Linear.Quaternion:$ctanh from linear-1.19.1.3

Percentage Accurate: 96.2% → 98.0%

Time: 6.5s

Alternatives: 9

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 96.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 3.4 \cdot 10^{-202}:\\ \;\;\;\;\frac{\frac{x\_m}{z}}{y} \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot \frac{\sin y}{y}}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= x_m 3.4e-202)
    (* (/ (/ x_m z) y) (sin y))
    (/ (* x_m (/ (sin y) y)) z))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (x_m <= 3.4e-202) {
		tmp = ((x_m / z) / y) * sin(y);
	} else {
		tmp = (x_m * (sin(y) / y)) / z;
	}
	return x_s * tmp;
}

Fortran

x\_m =     private
x\_s =     private
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_s, x_m, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x_m <= 3.4d-202) then
        tmp = ((x_m / z) / y) * sin(y)
    else
        tmp = (x_m * (sin(y) / y)) / z
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (x_m <= 3.4e-202) {
		tmp = ((x_m / z) / y) * Math.sin(y);
	} else {
		tmp = (x_m * (Math.sin(y) / y)) / z;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if x_m <= 3.4e-202:
		tmp = ((x_m / z) / y) * math.sin(y)
	else:
		tmp = (x_m * (math.sin(y) / y)) / z
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (x_m <= 3.4e-202)
		tmp = Float64(Float64(Float64(x_m / z) / y) * sin(y));
	else
		tmp = Float64(Float64(x_m * Float64(sin(y) / y)) / z);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if (x_m <= 3.4e-202)
		tmp = ((x_m / z) / y) * sin(y);
	else
		tmp = (x_m * (sin(y) / y)) / z;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[x$95$m, 3.4e-202], N[(N[(N[(x$95$m / z), $MachinePrecision] / y), $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 3.40000000000000012e-202

   1. Initial program 93.9%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 84.8

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

   4. Applied rewrites 84.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. *-commutative N/A

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

      5. associate-/r* N/A

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

      6. lift-/.f64 N/A

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

      7. lower-/.f64 89.0

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

   6. Applied rewrites 89.0%

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

   if 3.40000000000000012e-202 < x

   1. Initial program 99.2%

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

Alternative 2: 42.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \frac{x\_m \cdot \frac{\sin y}{y}}{z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-229}:\\ \;\;\;\;\frac{-x\_m}{z}\\ \mathbf{elif}\;t\_0 \leq 10^{-212}:\\ \;\;\;\;x\_m \cdot \frac{y}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z}\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (/ (* x_m (/ (sin y) y)) z)))
   (*
    x_s
    (if (<= t_0 -5e-229)
      (/ (- x_m) z)
      (if (<= t_0 1e-212) (* x_m (/ y (* z y))) (/ x_m z))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double t_0 = (x_m * (sin(y) / y)) / z;
	double tmp;
	if (t_0 <= -5e-229) {
		tmp = -x_m / z;
	} else if (t_0 <= 1e-212) {
		tmp = x_m * (y / (z * y));
	} else {
		tmp = x_m / z;
	}
	return x_s * tmp;
}

Fortran

x\_m =     private
x\_s =     private
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_s, x_m, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x_m * (sin(y) / y)) / z
    if (t_0 <= (-5d-229)) then
        tmp = -x_m / z
    else if (t_0 <= 1d-212) then
        tmp = x_m * (y / (z * y))
    else
        tmp = x_m / z
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double t_0 = (x_m * (Math.sin(y) / y)) / z;
	double tmp;
	if (t_0 <= -5e-229) {
		tmp = -x_m / z;
	} else if (t_0 <= 1e-212) {
		tmp = x_m * (y / (z * y));
	} else {
		tmp = x_m / z;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	t_0 = (x_m * (math.sin(y) / y)) / z
	tmp = 0
	if t_0 <= -5e-229:
		tmp = -x_m / z
	elif t_0 <= 1e-212:
		tmp = x_m * (y / (z * y))
	else:
		tmp = x_m / z
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	t_0 = Float64(Float64(x_m * Float64(sin(y) / y)) / z)
	tmp = 0.0
	if (t_0 <= -5e-229)
		tmp = Float64(Float64(-x_m) / z);
	elseif (t_0 <= 1e-212)
		tmp = Float64(x_m * Float64(y / Float64(z * y)));
	else
		tmp = Float64(x_m / z);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	t_0 = (x_m * (sin(y) / y)) / z;
	tmp = 0.0;
	if (t_0 <= -5e-229)
		tmp = -x_m / z;
	elseif (t_0 <= 1e-212)
		tmp = x_m * (y / (z * y));
	else
		tmp = x_m / z;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := Block[{t$95$0 = N[(N[(x$95$m * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$0, -5e-229], N[((-x$95$m) / z), $MachinePrecision], If[LessEqual[t$95$0, 1e-212], N[(x$95$m * N[(y / N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m / z), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z) < -5.00000000000000016e-229

   1. Initial program 99.8%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\sin y\right)}{\mathsf{neg}\left(y\right)}} \cdot \frac{x}{z} \]

      7. frac-times N/A

         \[\leadsto \color{blue}{\frac{\left(\mathsf{neg}\left(\sin y\right)\right) \cdot x}{\left(\mathsf{neg}\left(y\right)\right) \cdot z}} \]

