Linear.Quaternion:$ccos from linear-1.19.1.3

Percentage Accurate: 100.0% → 100.0%

Time: 4.8s

Alternatives: 12

Speedup: 0.7×

• 50.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

ReFlow

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

ReFlow

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

Alternative 1: 100.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{\sinh y}{y} \leq 1.0005:\\ \;\;\;\;\sin x \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y \cdot \sin x}{y}\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  :pre TRUE
  (if (<= (/ (sinh y) y) 1.0005)
  (* (sin x) 1.0)
  (/ (* (sinh y) (sin x)) y)))

C

double code(double x, double y) {
	double tmp;
	if ((sinh(y) / y) <= 1.0005) {
		tmp = sin(x) * 1.0;
	} else {
		tmp = (sinh(y) * sin(x)) / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((sinh(y) / y) <= 1.0005d0) then
        tmp = sin(x) * 1.0d0
    else
        tmp = (sinh(y) * sin(x)) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((Math.sinh(y) / y) <= 1.0005) {
		tmp = Math.sin(x) * 1.0;
	} else {
		tmp = (Math.sinh(y) * Math.sin(x)) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (math.sinh(y) / y) <= 1.0005:
		tmp = math.sin(x) * 1.0
	else:
		tmp = (math.sinh(y) * math.sin(x)) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(sinh(y) / y) <= 1.0005)
		tmp = Float64(sin(x) * 1.0);
	else
		tmp = Float64(Float64(sinh(y) * sin(x)) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((sinh(y) / y) <= 1.0005)
		tmp = sin(x) * 1.0;
	else
		tmp = (sinh(y) * sin(x)) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision], 1.0005], N[(N[Sin[x], $MachinePrecision] * 1.0), $MachinePrecision], N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / y), $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(y)) + ((- (1)) / (exp(y))))) / y) <= (10004999999999999449329379785922355949878692626953125e-52)) THEN ((sin(x)) * (1)) ELSE (((((1) / (2)) * ((exp(y)) + ((- (1)) / (exp(y))))) * (sin(x))) / y) ENDIF IN
	tmp
END code

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sinh.f64 y) y) < 1.0004999999999999

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0

      \[\leadsto \sin x \cdot 1 \]
   3. Step-by-step derivation

   4. Applied rewrites 50.4%

      \[\leadsto \sin x \cdot 1 \]

   if 1.0004999999999999 < (/.f64 (sinh.f64 y) y)

   1. Initial program 100.0%

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

   3. Applied rewrites 89.0%

      \[\leadsto \frac{\sinh y \cdot \sin x}{y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 99.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(\left|x\right|\right)\\ t_1 := \frac{\sinh y}{y}\\ t_2 := t\_0 \cdot t\_1\\ \mathsf{copysign}\left(1, x\right) \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\sinh y \cdot \frac{\mathsf{fma}\left(-0.16666666666666666 \cdot \left|x\right|, \left|x\right|, 1\right) \cdot \left|x\right|}{y}\\ \mathbf{elif}\;t\_2 \leq 1:\\ \;\;\;\;t\_0 \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\left|x\right| \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  :pre TRUE
  (let* ((t_0 (sin (fabs x))) (t_1 (/ (sinh y) y)) (t_2 (* t_0 t_1)))
  (*
   (copysign 1.0 x)
   (if (<= t_2 (- INFINITY))
     (*
      (sinh y)
      (/
       (*
        (fma (* -0.16666666666666666 (fabs x)) (fabs x) 1.0)
        (fabs x))
       y))
     (if (<= t_2 1.0) (* t_0 1.0) (* (fabs x) t_1))))))

C

double code(double x, double y) {
	double t_0 = sin(fabs(x));
	double t_1 = sinh(y) / y;
	double t_2 = t_0 * t_1;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = sinh(y) * ((fma((-0.16666666666666666 * fabs(x)), fabs(x), 1.0) * fabs(x)) / y);
	} else if (t_2 <= 1.0) {
		tmp = t_0 * 1.0;
	} else {
		tmp = fabs(x) * t_1;
	}
	return copysign(1.0, x) * tmp;
}

