Linear.Quaternion:$ctanh from linear-1.19.1.3

Percentage Accurate: 96.1% → 97.7%

Time: 3.6s

Alternatives: 11

Speedup: 0.5×

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: 96.1% 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: 97.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_0 := \frac{x\_m \cdot \frac{\sin y}{y}}{z\_m}\\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\frac{\sin y}{z\_m \cdot y} \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\right) \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x_s x_m y z_m)
 :precision binary64
 (let* ((t_0 (/ (* x_m (/ (sin y) y)) z_m)))
   (* z_s (* x_s (if (<= t_0 0.0) (* (/ (sin y) (* z_m y)) x_m) t_0)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x_s, double x_m, double y, double z_m) {
	double t_0 = (x_m * (sin(y) / y)) / z_m;
	double tmp;
	if (t_0 <= 0.0) {
		tmp = (sin(y) / (z_m * y)) * x_m;
	} else {
		tmp = t_0;
	}
	return z_s * (x_s * tmp);
}

Fortran

x\_m =     private
x\_s =     private
z\_m =     private
z\_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(z_s, x_s, x_m, y, z_m)
use fmin_fmax_functions
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x_m * (sin(y) / y)) / z_m
    if (t_0 <= 0.0d0) then
        tmp = (sin(y) / (z_m * y)) * x_m
    else
        tmp = t_0
    end if
    code = z_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x_s, double x_m, double y, double z_m) {
	double t_0 = (x_m * (Math.sin(y) / y)) / z_m;
	double tmp;
	if (t_0 <= 0.0) {
		tmp = (Math.sin(y) / (z_m * y)) * x_m;
	} else {
		tmp = t_0;
	}
	return z_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x_s, x_m, y, z_m):
	t_0 = (x_m * (math.sin(y) / y)) / z_m
	tmp = 0
	if t_0 <= 0.0:
		tmp = (math.sin(y) / (z_m * y)) * x_m
	else:
		tmp = t_0
	return z_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x_s, x_m, y, z_m)
	t_0 = Float64(Float64(x_m * Float64(sin(y) / y)) / z_m)
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(Float64(sin(y) / Float64(z_m * y)) * x_m);
	else
		tmp = t_0;
	end
	return Float64(z_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x_s, x_m, y, z_m)
	t_0 = (x_m * (sin(y) / y)) / z_m;
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = (sin(y) / (z_m * y)) * x_m;
	else
		tmp = t_0;
	end
	tmp_2 = z_s * (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]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x$95$s_, x$95$m_, y_, z$95$m_] := Block[{t$95$0 = N[(N[(x$95$m * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision]}, N[(z$95$s * N[(x$95$s * If[LessEqual[t$95$0, 0.0], N[(N[(N[Sin[y], $MachinePrecision] / N[(z$95$m * y), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision], t$95$0]), $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 90.7%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   2. 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. lift-sin.f64 N/A

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

      5. associate-/l* N/A

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

      6. associate-/r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 94.6

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

   3. Applied rewrites 94.6%

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

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

   1. Initial program 99.7%

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

Alternative 2: 77.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 7.8 \cdot 10^{-11}:\\ \;\;\;\;\frac{x\_m}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin y}{z\_m} \cdot \frac{x\_m}{y}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x_s x_m y z_m)
 :precision binary64
 (* z_s (* x_s (if (<= y 7.8e-11) (/ x_m z_m) (* (/ (sin y) z_m) (/ x_m y))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x_s, double x_m, double y, double z_m) {
	double tmp;
	if (y <= 7.8e-11) {
		tmp = x_m / z_m;
	} else {
		tmp = (sin(y) / z_m) * (x_m / y);
	}
	return z_s * (x_s * tmp);
}