      8. associate-/l* N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sin y\right)\right) \cdot \frac{x}{\left(\mathsf{neg}\left(y\right)\right) \cdot z}} \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sin y\right)\right) \cdot \frac{x}{\left(\mathsf{neg}\left(y\right)\right) \cdot z}} \]

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 84.5

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

   4. Applied rewrites 84.5%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(y\right)\right)} \cdot \frac{x}{\left(-y\right) \cdot z} \]

      2. lower-neg.f64 47.6

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

   7. Applied rewrites 47.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. associate-*l/ N/A

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

      8. unpow1 N/A

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

      9. metadata-eval N/A

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

      10. sqrt-pow1 N/A

          \[\leadsto \frac{\frac{x}{z} \cdot \left(-y\right)}{\color{blue}{\sqrt{{\left(-y\right)}^{2}}}} \]

      11. pow2 N/A

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

      12. lift-neg.f64 N/A

          \[\leadsto \frac{\frac{x}{z} \cdot \left(-y\right)}{\sqrt{\color{blue}{\left(\mathsf{neg}\left(y\right)\right)} \cdot \left(-y\right)}} \]

      13. lift-neg.f64 N/A

          \[\leadsto \frac{\frac{x}{z} \cdot \left(-y\right)}{\sqrt{\left(\mathsf{neg}\left(y\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}}} \]

      14. sqr-neg-rev N/A

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

      15. sqrt-prod N/A

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

      16. rem-square-sqrt N/A

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

      17. lower-/.f64 N/A

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

   9. Applied rewrites 55.0%

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

   10. Taylor expanded in y around 0

       \[\leadsto \color{blue}{-1 \cdot \frac{x}{z}} \]
   11. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \color{blue}{\frac{-1 \cdot x}{z}} \]

       2. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{-1 \cdot x}{z}} \]

       3. mul-1-neg N/A

          \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(x\right)}}{z} \]

       4. lower-neg.f64 2.2

          \[\leadsto \frac{\color{blue}{-x}}{z} \]

   12. Applied rewrites 2.2%

       \[\leadsto \color{blue}{\frac{-x}{z}} \]

   if -5.00000000000000016e-229 < (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z) < 9.99999999999999954e-213

   1. Initial program 88.5%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\sin y\right)}{\mathsf{neg}\left(y\right)}} \cdot \frac{x}{z} \]

      7. frac-times N/A

         \[\leadsto \color{blue}{\frac{\left(\mathsf{neg}\left(\sin y\right)\right) \cdot x}{\left(\mathsf{neg}\left(y\right)\right) \cdot z}} \]

      8. associate-/l* N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sin y\right)\right) \cdot \frac{x}{\left(\mathsf{neg}\left(y\right)\right) \cdot z}} \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sin y\right)\right) \cdot \frac{x}{\left(\mathsf{neg}\left(y\right)\right) \cdot z}} \]

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 92.7

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

   4. Applied rewrites 92.7%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(y\right)\right)} \cdot \frac{x}{\left(-y\right) \cdot z} \]

      2. lower-neg.f64 74.4

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

   7. Applied rewrites 74.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\frac{\left(-y\right) \cdot x}{z}\right)}{\mathsf{neg}\left(\left(-y\right)\right)}} \]

      8. lift-neg.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\frac{\left(-y\right) \cdot x}{z}\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)} \]

      9. remove-double-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(\frac{\left(-y\right) \cdot x}{z}\right)}{\color{blue}{y}} \]

      10. rem-square-sqrt N/A

          \[\leadsto \frac{\mathsf{neg}\left(\frac{\left(-y\right) \cdot x}{z}\right)}{\color{blue}{\sqrt{y} \cdot \sqrt{y}}} \]

      11. sqrt-prod N/A

          \[\leadsto \frac{\mathsf{neg}\left(\frac{\left(-y\right) \cdot x}{z}\right)}{\color{blue}{\sqrt{y \cdot y}}} \]

      12. sqr-neg-rev N/A

          \[\leadsto \frac{\mathsf{neg}\left(\frac{\left(-y\right) \cdot x}{z}\right)}{\sqrt{\color{blue}{\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(y\right)\right)}}} \]

      13. lift-neg.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\frac{\left(-y\right) \cdot x}{z}\right)}{\sqrt{\color{blue}{\left(-y\right)} \cdot \left(\mathsf{neg}\left(y\right)\right)}} \]

      14. lift-neg.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\frac{\left(-y\right) \cdot x}{z}\right)}{\sqrt{\left(-y\right) \cdot \color{blue}{\left(-y\right)}}} \]

      15. pow2 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\frac{\left(-y\right) \cdot x}{z}\right)}{\sqrt{\color{blue}{{\left(-y\right)}^{2}}}} \]

      16. sqrt-pow1 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\frac{\left(-y\right) \cdot x}{z}\right)}{\color{blue}{{\left(-y\right)}^{\left(\frac{2}{2}\right)}}} \]

      17. metadata-eval N/A

          \[\leadsto \frac{\mathsf{neg}\left(\frac{\left(-y\right) \cdot x}{z}\right)}{{\left(-y\right)}^{\color{blue}{1}}} \]

      18. unpow1 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\frac{\left(-y\right) \cdot x}{z}\right)}{\color{blue}{-y}} \]

      19. lift-neg.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\frac{\left(-y\right) \cdot x}{z}\right)}{\color{blue}{\mathsf{neg}\left(y\right)}} \]

      20. frac-2neg-rev N/A

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

      21. associate-/r* N/A

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

      22. lift-*.f64 N/A

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

   9. Applied rewrites 61.4%

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

   if 9.99999999999999954e-213 < (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z)

