Linear.Quaternion:$ccosh from linear-1.19.1.3

Percentage Accurate: 88.6% → 99.9%

Time: 2.9s

Alternatives: 13

Speedup: 1.0×

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

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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)) / x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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)) / x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.6%

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

   1. lift-/.f64 N/A

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

   2. mult-flip N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. mult-flip-rev N/A

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

   9. lower-/.f64 99.9

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

3. Applied rewrites 99.9%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/r/ N/A

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

   4. div-flip-rev N/A

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

   5. mult-flip N/A

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

   6. associate-*l/ N/A

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

   7. mult-flip N/A

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

   8. div-flip-rev N/A

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

   9. lower-/.f64 N/A

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

   10. lower-/.f64 99.9

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

5. Applied rewrites 99.9%

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

Alternative 2: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.6%

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

   1. lift-/.f64 N/A

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

   2. mult-flip N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. mult-flip-rev N/A

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

   9. lower-/.f64 99.9

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

3. Applied rewrites 99.9%

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

Alternative 3: 99.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \sinh \left(\left|y\right|\right)\\ t_1 := \frac{\sin x \cdot t\_0}{x}\\ \mathsf{copysign}\left(1, y\right) \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(1 + -0.16666666666666666 \cdot {x}^{2}\right) \cdot t\_0\\ \mathbf{elif}\;t\_1 \leq 10^{-11}:\\ \;\;\;\;\frac{\left|y\right|}{\frac{x}{\sin x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{x \cdot t\_0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (sinh (fabs y))) (t_1 (/ (* (sin x) t_0) x)))
   (*
    (copysign 1.0 y)
    (if (<= t_1 (- INFINITY))
      (* (+ 1.0 (* -0.16666666666666666 (pow x 2.0))) t_0)
      (if (<= t_1 1e-11)
        (/ (fabs y) (/ x (sin x)))
        (/ 1.0 (/ x (* x t_0))))))))

C

double code(double x, double y) {
	double t_0 = sinh(fabs(y));
	double t_1 = (sin(x) * t_0) / x;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (1.0 + (-0.16666666666666666 * pow(x, 2.0))) * t_0;
	} else if (t_1 <= 1e-11) {
		tmp = fabs(y) / (x / sin(x));
	} else {
		tmp = 1.0 / (x / (x * t_0));
	}
	return copysign(1.0, y) * tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = Math.sinh(Math.abs(y));
	double t_1 = (Math.sin(x) * t_0) / x;
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = (1.0 + (-0.16666666666666666 * Math.pow(x, 2.0))) * t_0;
	} else if (t_1 <= 1e-11) {
		tmp = Math.abs(y) / (x / Math.sin(x));
	} else {
		tmp = 1.0 / (x / (x * t_0));
	}
	return Math.copySign(1.0, y) * tmp;
}

Python

def code(x, y):
	t_0 = math.sinh(math.fabs(y))
	t_1 = (math.sin(x) * t_0) / x
	tmp = 0
	if t_1 <= -math.inf:
		tmp = (1.0 + (-0.16666666666666666 * math.pow(x, 2.0))) * t_0
	elif t_1 <= 1e-11:
		tmp = math.fabs(y) / (x / math.sin(x))
	else:
		tmp = 1.0 / (x / (x * t_0))
	return math.copysign(1.0, y) * tmp

Julia

function code(x, y)
	t_0 = sinh(abs(y))
	t_1 = Float64(Float64(sin(x) * t_0) / x)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(1.0 + Float64(-0.16666666666666666 * (x ^ 2.0))) * t_0);
	elseif (t_1 <= 1e-11)
		tmp = Float64(abs(y) / Float64(x / sin(x)));
	else
		tmp = Float64(1.0 / Float64(x / Float64(x * t_0)));
	end
	return Float64(copysign(1.0, y) * tmp)
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sinh(abs(y));
	t_1 = (sin(x) * t_0) / x;
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = (1.0 + (-0.16666666666666666 * (x ^ 2.0))) * t_0;
	elseif (t_1 <= 1e-11)
		tmp = abs(y) / (x / sin(x));
	else
		tmp = 1.0 / (x / (x * t_0));
	end
	tmp_2 = (sign(y) * abs(1.0)) * tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Sinh[N[Abs[y], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Sin[x], $MachinePrecision] * t$95$0), $MachinePrecision] / x), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$1, (-Infinity)], N[(N[(1.0 + N[(-0.16666666666666666 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[t$95$1, 1e-11], N[(N[Abs[y], $MachinePrecision] / N[(x / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(x / N[(x * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 88.6%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. mult-flip-rev N/A