Fortran

x\_m =     private
x\_s =     private
z\_m =     private
z\_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(z_s, x_s, x_m, y, z_m)
use fmin_fmax_functions
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (y <= 7.8d-11) then
        tmp = x_m / z_m
    else
        tmp = (sin(y) / z_m) * (x_m / y)
    end if
    code = z_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x_s, double x_m, double y, double z_m) {
	double tmp;
	if (y <= 7.8e-11) {
		tmp = x_m / z_m;
	} else {
		tmp = (Math.sin(y) / z_m) * (x_m / y);
	}
	return z_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x_s, x_m, y, z_m):
	tmp = 0
	if y <= 7.8e-11:
		tmp = x_m / z_m
	else:
		tmp = (math.sin(y) / z_m) * (x_m / y)
	return z_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x_s, x_m, y, z_m)
	tmp = 0.0
	if (y <= 7.8e-11)
		tmp = Float64(x_m / z_m);
	else
		tmp = Float64(Float64(sin(y) / z_m) * Float64(x_m / y));
	end
	return Float64(z_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x_s, x_m, y, z_m)
	tmp = 0.0;
	if (y <= 7.8e-11)
		tmp = x_m / z_m;
	else
		tmp = (sin(y) / z_m) * (x_m / y);
	end
	tmp_2 = z_s * (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]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x$95$s_, x$95$m_, y_, z$95$m_] := N[(z$95$s * N[(x$95$s * If[LessEqual[y, 7.8e-11], N[(x$95$m / z$95$m), $MachinePrecision], N[(N[(N[Sin[y], $MachinePrecision] / z$95$m), $MachinePrecision] * N[(x$95$m / y), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 7.80000000000000021e-11

   1. Initial program 97.5%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 72.4%

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

   if 7.80000000000000021e-11 < y

   1. Initial program 92.3%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   2. 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. lift-sin.f64 N/A

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

      5. associate-*r/ N/A

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

      6. associate-/r* N/A

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

      7. times-frac N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lower-/.f64 92.3

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

   3. Applied rewrites 92.3%

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

Alternative 3: 77.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 7.8 \cdot 10^{-11}:\\ \;\;\;\;\frac{x\_m}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin y}{z\_m \cdot y} \cdot x\_m\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x_s x_m y z_m)
 :precision binary64
 (* z_s (* x_s (if (<= y 7.8e-11) (/ x_m z_m) (* (/ (sin y) (* z_m y)) x_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x_s, double x_m, double y, double z_m) {
	double tmp;
	if (y <= 7.8e-11) {
		tmp = x_m / z_m;
	} else {
		tmp = (sin(y) / (z_m * y)) * x_m;
	}
	return z_s * (x_s * tmp);
}

Fortran

x\_m =     private
x\_s =     private
z\_m =     private
z\_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(z_s, x_s, x_m, y, z_m)
use fmin_fmax_functions
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (y <= 7.8d-11) then
        tmp = x_m / z_m
    else
        tmp = (sin(y) / (z_m * y)) * x_m
    end if
    code = z_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x_s, double x_m, double y, double z_m) {
	double tmp;
	if (y <= 7.8e-11) {
		tmp = x_m / z_m;
	} else {
		tmp = (Math.sin(y) / (z_m * y)) * x_m;
	}
	return z_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x_s, x_m, y, z_m):
	tmp = 0
	if y <= 7.8e-11:
		tmp = x_m / z_m
	else:
		tmp = (math.sin(y) / (z_m * y)) * x_m
	return z_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x_s, x_m, y, z_m)
	tmp = 0.0
	if (y <= 7.8e-11)
		tmp = Float64(x_m / z_m);
	else
		tmp = Float64(Float64(sin(y) / Float64(z_m * y)) * x_m);
	end
	return Float64(z_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x_s, x_m, y, z_m)
	tmp = 0.0;
	if (y <= 7.8e-11)
		tmp = x_m / z_m;
	else
		tmp = (sin(y) / (z_m * y)) * x_m;
	end
	tmp_2 = z_s * (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]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x$95$s_, x$95$m_, y_, z$95$m_] := N[(z$95$s * N[(x$95$s * If[LessEqual[y, 7.8e-11], N[(x$95$m / z$95$m), $MachinePrecision], N[(N[(N[Sin[y], $MachinePrecision] / N[(z$95$m * y), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 7.80000000000000021e-11

   1. Initial program 97.5%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 72.4%

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

   if 7.80000000000000021e-11 < y

   1. Initial program 92.3%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   2. 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. lift-sin.f64 N/A

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

      5. associate-/l* N/A

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

      6. associate-/r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 92.0

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

   3. Applied rewrites 92.0%

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

Alternative 4: 77.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 10^{-17}:\\ \;\;\;\;\frac{x\_m}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \frac{x\_m}{z\_m \cdot y}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x_s x_m y z_m)
 :precision binary64
 (* z_s (* x_s (if (<= y 1e-17) (/ x_m z_m) (* (sin y) (/ x_m (* z_m y)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x_s, double x_m, double y, double z_m) {
	double tmp;
	if (y <= 1e-17) {
		tmp = x_m / z_m;
	} else {
		tmp = sin(y) * (x_m / (z_m * y));
	}
	return z_s * (x_s * tmp);
}

