99.9%

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

Alternative 2: 87.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin y \cdot \left(\frac{1}{y} + \frac{0.5 \cdot \left(x \cdot x\right)}{y}\right)\\ \mathbf{if}\;x \leq 7000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{+55}:\\ \;\;\;\;\left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot \cosh x\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{+147}:\\ \;\;\;\;\cosh x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sin y) (+ (/ 1.0 y) (/ (* 0.5 (* x x)) y)))))
   (if (<= x 7000000.0)
     t_0
     (if (<= x 4.2e+55)
       (* (+ 1.0 (* -0.16666666666666666 (* y y))) (cosh x))
       (if (<= x 1.8e+147) (cosh x) t_0)))))

C

double code(double x, double y) {
	double t_0 = sin(y) * ((1.0 / y) + ((0.5 * (x * x)) / y));
	double tmp;
	if (x <= 7000000.0) {
		tmp = t_0;
	} else if (x <= 4.2e+55) {
		tmp = (1.0 + (-0.16666666666666666 * (y * y))) * cosh(x);
	} else if (x <= 1.8e+147) {
		tmp = cosh(x);
	} 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 = sin(y) * ((1.0d0 / y) + ((0.5d0 * (x * x)) / y))
    if (x <= 7000000.0d0) then
        tmp = t_0
    else if (x <= 4.2d+55) then
        tmp = (1.0d0 + ((-0.16666666666666666d0) * (y * y))) * cosh(x)
    else if (x <= 1.8d+147) then
        tmp = cosh(x)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sin(y) * ((1.0 / y) + ((0.5 * (x * x)) / y));
	double tmp;
	if (x <= 7000000.0) {
		tmp = t_0;
	} else if (x <= 4.2e+55) {
		tmp = (1.0 + (-0.16666666666666666 * (y * y))) * Math.cosh(x);
	} else if (x <= 1.8e+147) {
		tmp = Math.cosh(x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sin(y) * ((1.0 / y) + ((0.5 * (x * x)) / y))
	tmp = 0
	if x <= 7000000.0:
		tmp = t_0
	elif x <= 4.2e+55:
		tmp = (1.0 + (-0.16666666666666666 * (y * y))) * math.cosh(x)
	elif x <= 1.8e+147:
		tmp = math.cosh(x)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sin(y) * Float64(Float64(1.0 / y) + Float64(Float64(0.5 * Float64(x * x)) / y)))
	tmp = 0.0
	if (x <= 7000000.0)
		tmp = t_0;
	elseif (x <= 4.2e+55)
		tmp = Float64(Float64(1.0 + Float64(-0.16666666666666666 * Float64(y * y))) * cosh(x));
	elseif (x <= 1.8e+147)
		tmp = cosh(x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sin(y) * ((1.0 / y) + ((0.5 * (x * x)) / y));
	tmp = 0.0;
	if (x <= 7000000.0)
		tmp = t_0;
	elseif (x <= 4.2e+55)
		tmp = (1.0 + (-0.16666666666666666 * (y * y))) * cosh(x);
	elseif (x <= 1.8e+147)
		tmp = cosh(x);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] * N[(N[(1.0 / y), $MachinePrecision] + N[(N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 7000000.0], t$95$0, If[LessEqual[x, 4.2e+55], N[(N[(1.0 + N[(-0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cosh[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.8e+147], N[Cosh[x], $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < 7e6 or 1.8000000000000001e147 < x

   1. Initial program 99.9%

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

      1. add-log-exp 77.9%

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

      2. *-un-lft-identity 77.9%

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

      3. log-prod 77.9%

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

      4. metadata-eval 77.9%

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

      5. add-log-exp 99.9%

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

      6. *-commutative 99.9%

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

      7. associate-*l/ 99.9%

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

   3. Applied egg-rr 99.9%

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

      1. +-lft-identity 99.9%

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

      2. *-commutative 99.9%

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

      3. associate-*l/ 99.8%

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

      4. *-commutative 99.8%

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

   5. Simplified 99.8%

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

   6. Taylor expanded in x around 0 90.6%

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

      1. associate-*r/ 90.6%

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

      2. unpow2 90.6%

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

   8. Simplified 90.6%

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

   if 7e6 < x < 4.2000000000000001e55

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 93.3%

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

      1. unpow2 93.3%

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

   4. Simplified 93.3%

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

   if 4.2000000000000001e55 < x < 1.8000000000000001e147

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 76.2%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7000000:\\ \;\;\;\;\sin y \cdot \left(\frac{1}{y} + \frac{0.5 \cdot \left(x \cdot x\right)}{y}\right)\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{+55}:\\ \;\;\;\;\left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot \cosh x\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{+147}:\\ \;\;\;\;\cosh x\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \left(\frac{1}{y} + \frac{0.5 \cdot \left(x \cdot x\right)}{y}\right)\\ \end{array} \]

