Linear.Quaternion:$csinh from linear-1.19.1.3

Percentage Accurate: 99.9% → 99.9%

Time: 12.5s

Alternatives: 8

Speedup: 1.0×

• 49.0% 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

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

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(sin(y) / Float64(y / cosh(x)))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   2. associate-/r/ 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 2: 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]

Derivation

1. Initial program 99.9%

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

2. Final simplification 99.9%

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

Alternative 3: 63.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.3:\\ \;\;\;\;\frac{\sin y}{y + y \cdot \left(-0.5 \cdot \left(x \cdot x\right)\right)}\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{+170}:\\ \;\;\;\;\frac{\sin y}{y \cdot y} \cdot \left(y + 0.5 \cdot \left(y \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.3)
   (/ (sin y) (+ y (* y (* -0.5 (* x x)))))
   (if (<= x 1.05e+170)
     (* (/ (sin y) (* y y)) (+ y (* 0.5 (* y (* x x)))))
     (+ 1.0 (* (* x x) 0.5)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.3) {
		tmp = sin(y) / (y + (y * (-0.5 * (x * x))));
	} else if (x <= 1.05e+170) {
		tmp = (sin(y) / (y * y)) * (y + (0.5 * (y * (x * x))));
	} else {
		tmp = 1.0 + ((x * x) * 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 1.3d0) then
        tmp = sin(y) / (y + (y * ((-0.5d0) * (x * x))))
    else if (x <= 1.05d+170) then
        tmp = (sin(y) / (y * y)) * (y + (0.5d0 * (y * (x * x))))
    else
        tmp = 1.0d0 + ((x * x) * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 1.3) {
		tmp = Math.sin(y) / (y + (y * (-0.5 * (x * x))));
	} else if (x <= 1.05e+170) {
		tmp = (Math.sin(y) / (y * y)) * (y + (0.5 * (y * (x * x))));
	} else {
		tmp = 1.0 + ((x * x) * 0.5);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1.3:
		tmp = math.sin(y) / (y + (y * (-0.5 * (x * x))))
	elif x <= 1.05e+170:
		tmp = (math.sin(y) / (y * y)) * (y + (0.5 * (y * (x * x))))
	else:
		tmp = 1.0 + ((x * x) * 0.5)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.3)
		tmp = Float64(sin(y) / Float64(y + Float64(y * Float64(-0.5 * Float64(x * x)))));
	elseif (x <= 1.05e+170)
		tmp = Float64(Float64(sin(y) / Float64(y * y)) * Float64(y + Float64(0.5 * Float64(y * Float64(x * x)))));
	else
		tmp = Float64(1.0 + Float64(Float64(x * x) * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 1.3)
		tmp = sin(y) / (y + (y * (-0.5 * (x * x))));
	elseif (x <= 1.05e+170)
		tmp = (sin(y) / (y * y)) * (y + (0.5 * (y * (x * x))));
	else
		tmp = 1.0 + ((x * x) * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.3], N[(N[Sin[y], $MachinePrecision] / N[(y + N[(y * N[(-0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.05e+170], N[(N[(N[Sin[y], $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(y + N[(0.5 * N[(y * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.30000000000000004

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-/r/ 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 67.2%

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

      1. associate-*r* 67.2%

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

      2. unpow2 67.2%

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

   6. Simplified 67.2%

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

   if 1.30000000000000004 < x < 1.04999999999999999e170

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-/r/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 1.2%

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

      1. associate-*r* 1.2%

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

      2. unpow2 1.2%

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

   6. Simplified 1.2%

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

      1. flip-+ 3.9%

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

      2. associate-/r/ 3.9%

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

      3. *-commutative 3.9%

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

      4. *-commutative 3.9%

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

      5. swap-sqr 16.7%

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

      6. swap-sqr 16.7%

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

      7. metadata-eval 16.7%

         \[\leadsto \frac{\sin y}{y \cdot y - \left(y \cdot y\right) \cdot \left(\color{blue}{0.25} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)} \cdot \left(y - \left(-0.5 \cdot \left(x \cdot x\right)\right) \cdot y\right) \]