Fortran

x\_m =     private
x\_s =     private
z\_m =     private
z\_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(z_s, x_s, x_m, y, z_m)
use fmin_fmax_functions
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (y <= 1d-17) then
        tmp = x_m / z_m
    else
        tmp = sin(y) * (x_m / (z_m * y))
    end if
    code = z_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x_s, double x_m, double y, double z_m) {
	double tmp;
	if (y <= 1e-17) {
		tmp = x_m / z_m;
	} else {
		tmp = Math.sin(y) * (x_m / (z_m * y));
	}
	return z_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x_s, x_m, y, z_m):
	tmp = 0
	if y <= 1e-17:
		tmp = x_m / z_m
	else:
		tmp = math.sin(y) * (x_m / (z_m * y))
	return z_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x_s, x_m, y, z_m)
	tmp = 0.0
	if (y <= 1e-17)
		tmp = Float64(x_m / z_m);
	else
		tmp = Float64(sin(y) * Float64(x_m / Float64(z_m * y)));
	end
	return Float64(z_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x_s, x_m, y, z_m)
	tmp = 0.0;
	if (y <= 1e-17)
		tmp = x_m / z_m;
	else
		tmp = sin(y) * (x_m / (z_m * y));
	end
	tmp_2 = z_s * (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]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x$95$s_, x$95$m_, y_, z$95$m_] := N[(z$95$s * N[(x$95$s * If[LessEqual[y, 1e-17], N[(x$95$m / z$95$m), $MachinePrecision], N[(N[Sin[y], $MachinePrecision] * N[(x$95$m / N[(z$95$m * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 1.00000000000000007e-17

   1. Initial program 97.5%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 72.3%

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

   if 1.00000000000000007e-17 < y

   1. Initial program 92.4%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   2. 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. lift-sin.f64 N/A

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

      5. associate-*r/ N/A

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

      6. associate-/r* N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 92.1

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

   3. Applied rewrites 92.1%

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

Alternative 5: 66.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 8.6 \cdot 10^{-128}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(y \cdot y\right) \cdot x\_m, -0.16666666666666666, x\_m\right)}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-y}{\mathsf{fma}\left(-0.16666666666666666 \cdot y, y, -1\right)}}{z\_m \cdot y} \cdot x\_m\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x_s x_m y z_m)
 :precision binary64
 (*
  z_s
  (*
   x_s
   (if (<= z_m 8.6e-128)
     (/ (fma (* (* y y) x_m) -0.16666666666666666 x_m) z_m)
     (*
      (/ (/ (- y) (fma (* -0.16666666666666666 y) y -1.0)) (* z_m y))
      x_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x_s, double x_m, double y, double z_m) {
	double tmp;
	if (z_m <= 8.6e-128) {
		tmp = fma(((y * y) * x_m), -0.16666666666666666, x_m) / z_m;
	} else {
		tmp = ((-y / fma((-0.16666666666666666 * y), y, -1.0)) / (z_m * y)) * x_m;
	}
	return z_s * (x_s * tmp);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x_s, x_m, y, z_m)
	tmp = 0.0
	if (z_m <= 8.6e-128)
		tmp = Float64(fma(Float64(Float64(y * y) * x_m), -0.16666666666666666, x_m) / z_m);
	else
		tmp = Float64(Float64(Float64(Float64(-y) / fma(Float64(-0.16666666666666666 * y), y, -1.0)) / Float64(z_m * y)) * x_m);
	end
	return Float64(z_s * 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]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x$95$s_, x$95$m_, y_, z$95$m_] := N[(z$95$s * N[(x$95$s * If[LessEqual[z$95$m, 8.6e-128], N[(N[(N[(N[(y * y), $MachinePrecision] * x$95$m), $MachinePrecision] * -0.16666666666666666 + x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision], N[(N[(N[((-y) / N[(N[(-0.16666666666666666 * y), $MachinePrecision] * y + -1.0), $MachinePrecision]), $MachinePrecision] / N[(z$95$m * y), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 8.59999999999999988e-128