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

      9. lower-/.f64 99.9

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

         \[\leadsto \left(1 + \color{blue}{\frac{-1}{6} \cdot {x}^{2}}\right) \cdot \sinh y \]

      2. lower-*.f64 N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot \color{blue}{{x}^{2}}\right) \cdot \sinh y \]

      3. lower-pow.f64 62.0

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

   6. Applied rewrites 62.0%

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

   if -inf.0 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 9.99999999999999939e-12

   1. Initial program 88.6%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 41.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 63.9

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

   6. Applied rewrites 63.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. div-flip-rev N/A

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

      6. lift-/.f64 N/A

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

      7. mult-flip-rev N/A

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

      8. lower-/.f64 52.5

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

   8. Applied rewrites 52.5%

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

   if 9.99999999999999939e-12 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 88.6%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 51.3%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound-/.f64 51.3

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

   6. Applied rewrites 51.3%

      \[\leadsto \color{blue}{\frac{1}{\frac{x}{x \cdot \sinh y}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 99.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \sinh \left(\left|y\right|\right)\\ t_1 := \frac{\sin x \cdot t\_0}{x}\\ \mathsf{copysign}\left(1, y\right) \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(1 + -0.16666666666666666 \cdot {x}^{2}\right) \cdot t\_0\\ \mathbf{elif}\;t\_1 \leq 10^{-11}:\\ \;\;\;\;\frac{\sin x}{x} \cdot \left|y\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{x \cdot t\_0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (sinh (fabs y))) (t_1 (/ (* (sin x) t_0) x)))
   (*
    (copysign 1.0 y)
    (if (<= t_1 (- INFINITY))
      (* (+ 1.0 (* -0.16666666666666666 (pow x 2.0))) t_0)
      (if (<= t_1 1e-11)
        (* (/ (sin x) x) (fabs y))
        (/ 1.0 (/ x (* x t_0))))))))

C

double code(double x, double y) {
	double t_0 = sinh(fabs(y));
	double t_1 = (sin(x) * t_0) / x;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (1.0 + (-0.16666666666666666 * pow(x, 2.0))) * t_0;
	} else if (t_1 <= 1e-11) {
		tmp = (sin(x) / x) * fabs(y);
	} else {
		tmp = 1.0 / (x / (x * t_0));
	}
	return copysign(1.0, y) * tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = Math.sinh(Math.abs(y));
	double t_1 = (Math.sin(x) * t_0) / x;
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = (1.0 + (-0.16666666666666666 * Math.pow(x, 2.0))) * t_0;
	} else if (t_1 <= 1e-11) {
		tmp = (Math.sin(x) / x) * Math.abs(y);
	} else {
		tmp = 1.0 / (x / (x * t_0));
	}
	return Math.copySign(1.0, y) * tmp;
}

Python

def code(x, y):
	t_0 = math.sinh(math.fabs(y))
	t_1 = (math.sin(x) * t_0) / x
	tmp = 0
	if t_1 <= -math.inf:
		tmp = (1.0 + (-0.16666666666666666 * math.pow(x, 2.0))) * t_0
	elif t_1 <= 1e-11:
		tmp = (math.sin(x) / x) * math.fabs(y)
	else:
		tmp = 1.0 / (x / (x * t_0))
	return math.copysign(1.0, y) * tmp

Julia

function code(x, y)
	t_0 = sinh(abs(y))
	t_1 = Float64(Float64(sin(x) * t_0) / x)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(1.0 + Float64(-0.16666666666666666 * (x ^ 2.0))) * t_0);
	elseif (t_1 <= 1e-11)
		tmp = Float64(Float64(sin(x) / x) * abs(y));
	else
		tmp = Float64(1.0 / Float64(x / Float64(x * t_0)));
	end
	return Float64(copysign(1.0, y) * tmp)
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sinh(abs(y));
	t_1 = (sin(x) * t_0) / x;
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = (1.0 + (-0.16666666666666666 * (x ^ 2.0))) * t_0;
	elseif (t_1 <= 1e-11)
		tmp = (sin(x) / x) * abs(y);
	else
		tmp = 1.0 / (x / (x * t_0));
	end
	tmp_2 = (sign(y) * abs(1.0)) * tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Sinh[N[Abs[y], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Sin[x], $MachinePrecision] * t$95$0), $MachinePrecision] / x), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$1, (-Infinity)], N[(N[(1.0 + N[(-0.16666666666666666 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[t$95$1, 1e-11], N[(N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision] * N[Abs[y], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(x / N[(x * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 88.6%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. mult-flip-rev N/A

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

      9. lower-/.f64 99.9

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

         \[\leadsto \left(1 + \color{blue}{\frac{-1}{6} \cdot {x}^{2}}\right) \cdot \sinh y \]