      8. pow2 16.7%

         \[\leadsto \frac{\sin y}{y \cdot y - \left(y \cdot y\right) \cdot \left(0.25 \cdot \left(\color{blue}{{x}^{2}} \cdot \left(x \cdot x\right)\right)\right)} \cdot \left(y - \left(-0.5 \cdot \left(x \cdot x\right)\right) \cdot y\right) \]

      9. pow2 16.7%

         \[\leadsto \frac{\sin y}{y \cdot y - \left(y \cdot y\right) \cdot \left(0.25 \cdot \left({x}^{2} \cdot \color{blue}{{x}^{2}}\right)\right)} \cdot \left(y - \left(-0.5 \cdot \left(x \cdot x\right)\right) \cdot y\right) \]

      10. pow-prod-up 16.7%

          \[\leadsto \frac{\sin y}{y \cdot y - \left(y \cdot y\right) \cdot \left(0.25 \cdot \color{blue}{{x}^{\left(2 + 2\right)}}\right)} \cdot \left(y - \left(-0.5 \cdot \left(x \cdot x\right)\right) \cdot y\right) \]

      11. metadata-eval 16.7%

          \[\leadsto \frac{\sin y}{y \cdot y - \left(y \cdot y\right) \cdot \left(0.25 \cdot {x}^{\color{blue}{4}}\right)} \cdot \left(y - \left(-0.5 \cdot \left(x \cdot x\right)\right) \cdot y\right) \]

   8. Applied egg-rr 16.7%

      \[\leadsto \color{blue}{\frac{\sin y}{y \cdot y - \left(y \cdot y\right) \cdot \left(0.25 \cdot {x}^{4}\right)} \cdot \left(y - y \cdot \left(x \cdot \left(x \cdot -0.5\right)\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*l* 4.0%

         \[\leadsto \frac{\sin y}{y \cdot y - \color{blue}{y \cdot \left(y \cdot \left(0.25 \cdot {x}^{4}\right)\right)}} \cdot \left(y - y \cdot \left(x \cdot \left(x \cdot -0.5\right)\right)\right) \]

      2. distribute-lft-out-- 0.9%

         \[\leadsto \frac{\sin y}{\color{blue}{y \cdot \left(y - y \cdot \left(0.25 \cdot {x}^{4}\right)\right)}} \cdot \left(y - y \cdot \left(x \cdot \left(x \cdot -0.5\right)\right)\right) \]

      3. *-commutative 0.9%

         \[\leadsto \frac{\sin y}{y \cdot \left(y - y \cdot \color{blue}{\left({x}^{4} \cdot 0.25\right)}\right)} \cdot \left(y - y \cdot \left(x \cdot \left(x \cdot -0.5\right)\right)\right) \]

      4. associate-*r* 0.9%

         \[\leadsto \frac{\sin y}{y \cdot \left(y - \color{blue}{\left(y \cdot {x}^{4}\right) \cdot 0.25}\right)} \cdot \left(y - y \cdot \left(x \cdot \left(x \cdot -0.5\right)\right)\right) \]

      5. *-commutative 0.9%

         \[\leadsto \frac{\sin y}{y \cdot \left(y - \left(y \cdot {x}^{4}\right) \cdot 0.25\right)} \cdot \left(y - \color{blue}{\left(x \cdot \left(x \cdot -0.5\right)\right) \cdot y}\right) \]

      6. associate-*r* 0.9%

         \[\leadsto \frac{\sin y}{y \cdot \left(y - \left(y \cdot {x}^{4}\right) \cdot 0.25\right)} \cdot \left(y - \color{blue}{\left(\left(x \cdot x\right) \cdot -0.5\right)} \cdot y\right) \]

      7. unpow2 0.9%

         \[\leadsto \frac{\sin y}{y \cdot \left(y - \left(y \cdot {x}^{4}\right) \cdot 0.25\right)} \cdot \left(y - \left(\color{blue}{{x}^{2}} \cdot -0.5\right) \cdot y\right) \]