   1. Initial program 89.7%

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

   2. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 53.6

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

   4. Applied rewrites 53.6%

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

   if 8.59999999999999988e-128 < z

   1. Initial program 98.7%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   2. 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. lift-sin.f64 N/A

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

      5. associate-/l* N/A

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

      6. associate-/r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 91.9

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

   3. Applied rewrites 91.9%

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

   4. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 50.7

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

   6. Applied rewrites 50.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

         \[\leadsto \frac{\left(\left(y \cdot y\right) \cdot \frac{-1}{6} + 1\right) \cdot y}{z \cdot y} \cdot x \]

      4. *-commutative N/A

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

      5. flip-+ N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

   8. Applied rewrites 49.4%

      \[\leadsto \frac{\frac{y \cdot \mathsf{fma}\left(0.027777777777777776, \left(\left(y \cdot y\right) \cdot y\right) \cdot y, -1\right)}{\color{blue}{\mathsf{fma}\left(-0.16666666666666666 \cdot y, y, -1\right)}}}{z \cdot y} \cdot x \]

   9. Taylor expanded in y around 0

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 69.2

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

   11. Applied rewrites 69.2%

       \[\leadsto \frac{\frac{-y}{\mathsf{fma}\left(\color{blue}{-0.16666666666666666 \cdot y}, y, -1\right)}}{z \cdot y} \cdot x \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 66.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 0.04:\\ \;\;\;\;y \cdot \frac{\frac{x\_m}{y}}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot \mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right)}{z\_m}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x_s x_m y z_m)
 :precision binary64
 (*
  z_s
  (*
   x_s
   (if (<= (/ (sin y) y) 0.04)
     (* y (/ (/ x_m y) z_m))
     (/ (* x_m (fma (* y y) -0.16666666666666666 1.0)) z_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x_s, double x_m, double y, double z_m) {
	double tmp;
	if ((sin(y) / y) <= 0.04) {
		tmp = y * ((x_m / y) / z_m);
	} else {
		tmp = (x_m * fma((y * y), -0.16666666666666666, 1.0)) / z_m;
	}
	return z_s * (x_s * tmp);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x_s, x_m, y, z_m)
	tmp = 0.0
	if (Float64(sin(y) / y) <= 0.04)
		tmp = Float64(y * Float64(Float64(x_m / y) / z_m));
	else
		tmp = Float64(Float64(x_m * fma(Float64(y * y), -0.16666666666666666, 1.0)) / z_m);
	end
	return Float64(z_s * 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]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x$95$s_, x$95$m_, y_, z$95$m_] := N[(z$95$s * N[(x$95$s * If[LessEqual[N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], 0.04], N[(y * N[(N[(x$95$m / y), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * N[(N[(y * y), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 y) y) < 0.0400000000000000008

   1. Initial program 92.3%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   2. 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. lift-sin.f64 N/A

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

      5. associate-*r/ N/A

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

      6. associate-/r* N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 91.4

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

   3. Applied rewrites 91.4%

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

   4. Taylor expanded in y around 0

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

   6. Applied rewrites 33.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lift-/.f64 33.2

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

   8. Applied rewrites 33.2%

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

   if 0.0400000000000000008 < (/.f64 (sin.f64 y) y)

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 99.6

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

   4. Applied rewrites 99.6%

      \[\leadsto \frac{x \cdot \color{blue}{\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right)}}{z} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 66.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 0.04:\\ \;\;\;\;y \cdot \frac{\frac{x\_m}{y}}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(y \cdot y\right) \cdot x\_m, -0.16666666666666666, x\_m\right)}{z\_m}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x_s x_m y z_m)
 :precision binary64
 (*
  z_s
  (*
   x_s
   (if (<= (/ (sin y) y) 0.04)
     (* y (/ (/ x_m y) z_m))
     (/ (fma (* (* y y) x_m) -0.16666666666666666 x_m) z_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x_s, double x_m, double y, double z_m) {
	double tmp;
	if ((sin(y) / y) <= 0.04) {
		tmp = y * ((x_m / y) / z_m);
	} else {
		tmp = fma(((y * y) * x_m), -0.16666666666666666, x_m) / z_m;
	}
	return z_s * (x_s * tmp);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x_s, x_m, y, z_m)
	tmp = 0.0
	if (Float64(sin(y) / y) <= 0.04)
		tmp = Float64(y * Float64(Float64(x_m / y) / z_m));
	else
		tmp = Float64(fma(Float64(Float64(y * y) * x_m), -0.16666666666666666, x_m) / z_m);
	end
	return Float64(z_s * 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]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x$95$s_, x$95$m_, y_, z$95$m_] := N[(z$95$s * N[(x$95$s * If[LessEqual[N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], 0.04], N[(y * N[(N[(x$95$m / y), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(y * y), $MachinePrecision] * x$95$m), $MachinePrecision] * -0.16666666666666666 + x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 y) y) < 0.0400000000000000008