      2. lower-*.f64 N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot \color{blue}{{x}^{2}}\right) \cdot \sinh y \]

      3. lower-pow.f64 62.0

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

   6. Applied rewrites 62.0%

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

   if -inf.0 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 9.99999999999999939e-12

   1. Initial program 88.6%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 41.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lift-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 52.5

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

   6. Applied rewrites 52.5%

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

   if 9.99999999999999939e-12 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 88.6%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 51.3%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound-/.f64 51.3

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

   6. Applied rewrites 51.3%

      \[\leadsto \color{blue}{\frac{1}{\frac{x}{x \cdot \sinh y}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 98.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \sinh \left(\left|y\right|\right)\\ t_1 := \frac{\sin x \cdot t\_0}{x}\\ \mathsf{copysign}\left(1, y\right) \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(1 + -0.16666666666666666 \cdot {x}^{2}\right) \cdot t\_0\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-112}:\\ \;\;\;\;\frac{\left|y\right|}{x} \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (sinh (fabs y))) (t_1 (/ (* (sin x) t_0) x)))
   (*
    (copysign 1.0 y)
    (if (<= t_1 (- INFINITY))
      (* (+ 1.0 (* -0.16666666666666666 (pow x 2.0))) t_0)
      (if (<= t_1 2e-112) (* (/ (fabs y) x) (sin x)) t_0)))))

C

double code(double x, double y) {
	double t_0 = sinh(fabs(y));
	double t_1 = (sin(x) * t_0) / x;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (1.0 + (-0.16666666666666666 * pow(x, 2.0))) * t_0;
	} else if (t_1 <= 2e-112) {
		tmp = (fabs(y) / x) * sin(x);
	} else {
		tmp = t_0;
	}
	return copysign(1.0, y) * tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = Math.sinh(Math.abs(y));
	double t_1 = (Math.sin(x) * t_0) / x;
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = (1.0 + (-0.16666666666666666 * Math.pow(x, 2.0))) * t_0;
	} else if (t_1 <= 2e-112) {
		tmp = (Math.abs(y) / x) * Math.sin(x);
	} else {
		tmp = t_0;
	}
	return Math.copySign(1.0, y) * tmp;
}

Python

def code(x, y):
	t_0 = math.sinh(math.fabs(y))
	t_1 = (math.sin(x) * t_0) / x
	tmp = 0
	if t_1 <= -math.inf:
		tmp = (1.0 + (-0.16666666666666666 * math.pow(x, 2.0))) * t_0
	elif t_1 <= 2e-112:
		tmp = (math.fabs(y) / x) * math.sin(x)
	else:
		tmp = t_0
	return math.copysign(1.0, y) * tmp

Julia

function code(x, y)
	t_0 = sinh(abs(y))
	t_1 = Float64(Float64(sin(x) * t_0) / x)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(1.0 + Float64(-0.16666666666666666 * (x ^ 2.0))) * t_0);
	elseif (t_1 <= 2e-112)
		tmp = Float64(Float64(abs(y) / x) * sin(x));
	else
		tmp = t_0;
	end
	return Float64(copysign(1.0, y) * tmp)
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sinh(abs(y));
	t_1 = (sin(x) * t_0) / x;
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = (1.0 + (-0.16666666666666666 * (x ^ 2.0))) * t_0;
	elseif (t_1 <= 2e-112)
		tmp = (abs(y) / x) * sin(x);
	else
		tmp = t_0;
	end
	tmp_2 = (sign(y) * abs(1.0)) * tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Sinh[N[Abs[y], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Sin[x], $MachinePrecision] * t$95$0), $MachinePrecision] / x), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$1, (-Infinity)], N[(N[(1.0 + N[(-0.16666666666666666 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[t$95$1, 2e-112], N[(N[(N[Abs[y], $MachinePrecision] / x), $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision], t$95$0]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 88.6%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. mult-flip-rev N/A

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

      9. lower-/.f64 99.9

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

         \[\leadsto \left(1 + \color{blue}{\frac{-1}{6} \cdot {x}^{2}}\right) \cdot \sinh y \]

      2. lower-*.f64 N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot \color{blue}{{x}^{2}}\right) \cdot \sinh y \]

      3. lower-pow.f64 62.0

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

   6. Applied rewrites 62.0%

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

   if -inf.0 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 1.9999999999999999e-112

   1. Initial program 88.6%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 41.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 63.9

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

   6. Applied rewrites 63.9%

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

   if 1.9999999999999999e-112 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 88.6%