Julia

function code(x, y)
	t_0 = sin(abs(x))
	t_1 = Float64(sinh(y) / y)
	t_2 = Float64(t_0 * t_1)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(sinh(y) * Float64(Float64(fma(Float64(-0.16666666666666666 * abs(x)), abs(x), 1.0) * abs(x)) / y));
	elseif (t_2 <= 1.0)
		tmp = Float64(t_0 * 1.0);
	else
		tmp = Float64(abs(x) * t_1);
	end
	return Float64(copysign(1.0, x) * tmp)
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Sin[N[Abs[x], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 * t$95$1), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$2, (-Infinity)], N[(N[Sinh[y], $MachinePrecision] * N[(N[(N[(N[(-0.16666666666666666 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1.0], N[(t$95$0 * 1.0), $MachinePrecision], N[(N[Abs[x], $MachinePrecision] * t$95$1), $MachinePrecision]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 63.1%

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

   6. Applied rewrites 52.2%

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

   8. Applied rewrites 51.6%

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

   if -inf.0 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0

      \[\leadsto \sin x \cdot 1 \]
   3. Step-by-step derivation

   4. Applied rewrites 50.4%

      \[\leadsto \sin x \cdot 1 \]

   if 1 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 40.7%

      \[\leadsto 0.5 \cdot \frac{x \cdot \left(e^{y} - \frac{1}{e^{y}}\right)}{y} \]
   5. Step-by-step derivation

   6. Applied rewrites 63.4%

      \[\leadsto x \cdot \frac{\sinh y}{y} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 76.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \frac{\sinh y}{y}\\ \mathsf{copysign}\left(1, x\right) \cdot \begin{array}{l} \mathbf{if}\;\sin \left(\left|x\right|\right) \cdot t\_0 \leq 0.035:\\ \;\;\;\;\sinh y \cdot \frac{\mathsf{fma}\left(-0.16666666666666666 \cdot \left|x\right|, \left|x\right|, 1\right) \cdot \left|x\right|}{y}\\ \mathbf{else}:\\ \;\;\;\;\left|x\right| \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  :pre TRUE
  (let* ((t_0 (/ (sinh y) y)))
  (*
   (copysign 1.0 x)
   (if (<= (* (sin (fabs x)) t_0) 0.035)
     (*
      (sinh y)
      (/
       (*
        (fma (* -0.16666666666666666 (fabs x)) (fabs x) 1.0)
        (fabs x))
       y))
     (* (fabs x) t_0)))))

C

double code(double x, double y) {
	double t_0 = sinh(y) / y;
	double tmp;
	if ((sin(fabs(x)) * t_0) <= 0.035) {
		tmp = sinh(y) * ((fma((-0.16666666666666666 * fabs(x)), fabs(x), 1.0) * fabs(x)) / y);
	} else {
		tmp = fabs(x) * t_0;
	}
	return copysign(1.0, x) * tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sinh(y) / y)
	tmp = 0.0
	if (Float64(sin(abs(x)) * t_0) <= 0.035)
		tmp = Float64(sinh(y) * Float64(Float64(fma(Float64(-0.16666666666666666 * abs(x)), abs(x), 1.0) * abs(x)) / y));
	else
		tmp = Float64(abs(x) * t_0);
	end
	return Float64(copysign(1.0, x) * tmp)
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(N[Sin[N[Abs[x], $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], 0.035], N[(N[Sinh[y], $MachinePrecision] * N[(N[(N[(N[(-0.16666666666666666 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(N[Abs[x], $MachinePrecision] * t$95$0), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < 0.035000000000000003

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 63.1%

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

   6. Applied rewrites 52.2%

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

   8. Applied rewrites 51.6%

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

   if 0.035000000000000003 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 40.7%

      \[\leadsto 0.5 \cdot \frac{x \cdot \left(e^{y} - \frac{1}{e^{y}}\right)}{y} \]
   5. Step-by-step derivation

   6. Applied rewrites 63.4%

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

Alternative 4: 73.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \left|x\right| \cdot \left|x\right|\\ t_1 := \frac{\sinh y}{y}\\ \mathsf{copysign}\left(1, x\right) \cdot \begin{array}{l} \mathbf{if}\;\sin \left(\left|x\right|\right) \cdot t\_1 \leq 0.035:\\ \;\;\;\;\left(\left|x\right| \cdot \left(1 + -0.16666666666666666 \cdot \sqrt{t\_0 \cdot t\_0}\right)\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\left|x\right| \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  :pre TRUE
  (let* ((t_0 (* (fabs x) (fabs x))) (t_1 (/ (sinh y) y)))
  (*
   (copysign 1.0 x)
   (if (<= (* (sin (fabs x)) t_1) 0.035)
     (*
      (* (fabs x) (+ 1.0 (* -0.16666666666666666 (sqrt (* t_0 t_0)))))
      1.0)
     (* (fabs x) t_1)))))