Alternative 3: 69.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.15 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+52}:\\ \;\;\;\;\left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot \cosh x\\ \mathbf{else}:\\ \;\;\;\;\cosh x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.15e-5)
   (/ (sin y) y)
   (if (<= x 2e+52)
     (* (+ 1.0 (* -0.16666666666666666 (* y y))) (cosh x))
     (cosh x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.15e-5) {
		tmp = sin(y) / y;
	} else if (x <= 2e+52) {
		tmp = (1.0 + (-0.16666666666666666 * (y * y))) * cosh(x);
	} else {
		tmp = cosh(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 <= 1.15d-5) then
        tmp = sin(y) / y
    else if (x <= 2d+52) then
        tmp = (1.0d0 + ((-0.16666666666666666d0) * (y * y))) * cosh(x)
    else
        tmp = cosh(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 1.15e-5) {
		tmp = Math.sin(y) / y;
	} else if (x <= 2e+52) {
		tmp = (1.0 + (-0.16666666666666666 * (y * y))) * Math.cosh(x);
	} else {
		tmp = Math.cosh(x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1.15e-5:
		tmp = math.sin(y) / y
	elif x <= 2e+52:
		tmp = (1.0 + (-0.16666666666666666 * (y * y))) * math.cosh(x)
	else:
		tmp = math.cosh(x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.15e-5)
		tmp = Float64(sin(y) / y);
	elseif (x <= 2e+52)
		tmp = Float64(Float64(1.0 + Float64(-0.16666666666666666 * Float64(y * y))) * cosh(x));
	else
		tmp = cosh(x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 1.15e-5)
		tmp = sin(y) / y;
	elseif (x <= 2e+52)
		tmp = (1.0 + (-0.16666666666666666 * (y * y))) * cosh(x);
	else
		tmp = cosh(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.15e-5], N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], If[LessEqual[x, 2e+52], N[(N[(1.0 + N[(-0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cosh[x], $MachinePrecision]), $MachinePrecision], N[Cosh[x], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.15e-5

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 69.0%

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

   if 1.15e-5 < x < 2e52

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 93.3%

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

      1. unpow2 93.3%

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

   4. Simplified 93.3%

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

   if 2e52 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 79.2%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.15 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+52}:\\ \;\;\;\;\left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot \cosh x\\ \mathbf{else}:\\ \;\;\;\;\cosh x\\ \end{array} \]

Alternative 4: 69.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.15 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{else}:\\ \;\;\;\;\cosh x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x 1.15e-5) (/ (sin y) y) (cosh x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.15e-5) {
		tmp = sin(y) / y;
	} else {
		tmp = cosh(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 <= 1.15d-5) then
        tmp = sin(y) / y
    else
        tmp = cosh(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 1.15e-5) {
		tmp = Math.sin(y) / y;
	} else {
		tmp = Math.cosh(x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1.15e-5:
		tmp = math.sin(y) / y
	else:
		tmp = math.cosh(x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.15e-5)
		tmp = Float64(sin(y) / y);
	else
		tmp = cosh(x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 1.15e-5)
		tmp = sin(y) / y;
	else
		tmp = cosh(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.15e-5], N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], N[Cosh[x], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.15e-5

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 69.0%

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

   if 1.15e-5 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 76.5%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.15 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{else}:\\ \;\;\;\;\cosh x\\ \end{array} \]

Alternative 5: 63.2% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 (cosh x))

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return math.cosh(x)

Julia

function code(x, y)
	return cosh(x)
end

MATLAB

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

Wolfram

code[x_, y_] := N[Cosh[x], $MachinePrecision]

Derivation

1. Initial program 99.9%

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

2. Taylor expanded in y around 0 68.9%

   \[\leadsto \cosh x \cdot \color{blue}{1} \]

3. Final simplification 68.9%

   \[\leadsto \cosh x \]