      8. *-commutative 0.9%

         \[\leadsto \frac{\sin y}{y \cdot \left(y - \left(y \cdot {x}^{4}\right) \cdot 0.25\right)} \cdot \left(y - \color{blue}{\left(-0.5 \cdot {x}^{2}\right)} \cdot y\right) \]

      9. associate-*r* 0.9%

         \[\leadsto \frac{\sin y}{y \cdot \left(y - \left(y \cdot {x}^{4}\right) \cdot 0.25\right)} \cdot \left(y - \color{blue}{-0.5 \cdot \left({x}^{2} \cdot y\right)}\right) \]

      10. cancel-sign-sub-inv 0.9%

          \[\leadsto \frac{\sin y}{y \cdot \left(y - \left(y \cdot {x}^{4}\right) \cdot 0.25\right)} \cdot \color{blue}{\left(y + \left(--0.5\right) \cdot \left({x}^{2} \cdot y\right)\right)} \]

      11. metadata-eval 0.9%

          \[\leadsto \frac{\sin y}{y \cdot \left(y - \left(y \cdot {x}^{4}\right) \cdot 0.25\right)} \cdot \left(y + \color{blue}{0.5} \cdot \left({x}^{2} \cdot y\right)\right) \]

      12. *-commutative 0.9%

          \[\leadsto \frac{\sin y}{y \cdot \left(y - \left(y \cdot {x}^{4}\right) \cdot 0.25\right)} \cdot \left(y + 0.5 \cdot \color{blue}{\left(y \cdot {x}^{2}\right)}\right) \]

      13. unpow2 0.9%

          \[\leadsto \frac{\sin y}{y \cdot \left(y - \left(y \cdot {x}^{4}\right) \cdot 0.25\right)} \cdot \left(y + 0.5 \cdot \left(y \cdot \color{blue}{\left(x \cdot x\right)}\right)\right) \]

   10. Simplified 0.9%

       \[\leadsto \color{blue}{\frac{\sin y}{y \cdot \left(y - \left(y \cdot {x}^{4}\right) \cdot 0.25\right)} \cdot \left(y + 0.5 \cdot \left(y \cdot \left(x \cdot x\right)\right)\right)} \]

   11. Taylor expanded in x around 0 34.4%

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

       1. unpow2 34.4%

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

   13. Simplified 34.4%

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

   if 1.04999999999999999e170 < x

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-/r/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 1.6%

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

      1. associate-*r* 1.6%

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

      2. unpow2 1.6%

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

   6. Simplified 1.6%

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

   7. Taylor expanded in y around 0 1.6%

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

      1. +-commutative 1.6%

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

      2. unpow2 1.6%

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

      3. associate-*r* 1.6%

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

      4. *-commutative 1.6%

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

      5. fma-udef 1.6%

         \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(x \cdot -0.5, x, 1\right)}} \]

   9. Simplified 1.6%

      \[\leadsto \color{blue}{\frac{1}{\mathsf{fma}\left(x \cdot -0.5, x, 1\right)}} \]

   10. Taylor expanded in x around 0 78.6%

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

       1. unpow2 78.6%

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

   12. Simplified 78.6%

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

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.3:\\ \;\;\;\;\frac{\sin y}{y + y \cdot \left(-0.5 \cdot \left(x \cdot x\right)\right)}\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{+170}:\\ \;\;\;\;\frac{\sin y}{y \cdot y} \cdot \left(y + 0.5 \cdot \left(y \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot 0.5\\ \end{array} \]