C

double code(double x, double y) {
	double t_0 = fabs(x) * fabs(x);
	double t_1 = sinh(y) / y;
	double tmp;
	if ((sin(fabs(x)) * t_1) <= 0.035) {
		tmp = (fabs(x) * (1.0 + (-0.16666666666666666 * sqrt((t_0 * t_0))))) * 1.0;
	} else {
		tmp = fabs(x) * t_1;
	}
	return copysign(1.0, x) * tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = Math.abs(x) * Math.abs(x);
	double t_1 = Math.sinh(y) / y;
	double tmp;
	if ((Math.sin(Math.abs(x)) * t_1) <= 0.035) {
		tmp = (Math.abs(x) * (1.0 + (-0.16666666666666666 * Math.sqrt((t_0 * t_0))))) * 1.0;
	} else {
		tmp = Math.abs(x) * t_1;
	}
	return Math.copySign(1.0, x) * tmp;
}

Python

def code(x, y):
	t_0 = math.fabs(x) * math.fabs(x)
	t_1 = math.sinh(y) / y
	tmp = 0
	if (math.sin(math.fabs(x)) * t_1) <= 0.035:
		tmp = (math.fabs(x) * (1.0 + (-0.16666666666666666 * math.sqrt((t_0 * t_0))))) * 1.0
	else:
		tmp = math.fabs(x) * t_1
	return math.copysign(1.0, x) * tmp

Julia

function code(x, y)
	t_0 = Float64(abs(x) * abs(x))
	t_1 = Float64(sinh(y) / y)
	tmp = 0.0
	if (Float64(sin(abs(x)) * t_1) <= 0.035)
		tmp = Float64(Float64(abs(x) * Float64(1.0 + Float64(-0.16666666666666666 * sqrt(Float64(t_0 * t_0))))) * 1.0);
	else
		tmp = Float64(abs(x) * t_1);
	end
	return Float64(copysign(1.0, x) * tmp)
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = abs(x) * abs(x);
	t_1 = sinh(y) / y;
	tmp = 0.0;
	if ((sin(abs(x)) * t_1) <= 0.035)
		tmp = (abs(x) * (1.0 + (-0.16666666666666666 * sqrt((t_0 * t_0))))) * 1.0;
	else
		tmp = abs(x) * t_1;
	end
	tmp_2 = (sign(x) * abs(1.0)) * tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(N[Sin[N[Abs[x], $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision], 0.035], N[(N[(N[Abs[x], $MachinePrecision] * N[(1.0 + N[(-0.16666666666666666 * N[Sqrt[N[(t$95$0 * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[Abs[x], $MachinePrecision] * t$95$1), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < 0.035000000000000003

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0

      \[\leadsto \sin x \cdot 1 \]
   3. Step-by-step derivation

   4. Applied rewrites 50.4%

      \[\leadsto \sin x \cdot 1 \]

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 34.6%

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

   9. Applied rewrites 35.5%

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

   if 0.035000000000000003 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 40.7%

      \[\leadsto 0.5 \cdot \frac{x \cdot \left(e^{y} - \frac{1}{e^{y}}\right)}{y} \]
   5. Step-by-step derivation

   6. Applied rewrites 63.4%

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

Alternative 5: 73.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \frac{\sinh y}{y}\\ \mathsf{copysign}\left(1, x\right) \cdot \begin{array}{l} \mathbf{if}\;\sin \left(\left|x\right|\right) \cdot t\_0 \leq 0.035:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, \sqrt{\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|} \cdot \left|x\right|, \left|x\right|\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\left|x\right| \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  :pre TRUE
  (let* ((t_0 (/ (sinh y) y)))
  (*
   (copysign 1.0 x)
   (if (<= (* (sin (fabs x)) t_0) 0.035)
     (*
      (fma
       -0.16666666666666666
       (*
        (sqrt (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)))
        (fabs x))
       (fabs x))
      1.0)
     (* (fabs x) t_0)))))

