Linear.Quaternion:$csinh from linear-1.19.1.3

Percentage Accurate: 99.9% → 99.9%

Time: 2.6s

Alternatives: 8

Speedup: 1.0×

• 50.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

(FPCore (x y)
  :precision binary64
  :pre TRUE
  (* (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)
use fmin_fmax_functions
    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]

ReFlow

f(x, y):
	x in [-inf, +inf],
	y in [-inf, +inf]
code: THEORY
BEGIN
f(x, y: real): real =
	(((1) / (2)) * ((exp(x)) + ((1) / (exp(x))))) * ((sin(y)) / y)
END code

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y)
  :precision binary64
  :pre TRUE
  (* (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)
use fmin_fmax_functions
    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]

ReFlow

f(x, y):
	x in [-inf, +inf],
	y in [-inf, +inf]
code: THEORY
BEGIN
f(x, y: real): real =
	(((1) / (2)) * ((exp(x)) + ((1) / (exp(x))))) * ((sin(y)) / y)
END code

Alternative 1: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := \cosh x \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\cosh x \cdot \mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right)\\ \mathbf{elif}\;t\_1 \leq 0.9999999999999999:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\cosh x}}\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  :pre TRUE
  (let* ((t_0 (/ (sin y) y)) (t_1 (* (cosh x) t_0)))
  (if (<= t_1 (- INFINITY))
    (* (cosh x) (fma (* y y) -0.16666666666666666 1.0))
    (if (<= t_1 0.9999999999999999) t_0 (/ 1.0 (/ 1.0 (cosh x)))))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double t_1 = cosh(x) * t_0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = cosh(x) * fma((y * y), -0.16666666666666666, 1.0);
	} else if (t_1 <= 0.9999999999999999) {
		tmp = t_0;
	} else {
		tmp = 1.0 / (1.0 / cosh(x));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	t_1 = Float64(cosh(x) * t_0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(cosh(x) * fma(Float64(y * y), -0.16666666666666666, 1.0));
	elseif (t_1 <= 0.9999999999999999)
		tmp = t_0;
	else
		tmp = Float64(1.0 / Float64(1.0 / cosh(x)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cosh[x], $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[Cosh[x], $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.9999999999999999], t$95$0, N[(1.0 / N[(1.0 / N[Cosh[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -inf.0

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0

      \[\leadsto \cosh x \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \]
   3. Step-by-step derivation

   4. Applied rewrites 62.7%

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

   6. Applied rewrites 62.7%

      \[\leadsto \cosh x \cdot \mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right) \]

   if -inf.0 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < 0.99999999999999989

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

      \[\leadsto \frac{\sin y}{y} \]
   3. Step-by-step derivation

   4. Applied rewrites 51.0%

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

   if 0.99999999999999989 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 99.9%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in y around 0

      \[\leadsto \frac{1}{\frac{2}{e^{x} + \frac{1}{e^{x}}}} \]
   5. Step-by-step derivation

   6. Applied rewrites 62.4%

      \[\leadsto \frac{1}{\frac{2}{e^{x} + \frac{1}{e^{x}}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 62.4%

      \[\leadsto \frac{1}{\frac{1}{\cosh x}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 75.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -1 \cdot 10^{-148}:\\ \;\;\;\;\cosh x \cdot \mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\cosh x}}\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  :pre TRUE
  (if (<= (* (cosh x) (/ (sin y) y)) -1e-148)
  (* (cosh x) (fma (* y y) -0.16666666666666666 1.0))
  (/ 1.0 (/ 1.0 (cosh x)))))

C

double code(double x, double y) {
	double tmp;
	if ((cosh(x) * (sin(y) / y)) <= -1e-148) {
		tmp = cosh(x) * fma((y * y), -0.16666666666666666, 1.0);
	} else {
		tmp = 1.0 / (1.0 / cosh(x));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(sin(y) / y)) <= -1e-148)
		tmp = Float64(cosh(x) * fma(Float64(y * y), -0.16666666666666666, 1.0));
	else
		tmp = Float64(1.0 / Float64(1.0 / cosh(x)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Cosh[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -1e-148], N[(N[Cosh[x], $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 / N[Cosh[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