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

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(e^{y} - \frac{1}{e^{y}}\right)} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{y} - \color{blue}{\frac{1}{e^{y}}}\right) \]

      3. lower-exp.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{y} - \frac{\color{blue}{1}}{e^{y}}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{y} - \frac{1}{\color{blue}{e^{y}}}\right) \]

      5. lower-exp.f64 51.6

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

   4. Applied rewrites 51.6%

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

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(e^{y} - \frac{1}{e^{y}}\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(e^{y} - \frac{1}{e^{y}}\right) \cdot \color{blue}{\frac{1}{2}} \]

      3. metadata-eval N/A

         \[\leadsto \left(e^{y} - \frac{1}{e^{y}}\right) \cdot \frac{1}{\color{blue}{2}} \]

      4. mult-flip N/A

         \[\leadsto \frac{e^{y} - \frac{1}{e^{y}}}{\color{blue}{2}} \]

      5. lift--.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. rec-exp N/A

         \[\leadsto \frac{e^{y} - e^{\mathsf{neg}\left(y\right)}}{2} \]

      10. sinh-def N/A

          \[\leadsto \sinh y \]

      11. lift-sinh.f64 62.6

          \[\leadsto \sinh y \]

   6. Applied rewrites 62.6%

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

Alternative 6: 85.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \sinh \left(\left|y\right|\right)\\ \mathsf{copysign}\left(1, y\right) \cdot \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot t\_0}{x} \leq -2 \cdot 10^{-200}:\\ \;\;\;\;\left(1 + -0.16666666666666666 \cdot {x}^{2}\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{x} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (sinh (fabs y))))
   (*
    (copysign 1.0 y)
    (if (<= (/ (* (sin x) t_0) x) -2e-200)
      (* (+ 1.0 (* -0.16666666666666666 (pow x 2.0))) t_0)
      (* (/ t_0 x) x)))))

C

double code(double x, double y) {
	double t_0 = sinh(fabs(y));
	double tmp;
	if (((sin(x) * t_0) / x) <= -2e-200) {
		tmp = (1.0 + (-0.16666666666666666 * pow(x, 2.0))) * t_0;
	} else {
		tmp = (t_0 / x) * x;
	}
	return copysign(1.0, y) * tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = Math.sinh(Math.abs(y));
	double tmp;
	if (((Math.sin(x) * t_0) / x) <= -2e-200) {
		tmp = (1.0 + (-0.16666666666666666 * Math.pow(x, 2.0))) * t_0;
	} else {
		tmp = (t_0 / x) * x;
	}
	return Math.copySign(1.0, y) * tmp;
}

Python

def code(x, y):
	t_0 = math.sinh(math.fabs(y))
	tmp = 0
	if ((math.sin(x) * t_0) / x) <= -2e-200:
		tmp = (1.0 + (-0.16666666666666666 * math.pow(x, 2.0))) * t_0
	else:
		tmp = (t_0 / x) * x
	return math.copysign(1.0, y) * tmp

Julia

function code(x, y)
	t_0 = sinh(abs(y))
	tmp = 0.0
	if (Float64(Float64(sin(x) * t_0) / x) <= -2e-200)
		tmp = Float64(Float64(1.0 + Float64(-0.16666666666666666 * (x ^ 2.0))) * t_0);
	else
		tmp = Float64(Float64(t_0 / x) * x);
	end
	return Float64(copysign(1.0, y) * tmp)
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sinh(abs(y));
	tmp = 0.0;
	if (((sin(x) * t_0) / x) <= -2e-200)
		tmp = (1.0 + (-0.16666666666666666 * (x ^ 2.0))) * t_0;
	else
		tmp = (t_0 / x) * x;
	end
	tmp_2 = (sign(y) * abs(1.0)) * tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Sinh[N[Abs[y], $MachinePrecision]], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(N[(N[Sin[x], $MachinePrecision] * t$95$0), $MachinePrecision] / x), $MachinePrecision], -2e-200], N[(N[(1.0 + N[(-0.16666666666666666 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(t$95$0 / x), $MachinePrecision] * x), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -2e-200

   1. Initial program 88.6%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. mult-flip-rev N/A

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

      9. lower-/.f64 99.9

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

         \[\leadsto \left(1 + \color{blue}{\frac{-1}{6} \cdot {x}^{2}}\right) \cdot \sinh y \]

      2. lower-*.f64 N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot \color{blue}{{x}^{2}}\right) \cdot \sinh y \]

      3. lower-pow.f64 62.0

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

   6. Applied rewrites 62.0%

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

   if -2e-200 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 88.6%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 51.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 73.1