C

double code(double x, double y) {
	double t_0 = sinh(y) / y;
	double tmp;
	if ((sin(fabs(x)) * t_0) <= 0.035) {
		tmp = fma(-0.16666666666666666, (sqrt((((fabs(x) * fabs(x)) * fabs(x)) * fabs(x))) * fabs(x)), fabs(x)) * 1.0;
	} else {
		tmp = fabs(x) * t_0;
	}
	return copysign(1.0, x) * tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sinh(y) / y)
	tmp = 0.0
	if (Float64(sin(abs(x)) * t_0) <= 0.035)
		tmp = Float64(fma(-0.16666666666666666, Float64(sqrt(Float64(Float64(Float64(abs(x) * abs(x)) * abs(x)) * abs(x))) * abs(x)), abs(x)) * 1.0);
	else
		tmp = Float64(abs(x) * t_0);
	end
	return Float64(copysign(1.0, x) * tmp)
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(N[Sin[N[Abs[x], $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], 0.035], N[(N[(-0.16666666666666666 * N[(N[Sqrt[N[(N[(N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[Abs[x], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[Abs[x], $MachinePrecision] * t$95$0), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < 0.035000000000000003

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0

      \[\leadsto \sin x \cdot 1 \]
   3. Step-by-step derivation

   4. Applied rewrites 50.4%

      \[\leadsto \sin x \cdot 1 \]

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 34.6%

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

   9. Applied rewrites 34.6%

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

       1. rem-square-sqrt N/A

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

       2. sqrt-unprod N/A

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

       3. lift-*.f64 N/A

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

       4. lift-sqrt.f64 35.5%

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. associate-*r* N/A

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

       8. lift-*.f64 N/A

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

       9. lower-*.f64 35.5%

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

   11. Applied rewrites 35.5%

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

   if 0.035000000000000003 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 40.7%

      \[\leadsto 0.5 \cdot \frac{x \cdot \left(e^{y} - \frac{1}{e^{y}}\right)}{y} \]
   5. Step-by-step derivation

   6. Applied rewrites 63.4%

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

Alternative 6: 72.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \frac{\sinh y}{y}\\ \mathsf{copysign}\left(1, x\right) \cdot \begin{array}{l} \mathbf{if}\;\sin \left(\left|x\right|\right) \cdot t\_0 \leq 0.035:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, \left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|, \left|x\right|\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\left|x\right| \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  :pre TRUE
  (let* ((t_0 (/ (sinh y) y)))
  (*
   (copysign 1.0 x)
   (if (<= (* (sin (fabs x)) t_0) 0.035)
     (*
      (fma
       -0.16666666666666666
       (* (* (fabs x) (fabs x)) (fabs x))
       (fabs x))
      1.0)
     (* (fabs x) t_0)))))

C

double code(double x, double y) {
	double t_0 = sinh(y) / y;
	double tmp;
	if ((sin(fabs(x)) * t_0) <= 0.035) {
		tmp = fma(-0.16666666666666666, ((fabs(x) * fabs(x)) * fabs(x)), fabs(x)) * 1.0;
	} else {
		tmp = fabs(x) * t_0;
	}
	return copysign(1.0, x) * tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sinh(y) / y)
	tmp = 0.0
	if (Float64(sin(abs(x)) * t_0) <= 0.035)
		tmp = Float64(fma(-0.16666666666666666, Float64(Float64(abs(x) * abs(x)) * abs(x)), abs(x)) * 1.0);
	else
		tmp = Float64(abs(x) * t_0);
	end
	return Float64(copysign(1.0, x) * tmp)
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(N[Sin[N[Abs[x], $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], 0.035], N[(N[(-0.16666666666666666 * N[(N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[Abs[x], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[Abs[x], $MachinePrecision] * t$95$0), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < 0.035000000000000003

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0

      \[\leadsto \sin x \cdot 1 \]
   3. Step-by-step derivation

   4. Applied rewrites 50.4%

      \[\leadsto \sin x \cdot 1 \]

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 34.6%

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

   9. Applied rewrites 34.6%

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

   if 0.035000000000000003 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 40.7%

      \[\leadsto 0.5 \cdot \frac{x \cdot \left(e^{y} - \frac{1}{e^{y}}\right)}{y} \]
   5. Step-by-step derivation

   6. Applied rewrites 63.4%

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

Alternative 7: 64.2% accurate, 0.6× speedup

Math

\[\mathsf{copysign}\left(1, x\right) \cdot \begin{array}{l} \mathbf{if}\;\sin \left(\left|x\right|\right) \cdot \frac{\sinh y}{y} \leq 0.035:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, \left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|, \left|x\right|\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\left(y \cdot y\right) \cdot \left(y \cdot y\right)} \cdot \left|x\right|, 0.16666666666666666, \left|x\right|\right)\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  :pre TRUE
  (*
 (copysign 1.0 x)
 (if (<= (* (sin (fabs x)) (/ (sinh y) y)) 0.035)
   (*
    (fma
     -0.16666666666666666
     (* (* (fabs x) (fabs x)) (fabs x))
     (fabs x))
    1.0)
   (fma
    (* (sqrt (* (* y y) (* y y))) (fabs x))
    0.16666666666666666
    (fabs x)))))