ReFlow

f(x, y):
	x in [-inf, +inf],
	y in [-inf, +inf]
code: THEORY
BEGIN
f(x, y: real): real =
	LET tmp = IF (((((1) / (2)) * ((exp(x)) + ((1) / (exp(x))))) * ((sin(y)) / y)) <= (-999999999999999935748815890117282149663822731120577929628009867749049427642678029270005288335997592555778350731539802843803522676304070686845283750574203026982376971286717937832954133378800900225517256537491228018261850658864691187645294441546276508577338302627079226124410768716556281580123652602299929902354192051524133144221481909121761385664308924357650265601904493451002053916454315185546875e-544)) THEN ((((1) / (2)) * ((exp(x)) + ((1) / (exp(x))))) * (((y * y) * (-1666666666666666574148081281236954964697360992431640625e-55)) + (1))) ELSE ((1) / ((1) / (((1) / (2)) * ((exp(x)) + ((1) / (exp(x))))))) ENDIF IN
	tmp
END code

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -9.9999999999999994e-149

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0

      \[\leadsto \cosh x \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \]
   3. Step-by-step derivation

   4. Applied rewrites 62.7%

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

   6. Applied rewrites 62.7%

      \[\leadsto \cosh x \cdot \mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right) \]

   if -9.9999999999999994e-149 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 99.9%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in y around 0

      \[\leadsto \frac{1}{\frac{2}{e^{x} + \frac{1}{e^{x}}}} \]
   5. Step-by-step derivation

   6. Applied rewrites 62.4%

      \[\leadsto \frac{1}{\frac{2}{e^{x} + \frac{1}{e^{x}}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 62.4%

      \[\leadsto \frac{1}{\frac{1}{\cosh x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 72.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -1 \cdot 10^{-148}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\left(y \cdot y\right) \cdot \left(y \cdot y\right)}, -0.16666666666666666, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\cosh x}}\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  :pre TRUE
  (if (<= (* (cosh x) (/ (sin y) y)) -1e-148)
  (fma (sqrt (* (* y y) (* y y))) -0.16666666666666666 1.0)
  (/ 1.0 (/ 1.0 (cosh x)))))

C

double code(double x, double y) {
	double tmp;
	if ((cosh(x) * (sin(y) / y)) <= -1e-148) {
		tmp = fma(sqrt(((y * y) * (y * y))), -0.16666666666666666, 1.0);
	} else {
		tmp = 1.0 / (1.0 / cosh(x));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(sin(y) / y)) <= -1e-148)
		tmp = fma(sqrt(Float64(Float64(y * y) * Float64(y * y))), -0.16666666666666666, 1.0);
	else
		tmp = Float64(1.0 / Float64(1.0 / cosh(x)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Cosh[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -1e-148], N[(N[Sqrt[N[(N[(y * y), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision], N[(1.0 / N[(1.0 / N[Cosh[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

ReFlow

f(x, y):
	x in [-inf, +inf],
	y in [-inf, +inf]
code: THEORY
BEGIN
f(x, y: real): real =
	LET tmp = IF (((((1) / (2)) * ((exp(x)) + ((1) / (exp(x))))) * ((sin(y)) / y)) <= (-999999999999999935748815890117282149663822731120577929628009867749049427642678029270005288335997592555778350731539802843803522676304070686845283750574203026982376971286717937832954133378800900225517256537491228018261850658864691187645294441546276508577338302627079226124410768716556281580123652602299929902354192051524133144221481909121761385664308924357650265601904493451002053916454315185546875e-544)) THEN (((sqrt(((y * y) * (y * y)))) * (-1666666666666666574148081281236954964697360992431640625e-55)) + (1)) ELSE ((1) / ((1) / (((1) / (2)) * ((exp(x)) + ((1) / (exp(x))))))) ENDIF IN
	tmp
END code

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -9.9999999999999994e-149

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

      \[\leadsto \frac{\sin y}{y} \]
   3. Step-by-step derivation

   4. Applied rewrites 51.0%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 33.0%

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

   9. Applied rewrites 33.0%

      \[\leadsto \mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right) \]
   10. Step-by-step derivation

       1. rem-square-sqrt N/A

          \[\leadsto \mathsf{fma}\left(\sqrt{\left(\sqrt{y \cdot y} \cdot \sqrt{y \cdot y}\right) \cdot \left(\sqrt{y \cdot y} \cdot \sqrt{y \cdot y}\right)}, \frac{-1}{6}, 1\right) \]