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

   6. Applied rewrites 73.1%

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

Alternative 7: 83.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \sinh \left(\left|y\right|\right)\\ \mathsf{copysign}\left(1, y\right) \cdot \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot t\_0}{x} \leq -2 \cdot 10^{-200}:\\ \;\;\;\;\frac{\left(x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{2}\right)\right) \cdot \left|y\right|}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{x} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (sinh (fabs y))))
   (*
    (copysign 1.0 y)
    (if (<= (/ (* (sin x) t_0) x) -2e-200)
      (/ (* (* x (+ 1.0 (* -0.16666666666666666 (pow x 2.0)))) (fabs y)) x)
      (* (/ t_0 x) x)))))

C

double code(double x, double y) {
	double t_0 = sinh(fabs(y));
	double tmp;
	if (((sin(x) * t_0) / x) <= -2e-200) {
		tmp = ((x * (1.0 + (-0.16666666666666666 * pow(x, 2.0)))) * fabs(y)) / x;
	} else {
		tmp = (t_0 / x) * x;
	}
	return copysign(1.0, y) * tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = Math.sinh(Math.abs(y));
	double tmp;
	if (((Math.sin(x) * t_0) / x) <= -2e-200) {
		tmp = ((x * (1.0 + (-0.16666666666666666 * Math.pow(x, 2.0)))) * Math.abs(y)) / x;
	} else {
		tmp = (t_0 / x) * x;
	}
	return Math.copySign(1.0, y) * tmp;
}

Python

def code(x, y):
	t_0 = math.sinh(math.fabs(y))
	tmp = 0
	if ((math.sin(x) * t_0) / x) <= -2e-200:
		tmp = ((x * (1.0 + (-0.16666666666666666 * math.pow(x, 2.0)))) * math.fabs(y)) / x
	else:
		tmp = (t_0 / x) * x
	return math.copysign(1.0, y) * tmp

Julia

function code(x, y)
	t_0 = sinh(abs(y))
	tmp = 0.0
	if (Float64(Float64(sin(x) * t_0) / x) <= -2e-200)
		tmp = Float64(Float64(Float64(x * Float64(1.0 + Float64(-0.16666666666666666 * (x ^ 2.0)))) * abs(y)) / x);
	else
		tmp = Float64(Float64(t_0 / x) * x);
	end
	return Float64(copysign(1.0, y) * tmp)
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sinh(abs(y));
	tmp = 0.0;
	if (((sin(x) * t_0) / x) <= -2e-200)
		tmp = ((x * (1.0 + (-0.16666666666666666 * (x ^ 2.0)))) * abs(y)) / x;
	else
		tmp = (t_0 / x) * x;
	end
	tmp_2 = (sign(y) * abs(1.0)) * tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Sinh[N[Abs[y], $MachinePrecision]], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(N[(N[Sin[x], $MachinePrecision] * t$95$0), $MachinePrecision] / x), $MachinePrecision], -2e-200], N[(N[(N[(x * N[(1.0 + N[(-0.16666666666666666 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(t$95$0 / x), $MachinePrecision] * x), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -2e-200

   1. Initial program 88.6%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 41.3%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

         \[\leadsto \frac{\left(x \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {x}^{2}\right)}\right) \cdot y}{x} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\left(x \cdot \left(1 + \color{blue}{\frac{-1}{6} \cdot {x}^{2}}\right)\right) \cdot y}{x} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\left(x \cdot \left(1 + \frac{-1}{6} \cdot \color{blue}{{x}^{2}}\right)\right) \cdot y}{x} \]

      4. lower-pow.f64 25.8

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

   7. Applied rewrites 25.8%

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

   if -2e-200 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 88.6%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 51.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 73.1

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

   6. Applied rewrites 73.1%

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

Alternative 8: 83.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \sinh \left(\left|y\right|\right)\\ \mathsf{copysign}\left(1, y\right) \cdot \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot t\_0}{x} \leq -2 \cdot 10^{-200}:\\ \;\;\;\;\frac{x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{2}\right)}{x} \cdot \left|y\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{x} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (sinh (fabs y))))
   (*
    (copysign 1.0 y)
    (if (<= (/ (* (sin x) t_0) x) -2e-200)
      (* (/ (* x (+ 1.0 (* -0.16666666666666666 (pow x 2.0)))) x) (fabs y))
      (* (/ t_0 x) x)))))