Alternative 6: 39.7% accurate, 22.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 7.2 \cdot 10^{+140}:\\ \;\;\;\;1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;1 + 0.5 \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 7.2e+140)
   (+ 1.0 (* -0.16666666666666666 (* y y)))
   (+ 1.0 (* 0.5 (* x x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 7.2e+140) {
		tmp = 1.0 + (-0.16666666666666666 * (y * y));
	} else {
		tmp = 1.0 + (0.5 * (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 <= 7.2d+140) then
        tmp = 1.0d0 + ((-0.16666666666666666d0) * (y * y))
    else
        tmp = 1.0d0 + (0.5d0 * (x * x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 7.2e+140) {
		tmp = 1.0 + (-0.16666666666666666 * (y * y));
	} else {
		tmp = 1.0 + (0.5 * (x * x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 7.2e+140:
		tmp = 1.0 + (-0.16666666666666666 * (y * y))
	else:
		tmp = 1.0 + (0.5 * (x * x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 7.2e+140)
		tmp = Float64(1.0 + Float64(-0.16666666666666666 * Float64(y * y)));
	else
		tmp = Float64(1.0 + Float64(0.5 * Float64(x * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 7.2e+140)
		tmp = 1.0 + (-0.16666666666666666 * (y * y));
	else
		tmp = 1.0 + (0.5 * (x * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 7.2e+140], N[(1.0 + N[(-0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 7.1999999999999999e140

   1. Initial program 99.9%

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

      1. associate-*r/ 99.9%

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

      2. add-sqr-sqrt 53.6%

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

      3. associate-/r* 53.5%

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

      4. *-commutative 53.5%

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

   3. Applied egg-rr 53.5%

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

   4. Taylor expanded in x around 0 31.8%

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

   5. Taylor expanded in y around 0 40.6%

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

      1. *-commutative 40.6%

         \[\leadsto 1 + \color{blue}{{y}^{2} \cdot -0.16666666666666666} \]

      2. unpow2 40.6%

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

   7. Simplified 40.6%

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

   if 7.1999999999999999e140 < x

   1. Initial program 100.0%

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

      1. add-log-exp 100.0%

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

      2. *-un-lft-identity 100.0%

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

      3. log-prod 100.0%

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

      4. metadata-eval 100.0%

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

      5. add-log-exp 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-*l/ 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. +-lft-identity 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*l/ 100.0%

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

      4. *-commutative 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 91.9%

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

      1. associate-*r/ 91.9%

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

      2. unpow2 91.9%

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

   8. Simplified 91.9%

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

   9. Taylor expanded in y around 0 72.2%

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

       1. unpow2 72.2%

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

   11. Simplified 72.2%

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

4. Final simplification 44.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7.2 \cdot 10^{+140}:\\ \;\;\;\;1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;1 + 0.5 \cdot \left(x \cdot x\right)\\ \end{array} \]

Alternative 7: 45.3% accurate, 29.3× speedup

Math

\[\begin{array}{l} \\ 1 + 0.5 \cdot \left(x \cdot x\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ 1.0 (* 0.5 (* x x))))

C

double code(double x, double y) {
	return 1.0 + (0.5 * (x * x));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 + (0.5d0 * (x * x))
end function

Java

public static double code(double x, double y) {
	return 1.0 + (0.5 * (x * x));
}

Python

def code(x, y):
	return 1.0 + (0.5 * (x * x))

Julia

function code(x, y)
	return Float64(1.0 + Float64(0.5 * Float64(x * x)))
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 + (0.5 * (x * x));
end

Wolfram

code[x_, y_] := N[(1.0 + N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

   1. add-log-exp 81.0%

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

   2. *-un-lft-identity 81.0%

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

   3. log-prod 81.0%

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

   4. metadata-eval 81.0%

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

   5. add-log-exp 99.9%

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

   6. *-commutative 99.9%

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

   7. associate-*l/ 99.9%

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

3. Applied egg-rr 99.9%

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

   1. +-lft-identity 99.9%

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

   2. *-commutative 99.9%

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

   3. associate-*l/ 99.8%

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

   4. *-commutative 99.8%

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

5. Simplified 99.8%

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

6. Taylor expanded in x around 0 79.9%

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

   1. associate-*r/ 79.9%

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

   2. unpow2 79.9%

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

8. Simplified 79.9%

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

9. Taylor expanded in y around 0 50.4%

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

    1. unpow2 50.4%

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

11. Simplified 50.4%

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

12. Final simplification 50.4%

    \[\leadsto 1 + 0.5 \cdot \left(x \cdot x\right) \]

Alternative 8: 26.6% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 99.9%

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

   1. associate-*r/ 99.9%

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

   2. add-sqr-sqrt 53.3%

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

   3. associate-/r* 53.2%

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

   4. *-commutative 53.2%

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

3. Applied egg-rr 53.2%

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

4. Taylor expanded in x around 0 27.7%

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

5. Taylor expanded in y around 0 31.4%

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

6. Final simplification 31.4%

   \[\leadsto 1 \]

Developer target: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023242 
(FPCore (x y)
  :name "Linear.Quaternion:$csinh from linear-1.19.1.3"
  :precision binary64

  :herbie-target
  (/ (* (cosh x) (sin y)) y)

  (* (cosh x) (/ (sin y) y)))