   1. Initial program 99.0%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x}{z}} \]
   4. Step-by-step derivation

      1. lower-/.f64 64.8

         \[\leadsto \color{blue}{\frac{x}{z}} \]

   5. Applied rewrites 64.8%

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

Alternative 3: 44.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{x\_m \cdot \frac{\sin y}{y}}{z} \leq 0:\\ \;\;\;\;\left(-y\right) \cdot \frac{\frac{x\_m}{y}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x\_m \cdot \mathsf{fma}\left(0.008333333333333333 \cdot y, y, -0.16666666666666666\right), y \cdot y, x\_m\right)}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= (/ (* x_m (/ (sin y) y)) z) 0.0)
    (* (- y) (/ (/ x_m y) z))
    (/
     (fma
      (* x_m (fma (* 0.008333333333333333 y) y -0.16666666666666666))
      (* y y)
      x_m)
     z))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (((x_m * (sin(y) / y)) / z) <= 0.0) {
		tmp = -y * ((x_m / y) / z);
	} else {
		tmp = fma((x_m * fma((0.008333333333333333 * y), y, -0.16666666666666666)), (y * y), x_m) / z;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (Float64(Float64(x_m * Float64(sin(y) / y)) / z) <= 0.0)
		tmp = Float64(Float64(-y) * Float64(Float64(x_m / y) / z));
	else
		tmp = Float64(fma(Float64(x_m * fma(Float64(0.008333333333333333 * y), y, -0.16666666666666666)), Float64(y * y), x_m) / z);
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[N[(N[(x$95$m * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], 0.0], N[((-y) * N[(N[(x$95$m / y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x$95$m * N[(N[(0.008333333333333333 * y), $MachinePrecision] * y + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] * N[(y * y), $MachinePrecision] + x$95$m), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z) < 0.0

   1. Initial program 94.5%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\sin y\right)}{\mathsf{neg}\left(y\right)}} \cdot \frac{x}{z} \]

      7. frac-times N/A

         \[\leadsto \color{blue}{\frac{\left(\mathsf{neg}\left(\sin y\right)\right) \cdot x}{\left(\mathsf{neg}\left(y\right)\right) \cdot z}} \]

      8. associate-/l* N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sin y\right)\right) \cdot \frac{x}{\left(\mathsf{neg}\left(y\right)\right) \cdot z}} \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sin y\right)\right) \cdot \frac{x}{\left(\mathsf{neg}\left(y\right)\right) \cdot z}} \]

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 89.6

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

   4. Applied rewrites 89.6%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(y\right)\right)} \cdot \frac{x}{\left(-y\right) \cdot z} \]

      2. lower-neg.f64 61.3

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

   7. Applied rewrites 61.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. unpow1 N/A

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

      6. metadata-eval N/A

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

      7. sqrt-pow1 N/A

         \[\leadsto \left(-y\right) \cdot \frac{\frac{x}{\color{blue}{\sqrt{{\left(-y\right)}^{2}}}}}{z} \]

      8. pow2 N/A

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

      9. lift-neg.f64 N/A

         \[\leadsto \left(-y\right) \cdot \frac{\frac{x}{\sqrt{\color{blue}{\left(\mathsf{neg}\left(y\right)\right)} \cdot \left(-y\right)}}}{z} \]

      10. lift-neg.f64 N/A

          \[\leadsto \left(-y\right) \cdot \frac{\frac{x}{\sqrt{\left(\mathsf{neg}\left(y\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}}}}{z} \]

      11. sqr-neg-rev N/A

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

      12. sqrt-prod N/A

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

      13. rem-square-sqrt N/A

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

      14. lift-/.f64 33.4

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

   9. Applied rewrites 33.4%

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

   if 0.0 < (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z)

   1. Initial program 99.1%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{x + {y}^{2} \cdot \left(\frac{-1}{6} \cdot x + \frac{1}{120} \cdot \left(x \cdot {y}^{2}\right)\right)}}{z} \]
   4. Step-by-step derivation

      1. fp-cancel-sign-sub-inv N/A

         \[\leadsto \frac{\color{blue}{x - \left(\mathsf{neg}\left({y}^{2}\right)\right) \cdot \left(\frac{-1}{6} \cdot x + \frac{1}{120} \cdot \left(x \cdot {y}^{2}\right)\right)}}{z} \]

      2. fp-cancel-sub-sign-inv N/A

         \[\leadsto \frac{\color{blue}{x + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({y}^{2}\right)\right)\right)\right) \cdot \left(\frac{-1}{6} \cdot x + \frac{1}{120} \cdot \left(x \cdot {y}^{2}\right)\right)}}{z} \]

      3. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left({y}^{2}\right)\right)\right)\right) \cdot \left(\frac{-1}{6} \cdot x + \frac{1}{120} \cdot \left(x \cdot {y}^{2}\right)\right) + x}}{z} \]

      4. distribute-lft-neg-out N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left({y}^{2}\right)\right) \cdot \left(\frac{-1}{6} \cdot x + \frac{1}{120} \cdot \left(x \cdot {y}^{2}\right)\right)\right)\right)} + x}{z} \]

      5. distribute-lft-neg-out N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left({y}^{2} \cdot \left(\frac{-1}{6} \cdot x + \frac{1}{120} \cdot \left(x \cdot {y}^{2}\right)\right)\right)\right)}\right)\right) + x}{z} \]