   1. Initial program 92.3%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   2. 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. lift-sin.f64 N/A

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

      5. associate-*r/ N/A

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

      6. associate-/r* N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 91.4

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

   3. Applied rewrites 91.4%

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

   4. Taylor expanded in y around 0

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

   6. Applied rewrites 33.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lift-/.f64 33.2

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

   8. Applied rewrites 33.2%

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

   if 0.0400000000000000008 < (/.f64 (sin.f64 y) y)

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 99.6

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

   4. Applied rewrites 99.6%

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

Alternative 8: 66.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 0.04:\\ \;\;\;\;y \cdot \frac{\frac{x\_m}{y}}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right)}{z\_m} \cdot x\_m\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x_s x_m y z_m)
 :precision binary64
 (*
  z_s
  (*
   x_s
   (if (<= (/ (sin y) y) 0.04)
     (* y (/ (/ x_m y) z_m))
     (* (/ (fma (* y y) -0.16666666666666666 1.0) z_m) x_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x_s, double x_m, double y, double z_m) {
	double tmp;
	if ((sin(y) / y) <= 0.04) {
		tmp = y * ((x_m / y) / z_m);
	} else {
		tmp = (fma((y * y), -0.16666666666666666, 1.0) / z_m) * x_m;
	}
	return z_s * (x_s * tmp);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x_s, x_m, y, z_m)
	tmp = 0.0
	if (Float64(sin(y) / y) <= 0.04)
		tmp = Float64(y * Float64(Float64(x_m / y) / z_m));
	else
		tmp = Float64(Float64(fma(Float64(y * y), -0.16666666666666666, 1.0) / z_m) * x_m);
	end
	return Float64(z_s * 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]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x$95$s_, x$95$m_, y_, z$95$m_] := N[(z$95$s * N[(x$95$s * If[LessEqual[N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], 0.04], N[(y * N[(N[(x$95$m / y), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(y * y), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] / z$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 y) y) < 0.0400000000000000008

   1. Initial program 92.3%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   2. 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. lift-sin.f64 N/A

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

      5. associate-*r/ N/A

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

      6. associate-/r* N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 91.4

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

   3. Applied rewrites 91.4%

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

   4. Taylor expanded in y around 0

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

   6. Applied rewrites 33.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lift-/.f64 33.2

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

   8. Applied rewrites 33.2%

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

   if 0.0400000000000000008 < (/.f64 (sin.f64 y) y)

   1. Initial program 100.0%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   2. 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. lift-sin.f64 N/A

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

      5. associate-/l* N/A

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

      6. associate-/r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 85.1

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

   3. Applied rewrites 85.1%

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

   4. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. div-add-rev N/A

         \[\leadsto \frac{1 + \frac{-1}{6} \cdot {y}^{2}}{\color{blue}{z}} \cdot x \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{1 + \frac{-1}{6} \cdot {y}^{2}}{\color{blue}{z}} \cdot x \]

      5. +-commutative N/A

         \[\leadsto \frac{\frac{-1}{6} \cdot {y}^{2} + 1}{z} \cdot x \]

      6. *-commutative N/A

         \[\leadsto \frac{{y}^{2} \cdot \frac{-1}{6} + 1}{z} \cdot x \]

      7. lower-fma.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 99.3

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

   6. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right)}{z}} \cdot x \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 66.0% accurate, 0.9× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x_s x_m y z_m)
 :precision binary64
 (*
  z_s
  (* x_s (if (<= (/ (sin y) y) 5e-10) (* y (/ (/ x_m y) z_m)) (/ x_m z_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x_s, double x_m, double y, double z_m) {
	double tmp;
	if ((sin(y) / y) <= 5e-10) {
		tmp = y * ((x_m / y) / z_m);
	} else {
		tmp = x_m / z_m;
	}
	return z_s * (x_s * tmp);
}