C

double code(double x, double y) {
	double t_0 = sinh(fabs(y));
	double tmp;
	if (((sin(x) * t_0) / x) <= -2e-200) {
		tmp = ((x * (1.0 + (-0.16666666666666666 * pow(x, 2.0)))) / x) * fabs(y);
	} else {
		tmp = (t_0 / x) * x;
	}
	return copysign(1.0, y) * tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = Math.sinh(Math.abs(y));
	double tmp;
	if (((Math.sin(x) * t_0) / x) <= -2e-200) {
		tmp = ((x * (1.0 + (-0.16666666666666666 * Math.pow(x, 2.0)))) / x) * Math.abs(y);
	} else {
		tmp = (t_0 / x) * x;
	}
	return Math.copySign(1.0, y) * tmp;
}

Python

def code(x, y):
	t_0 = math.sinh(math.fabs(y))
	tmp = 0
	if ((math.sin(x) * t_0) / x) <= -2e-200:
		tmp = ((x * (1.0 + (-0.16666666666666666 * math.pow(x, 2.0)))) / x) * math.fabs(y)
	else:
		tmp = (t_0 / x) * x
	return math.copysign(1.0, y) * tmp

Julia

function code(x, y)
	t_0 = sinh(abs(y))
	tmp = 0.0
	if (Float64(Float64(sin(x) * t_0) / x) <= -2e-200)
		tmp = Float64(Float64(Float64(x * Float64(1.0 + Float64(-0.16666666666666666 * (x ^ 2.0)))) / x) * abs(y));
	else
		tmp = Float64(Float64(t_0 / x) * x);
	end
	return Float64(copysign(1.0, y) * tmp)
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sinh(abs(y));
	tmp = 0.0;
	if (((sin(x) * t_0) / x) <= -2e-200)
		tmp = ((x * (1.0 + (-0.16666666666666666 * (x ^ 2.0)))) / x) * abs(y);
	else
		tmp = (t_0 / x) * x;
	end
	tmp_2 = (sign(y) * abs(1.0)) * tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Sinh[N[Abs[y], $MachinePrecision]], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(N[(N[Sin[x], $MachinePrecision] * t$95$0), $MachinePrecision] / x), $MachinePrecision], -2e-200], N[(N[(N[(x * N[(1.0 + N[(-0.16666666666666666 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] * N[Abs[y], $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 / x), $MachinePrecision] * x), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -2e-200

   1. Initial program 88.6%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 41.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lift-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 52.5

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

   6. Applied rewrites 52.5%

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

   7. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {x}^{2}\right)}}{x} \cdot y \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{x \cdot \left(1 + \color{blue}{\frac{-1}{6} \cdot {x}^{2}}\right)}{x} \cdot y \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{x \cdot \left(1 + \frac{-1}{6} \cdot \color{blue}{{x}^{2}}\right)}{x} \cdot y \]

      4. lower-pow.f64 36.4

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

   9. Applied rewrites 36.4%

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

   if -2e-200 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 88.6%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 51.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 73.1

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

   6. Applied rewrites 73.1%

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

Alternative 9: 82.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \sinh \left(\left|y\right|\right)\\ \mathsf{copysign}\left(1, y\right) \cdot \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot t\_0}{x} \leq -2 \cdot 10^{-200}:\\ \;\;\;\;\left(1 + -0.16666666666666666 \cdot {x}^{2}\right) \cdot \left|y\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{x} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (sinh (fabs y))))
   (*
    (copysign 1.0 y)
    (if (<= (/ (* (sin x) t_0) x) -2e-200)
      (* (+ 1.0 (* -0.16666666666666666 (pow x 2.0))) (fabs y))
      (* (/ t_0 x) x)))))

C

double code(double x, double y) {
	double t_0 = sinh(fabs(y));
	double tmp;
	if (((sin(x) * t_0) / x) <= -2e-200) {
		tmp = (1.0 + (-0.16666666666666666 * pow(x, 2.0))) * fabs(y);
	} else {
		tmp = (t_0 / x) * x;
	}
	return copysign(1.0, y) * tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = Math.sinh(Math.abs(y));
	double tmp;
	if (((Math.sin(x) * t_0) / x) <= -2e-200) {
		tmp = (1.0 + (-0.16666666666666666 * Math.pow(x, 2.0))) * Math.abs(y);
	} else {
		tmp = (t_0 / x) * x;
	}
	return Math.copySign(1.0, y) * tmp;
}

Python

def code(x, y):
	t_0 = math.sinh(math.fabs(y))
	tmp = 0
	if ((math.sin(x) * t_0) / x) <= -2e-200:
		tmp = (1.0 + (-0.16666666666666666 * math.pow(x, 2.0))) * math.fabs(y)
	else:
		tmp = (t_0 / x) * x
	return math.copysign(1.0, y) * tmp