Alternative 4: 60.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6200:\\ \;\;\;\;\frac{\sin y}{y + y \cdot \left(-0.5 \cdot \left(x \cdot x\right)\right)}\\ \mathbf{elif}\;x \leq 6.9 \cdot 10^{+131}:\\ \;\;\;\;1 + \left(y \cdot y\right) \cdot \left|-0.16666666666666666\right|\\ \mathbf{else}:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 6200.0)
   (/ (sin y) (+ y (* y (* -0.5 (* x x)))))
   (if (<= x 6.9e+131)
     (+ 1.0 (* (* y y) (fabs -0.16666666666666666)))
     (+ 1.0 (* (* x x) 0.5)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 6200.0) {
		tmp = sin(y) / (y + (y * (-0.5 * (x * x))));
	} else if (x <= 6.9e+131) {
		tmp = 1.0 + ((y * y) * fabs(-0.16666666666666666));
	} else {
		tmp = 1.0 + ((x * x) * 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 6200.0d0) then
        tmp = sin(y) / (y + (y * ((-0.5d0) * (x * x))))
    else if (x <= 6.9d+131) then
        tmp = 1.0d0 + ((y * y) * abs((-0.16666666666666666d0)))
    else
        tmp = 1.0d0 + ((x * x) * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 6200.0) {
		tmp = Math.sin(y) / (y + (y * (-0.5 * (x * x))));
	} else if (x <= 6.9e+131) {
		tmp = 1.0 + ((y * y) * Math.abs(-0.16666666666666666));
	} else {
		tmp = 1.0 + ((x * x) * 0.5);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 6200.0:
		tmp = math.sin(y) / (y + (y * (-0.5 * (x * x))))
	elif x <= 6.9e+131:
		tmp = 1.0 + ((y * y) * math.fabs(-0.16666666666666666))
	else:
		tmp = 1.0 + ((x * x) * 0.5)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 6200.0)
		tmp = Float64(sin(y) / Float64(y + Float64(y * Float64(-0.5 * Float64(x * x)))));
	elseif (x <= 6.9e+131)
		tmp = Float64(1.0 + Float64(Float64(y * y) * abs(-0.16666666666666666)));
	else
		tmp = Float64(1.0 + Float64(Float64(x * x) * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 6200.0)
		tmp = sin(y) / (y + (y * (-0.5 * (x * x))));
	elseif (x <= 6.9e+131)
		tmp = 1.0 + ((y * y) * abs(-0.16666666666666666));
	else
		tmp = 1.0 + ((x * x) * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 6200.0], N[(N[Sin[y], $MachinePrecision] / N[(y + N[(y * N[(-0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 6.9e+131], N[(1.0 + N[(N[(y * y), $MachinePrecision] * N[Abs[-0.16666666666666666], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 6200

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-/r/ 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 66.9%

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

      1. associate-*r* 66.9%

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

      2. unpow2 66.9%

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

   6. Simplified 66.9%

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

   if 6200 < x < 6.9000000000000001e131

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-/r/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 2.8%

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

   5. Taylor expanded in y around 0 9.9%

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

      1. *-commutative 9.9%

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

      2. unpow2 9.9%

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

   7. Simplified 9.9%

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

   8. Taylor expanded in y around 0 9.9%

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

      1. *-commutative 9.9%

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

      2. unpow2 9.9%

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

      3. associate-*r* 9.9%

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

   10. Simplified 9.9%

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

       1. add-sqr-sqrt 0.8%

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

       2. sqrt-unprod 21.3%

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

       3. pow2 21.3%

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

       4. associate-*r* 21.3%

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

   12. Applied egg-rr 21.3%

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

       1. unpow2 21.3%

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

       2. rem-sqrt-square 14.1%

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

       3. unpow2 14.1%

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

       4. *-commutative 14.1%

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

       5. unpow2 14.1%

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

   14. Simplified 14.1%

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

   if 6.9000000000000001e131 < x

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-/r/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 1.6%

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

      1. associate-*r* 1.6%

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

      2. unpow2 1.6%

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

   6. Simplified 1.6%

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

   7. Taylor expanded in y around 0 1.6%

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

      1. +-commutative 1.6%

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

      2. unpow2 1.6%

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

      3. associate-*r* 1.6%

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

      4. *-commutative 1.6%

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

      5. fma-udef 1.6%

         \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(x \cdot -0.5, x, 1\right)}} \]