      6. remove-double-neg N/A

         \[\leadsto \frac{\color{blue}{{y}^{2} \cdot \left(\frac{-1}{6} \cdot x + \frac{1}{120} \cdot \left(x \cdot {y}^{2}\right)\right)} + x}{z} \]

      7. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{6} \cdot x + \frac{1}{120} \cdot \left(x \cdot {y}^{2}\right)\right) \cdot {y}^{2}} + x}{z} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{6} \cdot x + \frac{1}{120} \cdot \left(x \cdot {y}^{2}\right), {y}^{2}, x\right)}}{z} \]

   5. Applied rewrites 61.5%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot \mathsf{fma}\left(0.008333333333333333 \cdot y, y, -0.16666666666666666\right), y \cdot y, x\right)}}{z} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 44.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{x\_m \cdot \frac{\sin y}{y}}{z} \leq 0:\\ \;\;\;\;\left(-y\right) \cdot \frac{\frac{x\_m}{y}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= (/ (* x_m (/ (sin y) y)) z) 0.0)
    (* (- y) (/ (/ x_m y) z))
    (/ x_m z))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (((x_m * (sin(y) / y)) / z) <= 0.0) {
		tmp = -y * ((x_m / y) / z);
	} else {
		tmp = x_m / z;
	}
	return x_s * tmp;
}

Fortran

x\_m =     private
x\_s =     private
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_s, x_m, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (((x_m * (sin(y) / y)) / z) <= 0.0d0) then
        tmp = -y * ((x_m / y) / z)
    else
        tmp = x_m / z
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (((x_m * (Math.sin(y) / y)) / z) <= 0.0) {
		tmp = -y * ((x_m / y) / z);
	} else {
		tmp = x_m / z;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if ((x_m * (math.sin(y) / y)) / z) <= 0.0:
		tmp = -y * ((x_m / y) / z)
	else:
		tmp = x_m / z
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (Float64(Float64(x_m * Float64(sin(y) / y)) / z) <= 0.0)
		tmp = Float64(Float64(-y) * Float64(Float64(x_m / y) / z));
	else
		tmp = Float64(x_m / z);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if (((x_m * (sin(y) / y)) / z) <= 0.0)
		tmp = -y * ((x_m / y) / z);
	else
		tmp = x_m / z;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[N[(N[(x$95$m * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], 0.0], N[((-y) * N[(N[(x$95$m / y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(x$95$m / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z) < 0.0

   1. Initial program 94.5%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\sin y\right)}{\mathsf{neg}\left(y\right)}} \cdot \frac{x}{z} \]

      7. frac-times N/A

         \[\leadsto \color{blue}{\frac{\left(\mathsf{neg}\left(\sin y\right)\right) \cdot x}{\left(\mathsf{neg}\left(y\right)\right) \cdot z}} \]

      8. associate-/l* N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sin y\right)\right) \cdot \frac{x}{\left(\mathsf{neg}\left(y\right)\right) \cdot z}} \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sin y\right)\right) \cdot \frac{x}{\left(\mathsf{neg}\left(y\right)\right) \cdot z}} \]

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 89.6

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

   4. Applied rewrites 89.6%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(y\right)\right)} \cdot \frac{x}{\left(-y\right) \cdot z} \]

      2. lower-neg.f64 61.3

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

   7. Applied rewrites 61.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. unpow1 N/A

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

      6. metadata-eval N/A

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

      7. sqrt-pow1 N/A

         \[\leadsto \left(-y\right) \cdot \frac{\frac{x}{\color{blue}{\sqrt{{\left(-y\right)}^{2}}}}}{z} \]

      8. pow2 N/A

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

      9. lift-neg.f64 N/A

         \[\leadsto \left(-y\right) \cdot \frac{\frac{x}{\sqrt{\color{blue}{\left(\mathsf{neg}\left(y\right)\right)} \cdot \left(-y\right)}}}{z} \]

      10. lift-neg.f64 N/A

          \[\leadsto \left(-y\right) \cdot \frac{\frac{x}{\sqrt{\left(\mathsf{neg}\left(y\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}}}}{z} \]

      11. sqr-neg-rev N/A

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

      12. sqrt-prod N/A

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

      13. rem-square-sqrt N/A

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

      14. lift-/.f64 33.4

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

   9. Applied rewrites 33.4%

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

   if 0.0 < (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z)

   1. Initial program 99.1%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x}{z}} \]
   4. Step-by-step derivation

      1. lower-/.f64 60.5

         \[\leadsto \color{blue}{\frac{x}{z}} \]

   5. Applied rewrites 60.5%

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

Alternative 5: 44.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{x\_m \cdot \frac{\sin y}{y}}{z} \leq 0:\\ \;\;\;\;\left(-y\right) \cdot \frac{x\_m}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= (/ (* x_m (/ (sin y) y)) z) 0.0)
    (* (- y) (/ x_m (* z y)))
    (/ x_m z))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (((x_m * (sin(y) / y)) / z) <= 0.0) {
		tmp = -y * (x_m / (z * y));
	} else {
		tmp = x_m / z;
	}
	return x_s * tmp;
}