Julia

function code(x, y)
	t_0 = sinh(abs(y))
	tmp = 0.0
	if (Float64(Float64(sin(x) * t_0) / x) <= -2e-200)
		tmp = Float64(Float64(1.0 + Float64(-0.16666666666666666 * (x ^ 2.0))) * abs(y));
	else
		tmp = Float64(Float64(t_0 / x) * x);
	end
	return Float64(copysign(1.0, y) * tmp)
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sinh(abs(y));
	tmp = 0.0;
	if (((sin(x) * t_0) / x) <= -2e-200)
		tmp = (1.0 + (-0.16666666666666666 * (x ^ 2.0))) * abs(y);
	else
		tmp = (t_0 / x) * x;
	end
	tmp_2 = (sign(y) * abs(1.0)) * tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Sinh[N[Abs[y], $MachinePrecision]], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(N[(N[Sin[x], $MachinePrecision] * t$95$0), $MachinePrecision] / x), $MachinePrecision], -2e-200], N[(N[(1.0 + N[(-0.16666666666666666 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[y], $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 / x), $MachinePrecision] * x), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -2e-200

   1. Initial program 88.6%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 41.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lift-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 52.5

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

   6. Applied rewrites 52.5%

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

   7. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

         \[\leadsto \left(1 + \color{blue}{\frac{-1}{6} \cdot {x}^{2}}\right) \cdot y \]

      2. lower-*.f64 N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot \color{blue}{{x}^{2}}\right) \cdot y \]

      3. lower-pow.f64 36.2

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

   9. Applied rewrites 36.2%

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

   if -2e-200 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 88.6%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 51.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 73.1

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

   6. Applied rewrites 73.1%

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

Alternative 10: 73.1% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.6%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 51.3%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. lift-/.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 73.1

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

6. Applied rewrites 73.1%

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

Alternative 11: 65.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|x\right| \leq 9.2 \cdot 10^{+181}:\\ \;\;\;\;\sinh y\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(1 + y\right) - \frac{1}{1 + y}\right)\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (fabs x) 9.2e+181) (sinh y) (* 0.5 (- (+ 1.0 y) (/ 1.0 (+ 1.0 y))))))

C

double code(double x, double y) {
	double tmp;
	if (fabs(x) <= 9.2e+181) {
		tmp = sinh(y);
	} else {
		tmp = 0.5 * ((1.0 + y) - (1.0 / (1.0 + y)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (abs(x) <= 9.2d+181) then
        tmp = sinh(y)
    else
        tmp = 0.5d0 * ((1.0d0 + y) - (1.0d0 / (1.0d0 + y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (Math.abs(x) <= 9.2e+181) {
		tmp = Math.sinh(y);
	} else {
		tmp = 0.5 * ((1.0 + y) - (1.0 / (1.0 + y)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.fabs(x) <= 9.2e+181:
		tmp = math.sinh(y)
	else:
		tmp = 0.5 * ((1.0 + y) - (1.0 / (1.0 + y)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (abs(x) <= 9.2e+181)
		tmp = sinh(y);
	else
		tmp = Float64(0.5 * Float64(Float64(1.0 + y) - Float64(1.0 / Float64(1.0 + y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (abs(x) <= 9.2e+181)
		tmp = sinh(y);
	else
		tmp = 0.5 * ((1.0 + y) - (1.0 / (1.0 + y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[Abs[x], $MachinePrecision], 9.2e+181], N[Sinh[y], $MachinePrecision], N[(0.5 * N[(N[(1.0 + y), $MachinePrecision] - N[(1.0 / N[(1.0 + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 9.1999999999999995e181

   1. Initial program 88.6%

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

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(e^{y} - \frac{1}{e^{y}}\right)} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{y} - \color{blue}{\frac{1}{e^{y}}}\right) \]

      3. lower-exp.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{y} - \frac{\color{blue}{1}}{e^{y}}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{y} - \frac{1}{\color{blue}{e^{y}}}\right) \]

      5. lower-exp.f64 51.6

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

   4. Applied rewrites 51.6%

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

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(e^{y} - \frac{1}{e^{y}}\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(e^{y} - \frac{1}{e^{y}}\right) \cdot \color{blue}{\frac{1}{2}} \]

      3. metadata-eval N/A

         \[\leadsto \left(e^{y} - \frac{1}{e^{y}}\right) \cdot \frac{1}{\color{blue}{2}} \]

      4. mult-flip N/A

         \[\leadsto \frac{e^{y} - \frac{1}{e^{y}}}{\color{blue}{2}} \]

      5. lift--.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. rec-exp N/A

         \[\leadsto \frac{e^{y} - e^{\mathsf{neg}\left(y\right)}}{2} \]

      10. sinh-def N/A

          \[\leadsto \sinh y \]