C

double code(double x, double y) {
	double tmp;
	if ((sin(fabs(x)) * (sinh(y) / y)) <= 0.035) {
		tmp = fma(-0.16666666666666666, ((fabs(x) * fabs(x)) * fabs(x)), fabs(x)) * 1.0;
	} else {
		tmp = fma((sqrt(((y * y) * (y * y))) * fabs(x)), 0.16666666666666666, fabs(x));
	}
	return copysign(1.0, x) * tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(sin(abs(x)) * Float64(sinh(y) / y)) <= 0.035)
		tmp = Float64(fma(-0.16666666666666666, Float64(Float64(abs(x) * abs(x)) * abs(x)), abs(x)) * 1.0);
	else
		tmp = fma(Float64(sqrt(Float64(Float64(y * y) * Float64(y * y))) * abs(x)), 0.16666666666666666, abs(x));
	end
	return Float64(copysign(1.0, x) * tmp)
end

Wolfram

code[x_, y_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(N[Sin[N[Abs[x], $MachinePrecision]], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], 0.035], N[(N[(-0.16666666666666666 * N[(N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[Abs[x], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[(N[Sqrt[N[(N[(y * y), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * 0.16666666666666666 + N[Abs[x], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < 0.035000000000000003

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0

      \[\leadsto \sin x \cdot 1 \]
   3. Step-by-step derivation

   4. Applied rewrites 50.4%

      \[\leadsto \sin x \cdot 1 \]

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 34.6%

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

   9. Applied rewrites 34.6%

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

   if 0.035000000000000003 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 40.7%

      \[\leadsto 0.5 \cdot \frac{x \cdot \left(e^{y} - \frac{1}{e^{y}}\right)}{y} \]

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 48.1%

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

   9. Applied rewrites 48.1%

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

       1. rem-square-sqrt N/A

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

       2. sqrt-unprod N/A

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

       3. lower-sqrt.f64 N/A

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

       4. lower-*.f64 55.1%

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

   11. Applied rewrites 55.1%

       \[\leadsto \mathsf{fma}\left(\sqrt{\left(y \cdot y\right) \cdot \left(y \cdot y\right)} \cdot x, 0.16666666666666666, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 57.2% accurate, 0.7× speedup

Math

\[\mathsf{copysign}\left(1, x\right) \cdot \begin{array}{l} \mathbf{if}\;\sin \left(\left|x\right|\right) \cdot \frac{\sinh y}{y} \leq 0.035:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, \left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|, \left|x\right|\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(y \cdot y\right) \cdot \left|x\right|, 0.16666666666666666, \left|x\right|\right)\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  :pre TRUE
  (*
 (copysign 1.0 x)
 (if (<= (* (sin (fabs x)) (/ (sinh y) y)) 0.035)
   (*
    (fma
     -0.16666666666666666
     (* (* (fabs x) (fabs x)) (fabs x))
     (fabs x))
    1.0)
   (fma (* (* y y) (fabs x)) 0.16666666666666666 (fabs x)))))

C

double code(double x, double y) {
	double tmp;
	if ((sin(fabs(x)) * (sinh(y) / y)) <= 0.035) {
		tmp = fma(-0.16666666666666666, ((fabs(x) * fabs(x)) * fabs(x)), fabs(x)) * 1.0;
	} else {
		tmp = fma(((y * y) * fabs(x)), 0.16666666666666666, fabs(x));
	}
	return copysign(1.0, x) * tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(sin(abs(x)) * Float64(sinh(y) / y)) <= 0.035)
		tmp = Float64(fma(-0.16666666666666666, Float64(Float64(abs(x) * abs(x)) * abs(x)), abs(x)) * 1.0);
	else
		tmp = fma(Float64(Float64(y * y) * abs(x)), 0.16666666666666666, abs(x));
	end
	return Float64(copysign(1.0, x) * tmp)
end

Wolfram

code[x_, y_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(N[Sin[N[Abs[x], $MachinePrecision]], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], 0.035], N[(N[(-0.16666666666666666 * N[(N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[Abs[x], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[(N[(y * y), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * 0.16666666666666666 + N[Abs[x], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < 0.035000000000000003