Fortran

x\_m =     private
x\_s =     private
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_s, x_m, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (((x_m * (sin(y) / y)) / z) <= 0.0d0) then
        tmp = -y * (x_m / (z * y))
    else
        tmp = x_m / z
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (((x_m * (Math.sin(y) / y)) / z) <= 0.0) {
		tmp = -y * (x_m / (z * y));
	} else {
		tmp = x_m / z;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if ((x_m * (math.sin(y) / y)) / z) <= 0.0:
		tmp = -y * (x_m / (z * y))
	else:
		tmp = x_m / z
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (Float64(Float64(x_m * Float64(sin(y) / y)) / z) <= 0.0)
		tmp = Float64(Float64(-y) * Float64(x_m / Float64(z * y)));
	else
		tmp = Float64(x_m / z);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if (((x_m * (sin(y) / y)) / z) <= 0.0)
		tmp = -y * (x_m / (z * y));
	else
		tmp = x_m / z;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[N[(N[(x$95$m * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], 0.0], N[((-y) * N[(x$95$m / N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z) < 0.0

   1. Initial program 94.5%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\sin y\right)}{\mathsf{neg}\left(y\right)}} \cdot \frac{x}{z} \]

      7. frac-times N/A

         \[\leadsto \color{blue}{\frac{\left(\mathsf{neg}\left(\sin y\right)\right) \cdot x}{\left(\mathsf{neg}\left(y\right)\right) \cdot z}} \]

      8. associate-/l* N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sin y\right)\right) \cdot \frac{x}{\left(\mathsf{neg}\left(y\right)\right) \cdot z}} \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sin y\right)\right) \cdot \frac{x}{\left(\mathsf{neg}\left(y\right)\right) \cdot z}} \]

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 89.6

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

   4. Applied rewrites 89.6%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(y\right)\right)} \cdot \frac{x}{\left(-y\right) \cdot z} \]

      2. lower-neg.f64 61.3

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

   7. Applied rewrites 61.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lift-neg.f64 N/A

         \[\leadsto \left(-y\right) \cdot \frac{\frac{x}{z}}{\color{blue}{\mathsf{neg}\left(y\right)}} \]

      6. distribute-frac-neg2 N/A

         \[\leadsto \left(-y\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{\frac{x}{z}}{y}\right)\right)} \]

      7. distribute-neg-frac N/A

         \[\leadsto \left(-y\right) \cdot \color{blue}{\frac{\mathsf{neg}\left(\frac{x}{z}\right)}{y}} \]

      8. distribute-frac-neg N/A

         \[\leadsto \left(-y\right) \cdot \frac{\color{blue}{\frac{\mathsf{neg}\left(x\right)}{z}}}{y} \]

      9. lift-neg.f64 N/A

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

      10. frac-2neg-rev N/A

          \[\leadsto \left(-y\right) \cdot \color{blue}{\frac{\mathsf{neg}\left(\frac{-x}{z}\right)}{\mathsf{neg}\left(y\right)}} \]

      11. lift-neg.f64 N/A

          \[\leadsto \left(-y\right) \cdot \frac{\mathsf{neg}\left(\frac{\color{blue}{\mathsf{neg}\left(x\right)}}{z}\right)}{\mathsf{neg}\left(y\right)} \]

      12. distribute-frac-neg N/A

          \[\leadsto \left(-y\right) \cdot \frac{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\frac{x}{z}\right)\right)}\right)}{\mathsf{neg}\left(y\right)} \]

      13. lift-neg.f64 N/A

          \[\leadsto \left(-y\right) \cdot \frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{x}{z}\right)\right)\right)}{\color{blue}{-y}} \]

      14. unpow1 N/A

          \[\leadsto \left(-y\right) \cdot \frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{x}{z}\right)\right)\right)}{\color{blue}{{\left(-y\right)}^{1}}} \]

      15. metadata-eval N/A

          \[\leadsto \left(-y\right) \cdot \frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{x}{z}\right)\right)\right)}{{\left(-y\right)}^{\color{blue}{\left(\frac{2}{2}\right)}}} \]

      16. sqrt-pow1 N/A

          \[\leadsto \left(-y\right) \cdot \frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{x}{z}\right)\right)\right)}{\color{blue}{\sqrt{{\left(-y\right)}^{2}}}} \]

      17. pow2 N/A

          \[\leadsto \left(-y\right) \cdot \frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{x}{z}\right)\right)\right)}{\sqrt{\color{blue}{\left(-y\right) \cdot \left(-y\right)}}} \]

      18. lift-neg.f64 N/A

          \[\leadsto \left(-y\right) \cdot \frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{x}{z}\right)\right)\right)}{\sqrt{\color{blue}{\left(\mathsf{neg}\left(y\right)\right)} \cdot \left(-y\right)}} \]

      19. lift-neg.f64 N/A

          \[\leadsto \left(-y\right) \cdot \frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{x}{z}\right)\right)\right)}{\sqrt{\left(\mathsf{neg}\left(y\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}}} \]

      20. sqr-neg-rev N/A

          \[\leadsto \left(-y\right) \cdot \frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{x}{z}\right)\right)\right)}{\sqrt{\color{blue}{y \cdot y}}} \]

      21. sqrt-prod N/A

          \[\leadsto \left(-y\right) \cdot \frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{x}{z}\right)\right)\right)}{\color{blue}{\sqrt{y} \cdot \sqrt{y}}} \]

      22. rem-square-sqrt N/A

          \[\leadsto \left(-y\right) \cdot \frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{x}{z}\right)\right)\right)}{\color{blue}{y}} \]

      23. remove-double-neg N/A

          \[\leadsto \left(-y\right) \cdot \frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{x}{z}\right)\right)\right)}{\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)}} \]