       2. sqrt-unprod N/A

          \[\leadsto \mathsf{fma}\left(\sqrt{\sqrt{\left(y \cdot y\right) \cdot \left(y \cdot y\right)} \cdot \sqrt{\left(y \cdot y\right) \cdot \left(y \cdot y\right)}}, \frac{-1}{6}, 1\right) \]

       3. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\sqrt{\sqrt{\left(y \cdot y\right) \cdot \left(y \cdot y\right)} \cdot \sqrt{\left(y \cdot y\right) \cdot \left(y \cdot y\right)}}, \frac{-1}{6}, 1\right) \]

       4. lift-sqrt.f64 36.0%

          \[\leadsto \mathsf{fma}\left(\sqrt{\sqrt{\left(y \cdot y\right) \cdot \left(y \cdot y\right)} \cdot \sqrt{\left(y \cdot y\right) \cdot \left(y \cdot y\right)}}, -0.16666666666666666, 1\right) \]

   11. Applied rewrites 36.0%

       \[\leadsto \mathsf{fma}\left(\sqrt{\left(y \cdot y\right) \cdot \left(y \cdot y\right)}, -0.16666666666666666, 1\right) \]

   if -9.9999999999999994e-149 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 99.9%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in y around 0

      \[\leadsto \frac{1}{\frac{2}{e^{x} + \frac{1}{e^{x}}}} \]
   5. Step-by-step derivation

   6. Applied rewrites 62.4%

      \[\leadsto \frac{1}{\frac{2}{e^{x} + \frac{1}{e^{x}}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 62.4%

      \[\leadsto \frac{1}{\frac{1}{\cosh x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 48.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq 20:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\left(y \cdot y\right) \cdot \left(y \cdot y\right)}, -0.16666666666666666, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{y}}{-0.16666666666666666 \cdot y}\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  :pre TRUE
  (if (<= (* (cosh x) (/ (sin y) y)) 20.0)
  (fma (sqrt (* (* y y) (* y y))) -0.16666666666666666 1.0)
  (/ (/ -1.0 y) (* -0.16666666666666666 y))))

C

double code(double x, double y) {
	double tmp;
	if ((cosh(x) * (sin(y) / y)) <= 20.0) {
		tmp = fma(sqrt(((y * y) * (y * y))), -0.16666666666666666, 1.0);
	} else {
		tmp = (-1.0 / y) / (-0.16666666666666666 * y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(sin(y) / y)) <= 20.0)
		tmp = fma(sqrt(Float64(Float64(y * y) * Float64(y * y))), -0.16666666666666666, 1.0);
	else
		tmp = Float64(Float64(-1.0 / y) / Float64(-0.16666666666666666 * y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Cosh[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], 20.0], N[(N[Sqrt[N[(N[(y * y), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision], N[(N[(-1.0 / y), $MachinePrecision] / N[(-0.16666666666666666 * y), $MachinePrecision]), $MachinePrecision]]

ReFlow

f(x, y):
	x in [-inf, +inf],
	y in [-inf, +inf]
code: THEORY
BEGIN
f(x, y: real): real =
	LET tmp = IF (((((1) / (2)) * ((exp(x)) + ((1) / (exp(x))))) * ((sin(y)) / y)) <= (20)) THEN (((sqrt(((y * y) * (y * y)))) * (-1666666666666666574148081281236954964697360992431640625e-55)) + (1)) ELSE (((-1) / y) / ((-1666666666666666574148081281236954964697360992431640625e-55) * y)) ENDIF IN
	tmp
END code

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < 20

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

      \[\leadsto \frac{\sin y}{y} \]
   3. Step-by-step derivation

   4. Applied rewrites 51.0%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 33.0%

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

   9. Applied rewrites 33.0%

      \[\leadsto \mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right) \]
   10. Step-by-step derivation

       1. rem-square-sqrt N/A

          \[\leadsto \mathsf{fma}\left(\sqrt{\left(\sqrt{y \cdot y} \cdot \sqrt{y \cdot y}\right) \cdot \left(\sqrt{y \cdot y} \cdot \sqrt{y \cdot y}\right)}, \frac{-1}{6}, 1\right) \]