   9. Simplified 1.6%

      \[\leadsto \color{blue}{\frac{1}{\mathsf{fma}\left(x \cdot -0.5, x, 1\right)}} \]

   10. Taylor expanded in x around 0 81.3%

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

       1. unpow2 81.3%

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

   12. Simplified 81.3%

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

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6200:\\ \;\;\;\;\frac{\sin y}{y + y \cdot \left(-0.5 \cdot \left(x \cdot x\right)\right)}\\ \mathbf{elif}\;x \leq 6.9 \cdot 10^{+131}:\\ \;\;\;\;1 + \left(y \cdot y\right) \cdot \left|-0.16666666666666666\right|\\ \mathbf{else}:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot 0.5\\ \end{array} \]

Alternative 5: 60.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6400:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{+133}:\\ \;\;\;\;1 + \left(y \cdot y\right) \cdot \left|-0.16666666666666666\right|\\ \mathbf{else}:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 6400.0)
   (/ (sin y) y)
   (if (<= x 2.4e+133)
     (+ 1.0 (* (* y y) (fabs -0.16666666666666666)))
     (+ 1.0 (* (* x x) 0.5)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 6400.0) {
		tmp = sin(y) / y;
	} else if (x <= 2.4e+133) {
		tmp = 1.0 + ((y * y) * fabs(-0.16666666666666666));
	} else {
		tmp = 1.0 + ((x * x) * 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 6400.0d0) then
        tmp = sin(y) / y
    else if (x <= 2.4d+133) then
        tmp = 1.0d0 + ((y * y) * abs((-0.16666666666666666d0)))
    else
        tmp = 1.0d0 + ((x * x) * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 6400.0) {
		tmp = Math.sin(y) / y;
	} else if (x <= 2.4e+133) {
		tmp = 1.0 + ((y * y) * Math.abs(-0.16666666666666666));
	} else {
		tmp = 1.0 + ((x * x) * 0.5);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 6400.0:
		tmp = math.sin(y) / y
	elif x <= 2.4e+133:
		tmp = 1.0 + ((y * y) * math.fabs(-0.16666666666666666))
	else:
		tmp = 1.0 + ((x * x) * 0.5)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 6400.0)
		tmp = Float64(sin(y) / y);
	elseif (x <= 2.4e+133)
		tmp = Float64(1.0 + Float64(Float64(y * y) * abs(-0.16666666666666666)));
	else
		tmp = Float64(1.0 + Float64(Float64(x * x) * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 6400.0)
		tmp = sin(y) / y;
	elseif (x <= 2.4e+133)
		tmp = 1.0 + ((y * y) * abs(-0.16666666666666666));
	else
		tmp = 1.0 + ((x * x) * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 6400.0], N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], If[LessEqual[x, 2.4e+133], N[(1.0 + N[(N[(y * y), $MachinePrecision] * N[Abs[-0.16666666666666666], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 6400

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-/r/ 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 67.2%

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

   if 6400 < x < 2.3999999999999999e133

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-/r/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 2.8%

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

   5. Taylor expanded in y around 0 9.9%

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

      1. *-commutative 9.9%

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

      2. unpow2 9.9%

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

   7. Simplified 9.9%

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

   8. Taylor expanded in y around 0 9.9%

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

      1. *-commutative 9.9%

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

      2. unpow2 9.9%

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

      3. associate-*r* 9.9%

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

   10. Simplified 9.9%

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

       1. add-sqr-sqrt 0.8%

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

       2. sqrt-unprod 21.3%

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

       3. pow2 21.3%

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

       4. associate-*r* 21.3%

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

   12. Applied egg-rr 21.3%

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

       1. unpow2 21.3%

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

       2. rem-sqrt-square 14.1%

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

       3. unpow2 14.1%

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

       4. *-commutative 14.1%

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

       5. unpow2 14.1%

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

   14. Simplified 14.1%

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

   if 2.3999999999999999e133 < x

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-/r/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 1.6%