      11. lift-sinh.f64 62.6

          \[\leadsto \sinh y \]

   6. Applied rewrites 62.6%

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

   if 9.1999999999999995e181 < x

   1. Initial program 88.6%

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

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(e^{y} - \frac{1}{e^{y}}\right)} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{y} - \color{blue}{\frac{1}{e^{y}}}\right) \]

      3. lower-exp.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{y} - \frac{\color{blue}{1}}{e^{y}}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{y} - \frac{1}{\color{blue}{e^{y}}}\right) \]

      5. lower-exp.f64 51.6

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

   4. Applied rewrites 51.6%

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

   5. Taylor expanded in y around 0

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

      1. lower-+.f64 33.9

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

   7. Applied rewrites 33.9%

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

   8. Taylor expanded in y around 0

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

      1. lower-+.f64 16.9

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

   10. Applied rewrites 16.9%

       \[\leadsto 0.5 \cdot \left(\left(1 + y\right) - \frac{1}{1 + \color{blue}{y}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 62.6% accurate, 3.9× speedup

Math

\[\sinh y \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[Sinh[y], $MachinePrecision]

Derivation

1. Initial program 88.6%

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

2. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(e^{y} - \frac{1}{e^{y}}\right)} \]

   2. lower--.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(e^{y} - \color{blue}{\frac{1}{e^{y}}}\right) \]

   3. lower-exp.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(e^{y} - \frac{\color{blue}{1}}{e^{y}}\right) \]

   4. lower-/.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(e^{y} - \frac{1}{\color{blue}{e^{y}}}\right) \]

   5. lower-exp.f64 51.6

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

4. Applied rewrites 51.6%

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

   1. lift-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(e^{y} - \frac{1}{e^{y}}\right)} \]

   2. *-commutative N/A

      \[\leadsto \left(e^{y} - \frac{1}{e^{y}}\right) \cdot \color{blue}{\frac{1}{2}} \]

   3. metadata-eval N/A

      \[\leadsto \left(e^{y} - \frac{1}{e^{y}}\right) \cdot \frac{1}{\color{blue}{2}} \]

   4. mult-flip N/A

      \[\leadsto \frac{e^{y} - \frac{1}{e^{y}}}{\color{blue}{2}} \]

   5. lift--.f64 N/A

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

   6. lift-exp.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. lift-exp.f64 N/A

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

   9. rec-exp N/A

      \[\leadsto \frac{e^{y} - e^{\mathsf{neg}\left(y\right)}}{2} \]

   10. sinh-def N/A

       \[\leadsto \sinh y \]

   11. lift-sinh.f64 62.6

       \[\leadsto \sinh y \]

6. Applied rewrites 62.6%

   \[\leadsto \color{blue}{\sinh y} \]
7. Add Preprocessing

Alternative 13: 28.0% accurate, 7.7× speedup

Math

\[\left(y + y\right) \cdot 0.5 \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.6%

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

2. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(e^{y} - \frac{1}{e^{y}}\right)} \]

   2. lower--.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(e^{y} - \color{blue}{\frac{1}{e^{y}}}\right) \]

   3. lower-exp.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(e^{y} - \frac{\color{blue}{1}}{e^{y}}\right) \]

   4. lower-/.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(e^{y} - \frac{1}{\color{blue}{e^{y}}}\right) \]

   5. lower-exp.f64 51.6

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

4. Applied rewrites 51.6%

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

5. Taylor expanded in y around 0

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

   1. lower-*.f64 28.0

      \[\leadsto 0.5 \cdot \left(2 \cdot y\right) \]

7. Applied rewrites 28.0%

   \[\leadsto 0.5 \cdot \left(2 \cdot \color{blue}{y}\right) \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(2 \cdot y\right)} \]

   2. *-commutative N/A

      \[\leadsto \left(2 \cdot y\right) \cdot \color{blue}{\frac{1}{2}} \]

   3. lower-*.f64 28.0

      \[\leadsto \left(2 \cdot y\right) \cdot \color{blue}{0.5} \]

   4. lift-*.f64 N/A

      \[\leadsto \left(2 \cdot y\right) \cdot \frac{1}{2} \]

   5. count-2-rev N/A

      \[\leadsto \left(y + y\right) \cdot \frac{1}{2} \]

   6. lower-+.f64 28.0

      \[\leadsto \left(y + y\right) \cdot 0.5 \]

9. Applied rewrites 28.0%

   \[\leadsto \left(y + y\right) \cdot \color{blue}{0.5} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025172 
(FPCore (x y)
  :name "Linear.Quaternion:$ccosh from linear-1.19.1.3"
  :precision binary64
  (/ (* (sin x) (sinh y)) x))