       2. sqrt-unprod N/A

          \[\leadsto \mathsf{fma}\left(\sqrt{\sqrt{\left(y \cdot y\right) \cdot \left(y \cdot y\right)} \cdot \sqrt{\left(y \cdot y\right) \cdot \left(y \cdot y\right)}}, \frac{-1}{6}, 1\right) \]

       3. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\sqrt{\sqrt{\left(y \cdot y\right) \cdot \left(y \cdot y\right)} \cdot \sqrt{\left(y \cdot y\right) \cdot \left(y \cdot y\right)}}, \frac{-1}{6}, 1\right) \]

       4. lift-sqrt.f64 36.0%

          \[\leadsto \mathsf{fma}\left(\sqrt{\sqrt{\left(y \cdot y\right) \cdot \left(y \cdot y\right)} \cdot \sqrt{\left(y \cdot y\right) \cdot \left(y \cdot y\right)}}, -0.16666666666666666, 1\right) \]

   11. Applied rewrites 36.0%

       \[\leadsto \mathsf{fma}\left(\sqrt{\left(y \cdot y\right) \cdot \left(y \cdot y\right)}, -0.16666666666666666, 1\right) \]

   if 20 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

      \[\leadsto \frac{\sin y}{y} \]
   3. Step-by-step derivation

   4. Applied rewrites 51.0%

      \[\leadsto \frac{\sin y}{y} \]
   5. Step-by-step derivation

   6. Applied rewrites 50.9%

      \[\leadsto \frac{\frac{-1}{y}}{\frac{-1}{\sin y}} \]

   7. Taylor expanded in y around 0

      \[\leadsto \frac{\frac{-1}{y}}{\frac{\frac{-1}{6} \cdot {y}^{2} - 1}{y}} \]
   8. Step-by-step derivation

   9. Applied rewrites 27.3%

      \[\leadsto \frac{\frac{-1}{y}}{\frac{-0.16666666666666666 \cdot {y}^{2} - 1}{y}} \]

   10. Taylor expanded in y around inf

       \[\leadsto \frac{\frac{-1}{y}}{\frac{-1}{6} \cdot y} \]
   11. Step-by-step derivation

   12. Applied rewrites 15.5%

       \[\leadsto \frac{\frac{-1}{y}}{-0.16666666666666666 \cdot y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 45.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq 20:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{y}}{-0.16666666666666666 \cdot y}\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  :pre TRUE
  (if (<= (* (cosh x) (/ (sin y) y)) 20.0)
  (fma (* y y) -0.16666666666666666 1.0)
  (/ (/ -1.0 y) (* -0.16666666666666666 y))))

C

double code(double x, double y) {
	double tmp;
	if ((cosh(x) * (sin(y) / y)) <= 20.0) {
		tmp = fma((y * y), -0.16666666666666666, 1.0);
	} else {
		tmp = (-1.0 / y) / (-0.16666666666666666 * y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(sin(y) / y)) <= 20.0)
		tmp = fma(Float64(y * y), -0.16666666666666666, 1.0);
	else
		tmp = Float64(Float64(-1.0 / y) / Float64(-0.16666666666666666 * y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Cosh[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], 20.0], N[(N[(y * y), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision], N[(N[(-1.0 / y), $MachinePrecision] / N[(-0.16666666666666666 * y), $MachinePrecision]), $MachinePrecision]]

ReFlow

f(x, y):
	x in [-inf, +inf],
	y in [-inf, +inf]
code: THEORY
BEGIN
f(x, y: real): real =
	LET tmp = IF (((((1) / (2)) * ((exp(x)) + ((1) / (exp(x))))) * ((sin(y)) / y)) <= (20)) THEN (((y * y) * (-1666666666666666574148081281236954964697360992431640625e-55)) + (1)) ELSE (((-1) / y) / ((-1666666666666666574148081281236954964697360992431640625e-55) * y)) ENDIF IN
	tmp
END code