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

      1. associate-*r* 1.6%

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

      2. unpow2 1.6%

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

   6. Simplified 1.6%

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

   7. Taylor expanded in y around 0 1.6%

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

      1. +-commutative 1.6%

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

      2. unpow2 1.6%

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

      3. associate-*r* 1.6%

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

      4. *-commutative 1.6%

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

      5. fma-udef 1.6%

         \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(x \cdot -0.5, x, 1\right)}} \]

   9. Simplified 1.6%

      \[\leadsto \color{blue}{\frac{1}{\mathsf{fma}\left(x \cdot -0.5, x, 1\right)}} \]

   10. Taylor expanded in x around 0 81.3%

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

       1. unpow2 81.3%

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

   12. Simplified 81.3%

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

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6400:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{+133}:\\ \;\;\;\;1 + \left(y \cdot y\right) \cdot \left|-0.16666666666666666\right|\\ \mathbf{else}:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot 0.5\\ \end{array} \]

Alternative 6: 59.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.2 \cdot 10^{-12}:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.2e-12) (/ (sin y) y) (+ 1.0 (* (* x x) 0.5))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.2e-12) {
		tmp = sin(y) / y;
	} else {
		tmp = 1.0 + ((x * x) * 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 1.2d-12) then
        tmp = sin(y) / y
    else
        tmp = 1.0d0 + ((x * x) * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 1.2e-12) {
		tmp = Math.sin(y) / y;
	} else {
		tmp = 1.0 + ((x * x) * 0.5);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1.2e-12:
		tmp = math.sin(y) / y
	else:
		tmp = 1.0 + ((x * x) * 0.5)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.2e-12)
		tmp = Float64(sin(y) / y);
	else
		tmp = Float64(1.0 + Float64(Float64(x * x) * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 1.2e-12)
		tmp = sin(y) / y;
	else
		tmp = 1.0 + ((x * x) * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.2e-12], N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], N[(1.0 + N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.19999999999999994e-12

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-/r/ 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 67.5%

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

   if 1.19999999999999994e-12 < x

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-/r/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 4.1%

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

      1. associate-*r* 4.1%

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

      2. unpow2 4.1%

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

   6. Simplified 4.1%

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

   7. Taylor expanded in y around 0 4.2%

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

      1. +-commutative 4.2%

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

      2. unpow2 4.2%

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

      3. associate-*r* 4.2%

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

      4. *-commutative 4.2%

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

      5. fma-udef 4.2%

         \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(x \cdot -0.5, x, 1\right)}} \]

   9. Simplified 4.2%

      \[\leadsto \color{blue}{\frac{1}{\mathsf{fma}\left(x \cdot -0.5, x, 1\right)}} \]

   10. Taylor expanded in x around 0 47.2%

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

       1. unpow2 47.2%

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

   12. Simplified 47.2%

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

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.2 \cdot 10^{-12}:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot 0.5\\ \end{array} \]

Alternative 7: 44.9% accurate, 29.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   2. associate-/r/ 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in x around 0 52.0%

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

   1. associate-*r* 52.0%

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

   2. unpow2 52.0%

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

6. Simplified 52.0%

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

7. Taylor expanded in y around 0 27.5%

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

   1. +-commutative 27.5%

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

   2. unpow2 27.5%

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

   3. associate-*r* 27.5%

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

   4. *-commutative 27.5%

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

   5. fma-udef 27.5%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(x \cdot -0.5, x, 1\right)}} \]

9. Simplified 27.5%

   \[\leadsto \color{blue}{\frac{1}{\mathsf{fma}\left(x \cdot -0.5, x, 1\right)}} \]

10. Taylor expanded in x around 0 47.6%

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

    1. unpow2 47.6%

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

12. Simplified 47.6%

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

13. Final simplification 47.6%

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

Alternative 8: 27.1% 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. *-commutative 99.9%

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

   2. associate-/r/ 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in x around 0 52.6%

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

5. Taylor expanded in y around 0 27.9%

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

6. Final simplification 27.9%

   \[\leadsto 1 \]

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 2023279 
(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)))