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0

      \[\leadsto \sin x \cdot 1 \]
   3. Step-by-step derivation

   4. Applied rewrites 50.4%

      \[\leadsto \sin x \cdot 1 \]

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 34.6%

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

   9. Applied rewrites 34.6%

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

   if 0.035000000000000003 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 40.7%

      \[\leadsto 0.5 \cdot \frac{x \cdot \left(e^{y} - \frac{1}{e^{y}}\right)}{y} \]

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 48.1%

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

   9. Applied rewrites 48.1%

      \[\leadsto \mathsf{fma}\left(\left(y \cdot y\right) \cdot x, 0.16666666666666666, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 57.2% accurate, 0.7× speedup

Math

\[\mathsf{copysign}\left(1, x\right) \cdot \begin{array}{l} \mathbf{if}\;\sin \left(\left|x\right|\right) \cdot \frac{\sinh y}{y} \leq 0.035:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.16666666666666666 \cdot \left|x\right|, \left|x\right|, 1\right) \cdot \left|x\right|\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(y \cdot y\right) \cdot \left|x\right|, 0.16666666666666666, \left|x\right|\right)\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  :pre TRUE
  (*
 (copysign 1.0 x)
 (if (<= (* (sin (fabs x)) (/ (sinh y) y)) 0.035)
   (*
    (* (fma (* -0.16666666666666666 (fabs x)) (fabs x) 1.0) (fabs x))
    1.0)
   (fma (* (* y y) (fabs x)) 0.16666666666666666 (fabs x)))))

C

double code(double x, double y) {
	double tmp;
	if ((sin(fabs(x)) * (sinh(y) / y)) <= 0.035) {
		tmp = (fma((-0.16666666666666666 * fabs(x)), fabs(x), 1.0) * fabs(x)) * 1.0;
	} else {
		tmp = fma(((y * y) * fabs(x)), 0.16666666666666666, fabs(x));
	}
	return copysign(1.0, x) * tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(sin(abs(x)) * Float64(sinh(y) / y)) <= 0.035)
		tmp = Float64(Float64(fma(Float64(-0.16666666666666666 * abs(x)), abs(x), 1.0) * abs(x)) * 1.0);
	else
		tmp = fma(Float64(Float64(y * y) * abs(x)), 0.16666666666666666, abs(x));
	end
	return Float64(copysign(1.0, x) * tmp)
end

Wolfram

code[x_, y_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(N[Sin[N[Abs[x], $MachinePrecision]], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], 0.035], N[(N[(N[(N[(-0.16666666666666666 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[(N[(y * y), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * 0.16666666666666666 + N[Abs[x], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < 0.035000000000000003

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0

      \[\leadsto \sin x \cdot 1 \]
   3. Step-by-step derivation

   4. Applied rewrites 50.4%

      \[\leadsto \sin x \cdot 1 \]

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 34.6%

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

   9. Applied rewrites 34.6%

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

   if 0.035000000000000003 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 40.7%

      \[\leadsto 0.5 \cdot \frac{x \cdot \left(e^{y} - \frac{1}{e^{y}}\right)}{y} \]

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 48.1%

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

   9. Applied rewrites 48.1%

      \[\leadsto \mathsf{fma}\left(\left(y \cdot y\right) \cdot x, 0.16666666666666666, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 48.1% accurate, 4.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

ReFlow

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

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 40.7%

   \[\leadsto 0.5 \cdot \frac{x \cdot \left(e^{y} - \frac{1}{e^{y}}\right)}{y} \]

5. Taylor expanded in y around 0

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

7. Applied rewrites 48.1%

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

9. Applied rewrites 48.1%

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

Alternative 11: 26.6% accurate, 7.9× speedup

Math

\[\left(x \cdot 1\right) \cdot 1 \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

ReFlow

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

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in y around 0

   \[\leadsto \sin x \cdot 1 \]
3. Step-by-step derivation

4. Applied rewrites 50.4%

   \[\leadsto \sin x \cdot 1 \]

5. Taylor expanded in x around 0

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

7. Applied rewrites 34.6%

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

8. Taylor expanded in x around 0

   \[\leadsto \left(x \cdot 1\right) \cdot 1 \]
9. Step-by-step derivation

10. Applied rewrites 26.6%

    \[\leadsto \left(x \cdot 1\right) \cdot 1 \]
11. Add Preprocessing

Reproduce

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