      24. lift-neg.f64 N/A

          \[\leadsto \left(-y\right) \cdot \frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{x}{z}\right)\right)\right)}{\mathsf{neg}\left(\color{blue}{\left(-y\right)}\right)} \]

      25. frac-2neg N/A

          \[\leadsto \left(-y\right) \cdot \color{blue}{\frac{\mathsf{neg}\left(\frac{x}{z}\right)}{-y}} \]

      26. lift-neg.f64 N/A

          \[\leadsto \left(-y\right) \cdot \frac{\mathsf{neg}\left(\frac{x}{z}\right)}{\color{blue}{\mathsf{neg}\left(y\right)}} \]

      27. frac-2neg N/A

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

      28. associate-/r* N/A

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

      29. lift-*.f64 N/A

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

   9. Applied rewrites 33.1%

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

   if 0.0 < (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z)

   1. Initial program 99.1%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x}{z}} \]
   4. Step-by-step derivation

      1. lower-/.f64 60.5

         \[\leadsto \color{blue}{\frac{x}{z}} \]

   5. Applied rewrites 60.5%

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

Alternative 6: 60.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq -1.85 \cdot 10^{-305}:\\ \;\;\;\;\frac{-x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (* x_s (if (<= (/ (sin y) y) -1.85e-305) (/ (- x_m) z) (/ x_m z))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((sin(y) / y) <= -1.85e-305) {
		tmp = -x_m / z;
	} else {
		tmp = x_m / z;
	}
	return x_s * tmp;
}

Fortran

x\_m =     private
x\_s =     private
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_s, x_m, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((sin(y) / y) <= (-1.85d-305)) then
        tmp = -x_m / z
    else
        tmp = x_m / z
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((Math.sin(y) / y) <= -1.85e-305) {
		tmp = -x_m / z;
	} else {
		tmp = x_m / z;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if (math.sin(y) / y) <= -1.85e-305:
		tmp = -x_m / z
	else:
		tmp = x_m / z
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (Float64(sin(y) / y) <= -1.85e-305)
		tmp = Float64(Float64(-x_m) / z);
	else
		tmp = Float64(x_m / z);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if ((sin(y) / y) <= -1.85e-305)
		tmp = -x_m / z;
	else
		tmp = x_m / z;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], -1.85e-305], N[((-x$95$m) / z), $MachinePrecision], N[(x$95$m / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 y) y) < -1.84999999999999989e-305

   1. Initial program 92.6%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\sin y\right)}{\mathsf{neg}\left(y\right)}} \cdot \frac{x}{z} \]

      7. frac-times N/A

         \[\leadsto \color{blue}{\frac{\left(\mathsf{neg}\left(\sin y\right)\right) \cdot x}{\left(\mathsf{neg}\left(y\right)\right) \cdot z}} \]

      8. associate-/l* N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sin y\right)\right) \cdot \frac{x}{\left(\mathsf{neg}\left(y\right)\right) \cdot z}} \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sin y\right)\right) \cdot \frac{x}{\left(\mathsf{neg}\left(y\right)\right) \cdot z}} \]

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 93.3

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

   4. Applied rewrites 93.3%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(y\right)\right)} \cdot \frac{x}{\left(-y\right) \cdot z} \]

      2. lower-neg.f64 28.1

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

   7. Applied rewrites 28.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. associate-*l/ N/A

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

      8. unpow1 N/A

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

      9. metadata-eval N/A

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

      10. sqrt-pow1 N/A

          \[\leadsto \frac{\frac{x}{z} \cdot \left(-y\right)}{\color{blue}{\sqrt{{\left(-y\right)}^{2}}}} \]

      11. pow2 N/A

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

      12. lift-neg.f64 N/A

          \[\leadsto \frac{\frac{x}{z} \cdot \left(-y\right)}{\sqrt{\color{blue}{\left(\mathsf{neg}\left(y\right)\right)} \cdot \left(-y\right)}} \]

      13. lift-neg.f64 N/A

          \[\leadsto \frac{\frac{x}{z} \cdot \left(-y\right)}{\sqrt{\left(\mathsf{neg}\left(y\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}}} \]

      14. sqr-neg-rev N/A

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

      15. sqrt-prod N/A

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

      16. rem-square-sqrt N/A

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

      17. lower-/.f64 N/A

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

   9. Applied rewrites 13.3%

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

   10. Taylor expanded in y around 0

       \[\leadsto \color{blue}{-1 \cdot \frac{x}{z}} \]
   11. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \color{blue}{\frac{-1 \cdot x}{z}} \]

       2. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{-1 \cdot x}{z}} \]

       3. mul-1-neg N/A

          \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(x\right)}}{z} \]

       4. lower-neg.f64 20.8

          \[\leadsto \frac{\color{blue}{-x}}{z} \]

   12. Applied rewrites 20.8%

       \[\leadsto \color{blue}{\frac{-x}{z}} \]

   if -1.84999999999999989e-305 < (/.f64 (sin.f64 y) y)

   1. Initial program 97.2%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x}{z}} \]
   4. Step-by-step derivation

      1. lower-/.f64 69.5

         \[\leadsto \color{blue}{\frac{x}{z}} \]