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < 20

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

      \[\leadsto \frac{\sin y}{y} \]
   3. Step-by-step derivation

   4. Applied rewrites 51.0%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 33.0%

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

   9. Applied rewrites 33.0%

      \[\leadsto \mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right) \]

   if 20 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

      \[\leadsto \frac{\sin y}{y} \]
   3. Step-by-step derivation

   4. Applied rewrites 51.0%

      \[\leadsto \frac{\sin y}{y} \]
   5. Step-by-step derivation

   6. Applied rewrites 50.9%

      \[\leadsto \frac{\frac{-1}{y}}{\frac{-1}{\sin y}} \]

   7. Taylor expanded in y around 0

      \[\leadsto \frac{\frac{-1}{y}}{\frac{\frac{-1}{6} \cdot {y}^{2} - 1}{y}} \]
   8. Step-by-step derivation

   9. Applied rewrites 27.3%

      \[\leadsto \frac{\frac{-1}{y}}{\frac{-0.16666666666666666 \cdot {y}^{2} - 1}{y}} \]

   10. Taylor expanded in y around inf

       \[\leadsto \frac{\frac{-1}{y}}{\frac{-1}{6} \cdot y} \]
   11. Step-by-step derivation

   12. Applied rewrites 15.5%

       \[\leadsto \frac{\frac{-1}{y}}{-0.16666666666666666 \cdot y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 33.0% accurate, 6.1× speedup

Math

\[\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right) \]

FPCore

(FPCore (x y)
  :precision binary64
  :pre TRUE
  (fma (* y y) -0.16666666666666666 1.0))

C

double code(double x, double y) {
	return fma((y * y), -0.16666666666666666, 1.0);
}

Julia

function code(x, y)
	return fma(Float64(y * y), -0.16666666666666666, 1.0)
end

Wolfram

code[x_, y_] := N[(N[(y * y), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision]

ReFlow

f(x, y):
	x in [-inf, +inf],
	y in [-inf, +inf]
code: THEORY
BEGIN
f(x, y: real): real =
	((y * y) * (-1666666666666666574148081281236954964697360992431640625e-55)) + (1)
END code

Derivation

1. Initial program 99.9%

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

2. Taylor expanded in x around 0

   \[\leadsto \frac{\sin y}{y} \]
3. Step-by-step derivation

4. Applied rewrites 51.0%

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

5. Taylor expanded in y around 0

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

7. Applied rewrites 33.0%

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

9. Applied rewrites 33.0%

   \[\leadsto \mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right) \]
10. Add Preprocessing

Alternative 7: 26.7% accurate, 6.3× speedup

Math

\[\frac{1}{\frac{2}{2}} \]

FPCore

(FPCore (x y)
  :precision binary64
  :pre TRUE
  (/ 1.0 (/ 2.0 2.0)))

C

double code(double x, double y) {
	return 1.0 / (2.0 / 2.0);
}

Fortran

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 / (2.0d0 / 2.0d0)
end function

Java

public static double code(double x, double y) {
	return 1.0 / (2.0 / 2.0);
}

Python

def code(x, y):
	return 1.0 / (2.0 / 2.0)

Julia

function code(x, y)
	return Float64(1.0 / Float64(2.0 / 2.0))
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 / (2.0 / 2.0);
end

Wolfram

code[x_, y_] := N[(1.0 / N[(2.0 / 2.0), $MachinePrecision]), $MachinePrecision]

ReFlow

f(x, y):
	x in [-inf, +inf],
	y in [-inf, +inf]
code: THEORY
BEGIN
f(x, y: real): real =
	(1) / ((2) / (2))
END code

Derivation

1. Initial program 99.9%

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

3. Applied rewrites 99.8%

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

4. Taylor expanded in y around 0

   \[\leadsto \frac{1}{\frac{2}{e^{x} + \frac{1}{e^{x}}}} \]
5. Step-by-step derivation

6. Applied rewrites 62.4%

   \[\leadsto \frac{1}{\frac{2}{e^{x} + \frac{1}{e^{x}}}} \]

7. Taylor expanded in x around 0

   \[\leadsto \frac{1}{\frac{2}{2}} \]
8. Step-by-step derivation

9. Applied rewrites 26.7%

   \[\leadsto \frac{1}{\frac{2}{2}} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2026092 
(FPCore (x y)
  :name "Linear.Quaternion:$csinh from linear-1.19.1.3"
  :precision binary64
  (* (cosh x) (/ (sin y) y)))