   5. Applied rewrites 69.5%

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

Alternative 7: 73.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 0.008:\\ \;\;\;\;\frac{\mathsf{fma}\left(x\_m \cdot \mathsf{fma}\left(0.008333333333333333 \cdot y, y, -0.16666666666666666\right), y \cdot y, x\_m\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin y \cdot x\_m}{z \cdot y}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= y 0.008)
    (/
     (fma
      (* x_m (fma (* 0.008333333333333333 y) y -0.16666666666666666))
      (* y y)
      x_m)
     z)
    (/ (* (sin y) x_m) (* z y)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (y <= 0.008) {
		tmp = fma((x_m * fma((0.008333333333333333 * y), y, -0.16666666666666666)), (y * y), x_m) / z;
	} else {
		tmp = (sin(y) * x_m) / (z * y);
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (y <= 0.008)
		tmp = Float64(fma(Float64(x_m * fma(Float64(0.008333333333333333 * y), y, -0.16666666666666666)), Float64(y * y), x_m) / z);
	else
		tmp = Float64(Float64(sin(y) * x_m) / Float64(z * y));
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[y, 0.008], N[(N[(N[(x$95$m * N[(N[(0.008333333333333333 * y), $MachinePrecision] * y + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] * N[(y * y), $MachinePrecision] + x$95$m), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[Sin[y], $MachinePrecision] * x$95$m), $MachinePrecision] / N[(z * y), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 0.0080000000000000002

   1. Initial program 96.2%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{x + {y}^{2} \cdot \left(\frac{-1}{6} \cdot x + \frac{1}{120} \cdot \left(x \cdot {y}^{2}\right)\right)}}{z} \]
   4. Step-by-step derivation

      1. fp-cancel-sign-sub-inv N/A

         \[\leadsto \frac{\color{blue}{x - \left(\mathsf{neg}\left({y}^{2}\right)\right) \cdot \left(\frac{-1}{6} \cdot x + \frac{1}{120} \cdot \left(x \cdot {y}^{2}\right)\right)}}{z} \]

      2. fp-cancel-sub-sign-inv N/A

         \[\leadsto \frac{\color{blue}{x + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({y}^{2}\right)\right)\right)\right) \cdot \left(\frac{-1}{6} \cdot x + \frac{1}{120} \cdot \left(x \cdot {y}^{2}\right)\right)}}{z} \]

      3. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left({y}^{2}\right)\right)\right)\right) \cdot \left(\frac{-1}{6} \cdot x + \frac{1}{120} \cdot \left(x \cdot {y}^{2}\right)\right) + x}}{z} \]

      4. distribute-lft-neg-out N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left({y}^{2}\right)\right) \cdot \left(\frac{-1}{6} \cdot x + \frac{1}{120} \cdot \left(x \cdot {y}^{2}\right)\right)\right)\right)} + x}{z} \]

      5. distribute-lft-neg-out N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left({y}^{2} \cdot \left(\frac{-1}{6} \cdot x + \frac{1}{120} \cdot \left(x \cdot {y}^{2}\right)\right)\right)\right)}\right)\right) + x}{z} \]

      6. remove-double-neg N/A

         \[\leadsto \frac{\color{blue}{{y}^{2} \cdot \left(\frac{-1}{6} \cdot x + \frac{1}{120} \cdot \left(x \cdot {y}^{2}\right)\right)} + x}{z} \]

      7. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{6} \cdot x + \frac{1}{120} \cdot \left(x \cdot {y}^{2}\right)\right) \cdot {y}^{2}} + x}{z} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{6} \cdot x + \frac{1}{120} \cdot \left(x \cdot {y}^{2}\right), {y}^{2}, x\right)}}{z} \]

   5. Applied rewrites 65.1%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot \mathsf{fma}\left(0.008333333333333333 \cdot y, y, -0.16666666666666666\right), y \cdot y, x\right)}}{z} \]

   if 0.0080000000000000002 < y

   1. Initial program 95.8%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(x \cdot \sin y\right)}{\mathsf{neg}\left(y \cdot z\right)}} \]

      7. distribute-frac-neg2 N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\mathsf{neg}\left(x \cdot \sin y\right)}{y \cdot z}\right)} \]

      8. distribute-neg-frac N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(x \cdot \sin y\right)\right)\right)}{y \cdot z}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(x \cdot \sin y\right)\right)\right)}{y \cdot z}} \]

      10. distribute-rgt-neg-in N/A

          \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{x \cdot \left(\mathsf{neg}\left(\sin y\right)\right)}\right)}{y \cdot z} \]

      11. distribute-rgt-neg-in N/A

          \[\leadsto \frac{\color{blue}{x \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\sin y\right)\right)\right)\right)}}{y \cdot z} \]

      12. remove-double-neg N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 92.1

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

   4. Applied rewrites 92.1%

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

Alternative 8: 76.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 1.5 \cdot 10^{-8}:\\ \;\;\;\;\frac{x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z \cdot y} \cdot \sin y\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (* x_s (if (<= y 1.5e-8) (/ x_m z) (* (/ x_m (* z y)) (sin y)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (y <= 1.5e-8) {
		tmp = x_m / z;
	} else {
		tmp = (x_m / (z * y)) * sin(y);
	}
	return x_s * tmp;
}

Fortran

x\_m =     private
x\_s =     private
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_s, x_m, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 1.5d-8) then
        tmp = x_m / z
    else
        tmp = (x_m / (z * y)) * sin(y)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (y <= 1.5e-8) {
		tmp = x_m / z;
	} else {
		tmp = (x_m / (z * y)) * Math.sin(y);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if y <= 1.5e-8:
		tmp = x_m / z
	else:
		tmp = (x_m / (z * y)) * math.sin(y)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (y <= 1.5e-8)
		tmp = Float64(x_m / z);
	else
		tmp = Float64(Float64(x_m / Float64(z * y)) * sin(y));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if (y <= 1.5e-8)
		tmp = x_m / z;
	else
		tmp = (x_m / (z * y)) * sin(y);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[y, 1.5e-8], N[(x$95$m / z), $MachinePrecision], N[(N[(x$95$m / N[(z * y), $MachinePrecision]), $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 1.49999999999999987e-8

   1. Initial program 96.1%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x}{z}} \]
   4. Step-by-step derivation

      1. lower-/.f64 68.0

         \[\leadsto \color{blue}{\frac{x}{z}} \]

   5. Applied rewrites 68.0%

      \[\leadsto \color{blue}{\frac{x}{z}} \]

   if 1.49999999999999987e-8 < y

   1. Initial program 96.1%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 96.1

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

   4. Applied rewrites 96.1%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. remove-double-neg N/A

         \[\leadsto \frac{x}{\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(y \cdot z\right)\right)\right)}} \cdot \sin y \]

      5. distribute-lft-neg-out N/A

         \[\leadsto \frac{x}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(y\right)\right) \cdot z}\right)} \cdot \sin y \]

      6. lift-neg.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-neg.f64 N/A

          \[\leadsto \frac{x}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(y\right)\right)} \cdot z\right)} \cdot \sin y \]

      11. distribute-lft-neg-out N/A

          \[\leadsto \frac{x}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(y \cdot z\right)\right)}\right)} \cdot \sin y \]

      12. remove-double-neg N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 92.7

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

   6. Applied rewrites 92.7%

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

Alternative 9: 59.2% accurate, 10.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \frac{x\_m}{z} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z) :precision binary64 (* x_s (/ x_m z)))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	return x_s * (x_m / z);
}

Fortran

x\_m =     private
x\_s =     private
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_s, x_m, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x_s * (x_m / z)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	return x_s * (x_m / z);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	return x_s * (x_m / z)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	return Float64(x_s * Float64(x_m / z))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m, y, z)
	tmp = x_s * (x_m / z);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.1%

   \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\frac{x}{z}} \]
4. Step-by-step derivation

   1. lower-/.f64 56.4

      \[\leadsto \color{blue}{\frac{x}{z}} \]

5. Applied rewrites 56.4%

   \[\leadsto \color{blue}{\frac{x}{z}} \]
6. Add Preprocessing

Developer Target 1: 99.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{\sin y}\\ t_1 := \frac{x \cdot \frac{1}{t\_0}}{z}\\ \mathbf{if}\;z < -4.2173720203427147 \cdot 10^{-29}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z < 4.446702369113811 \cdot 10^{+64}:\\ \;\;\;\;\frac{x}{z \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ y (sin y))) (t_1 (/ (* x (/ 1.0 t_0)) z)))
   (if (< z -4.2173720203427147e-29)
     t_1
     (if (< z 4.446702369113811e+64) (/ x (* z t_0)) t_1))))

C

double code(double x, double y, double z) {
	double t_0 = y / sin(y);
	double t_1 = (x * (1.0 / t_0)) / z;
	double tmp;
	if (z < -4.2173720203427147e-29) {
		tmp = t_1;
	} else if (z < 4.446702369113811e+64) {
		tmp = x / (z * t_0);
	} else {
		tmp = t_1;
	}
	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, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = y / sin(y)
    t_1 = (x * (1.0d0 / t_0)) / z
    if (z < (-4.2173720203427147d-29)) then
        tmp = t_1
    else if (z < 4.446702369113811d+64) then
        tmp = x / (z * t_0)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y / Math.sin(y);
	double t_1 = (x * (1.0 / t_0)) / z;
	double tmp;
	if (z < -4.2173720203427147e-29) {
		tmp = t_1;
	} else if (z < 4.446702369113811e+64) {
		tmp = x / (z * t_0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y / math.sin(y)
	t_1 = (x * (1.0 / t_0)) / z
	tmp = 0
	if z < -4.2173720203427147e-29:
		tmp = t_1
	elif z < 4.446702369113811e+64:
		tmp = x / (z * t_0)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y / sin(y))
	t_1 = Float64(Float64(x * Float64(1.0 / t_0)) / z)
	tmp = 0.0
	if (z < -4.2173720203427147e-29)
		tmp = t_1;
	elseif (z < 4.446702369113811e+64)
		tmp = Float64(x / Float64(z * t_0));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y / sin(y);
	t_1 = (x * (1.0 / t_0)) / z;
	tmp = 0.0;
	if (z < -4.2173720203427147e-29)
		tmp = t_1;
	elseif (z < 4.446702369113811e+64)
		tmp = x / (z * t_0);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y / N[Sin[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, If[Less[z, -4.2173720203427147e-29], t$95$1, If[Less[z, 4.446702369113811e+64], N[(x / N[(z * t$95$0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Reproduce

herbie shell --seed 2024350 
(FPCore (x y z)
  :name "Linear.Quaternion:$ctanh from linear-1.19.1.3"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< z -42173720203427147/1000000000000000000000000000000000000000000000) (/ (* x (/ 1 (/ y (sin y)))) z) (if (< z 44467023691138110000000000000000000000000000000000000000000000000) (/ x (* z (/ y (sin y)))) (/ (* x (/ 1 (/ y (sin y)))) z))))

  (/ (* x (/ (sin y) y)) z))