Fortran

x\_m =     private
x\_s =     private
z\_m =     private
z\_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(z_s, x_s, x_m, y, z_m)
use fmin_fmax_functions
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if ((sin(y) / y) <= 5d-10) then
        tmp = y * ((x_m / y) / z_m)
    else
        tmp = x_m / z_m
    end if
    code = z_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x_s, double x_m, double y, double z_m) {
	double tmp;
	if ((Math.sin(y) / y) <= 5e-10) {
		tmp = y * ((x_m / y) / z_m);
	} else {
		tmp = x_m / z_m;
	}
	return z_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x_s, x_m, y, z_m):
	tmp = 0
	if (math.sin(y) / y) <= 5e-10:
		tmp = y * ((x_m / y) / z_m)
	else:
		tmp = x_m / z_m
	return z_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x_s, x_m, y, z_m)
	tmp = 0.0
	if (Float64(sin(y) / y) <= 5e-10)
		tmp = Float64(y * Float64(Float64(x_m / y) / z_m));
	else
		tmp = Float64(x_m / z_m);
	end
	return Float64(z_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x_s, x_m, y, z_m)
	tmp = 0.0;
	if ((sin(y) / y) <= 5e-10)
		tmp = y * ((x_m / y) / z_m);
	else
		tmp = x_m / z_m;
	end
	tmp_2 = z_s * (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]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x$95$s_, x$95$m_, y_, z$95$m_] := N[(z$95$s * N[(x$95$s * If[LessEqual[N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], 5e-10], N[(y * N[(N[(x$95$m / y), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision], N[(x$95$m / z$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 y) y) < 5.00000000000000031e-10

   1. Initial program 92.2%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   2. 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. lift-sin.f64 N/A

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

      5. associate-*r/ N/A

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

      6. associate-/r* N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 91.3

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

   3. Applied rewrites 91.3%

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

   4. Taylor expanded in y around 0

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

   6. Applied rewrites 33.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lift-/.f64 33.2

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

   8. Applied rewrites 33.2%

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

   if 5.00000000000000031e-10 < (/.f64 (sin.f64 y) y)

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 98.3%

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

Alternative 10: 64.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 0.999999:\\ \;\;\;\;y \cdot \frac{x\_m}{z\_m \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z\_m}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x_s x_m y z_m)
 :precision binary64
 (*
  z_s
  (*
   x_s
   (if (<= (/ (sin y) y) 0.999999) (* y (/ x_m (* z_m y))) (/ x_m z_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x_s, double x_m, double y, double z_m) {
	double tmp;
	if ((sin(y) / y) <= 0.999999) {
		tmp = y * (x_m / (z_m * y));
	} else {
		tmp = x_m / z_m;
	}
	return z_s * (x_s * tmp);
}

Fortran

x\_m =     private
x\_s =     private
z\_m =     private
z\_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(z_s, x_s, x_m, y, z_m)
use fmin_fmax_functions
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if ((sin(y) / y) <= 0.999999d0) then
        tmp = y * (x_m / (z_m * y))
    else
        tmp = x_m / z_m
    end if
    code = z_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x_s, double x_m, double y, double z_m) {
	double tmp;
	if ((Math.sin(y) / y) <= 0.999999) {
		tmp = y * (x_m / (z_m * y));
	} else {
		tmp = x_m / z_m;
	}
	return z_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x_s, x_m, y, z_m):
	tmp = 0
	if (math.sin(y) / y) <= 0.999999:
		tmp = y * (x_m / (z_m * y))
	else:
		tmp = x_m / z_m
	return z_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x_s, x_m, y, z_m)
	tmp = 0.0
	if (Float64(sin(y) / y) <= 0.999999)
		tmp = Float64(y * Float64(x_m / Float64(z_m * y)));
	else
		tmp = Float64(x_m / z_m);
	end
	return Float64(z_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x_s, x_m, y, z_m)
	tmp = 0.0;
	if ((sin(y) / y) <= 0.999999)
		tmp = y * (x_m / (z_m * y));
	else
		tmp = x_m / z_m;
	end
	tmp_2 = z_s * (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]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x$95$s_, x$95$m_, y_, z$95$m_] := N[(z$95$s * N[(x$95$s * If[LessEqual[N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], 0.999999], N[(y * N[(x$95$m / N[(z$95$m * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m / z$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 y) y) < 0.999998999999999971

   1. Initial program 92.4%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   2. 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. lift-sin.f64 N/A

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

      5. associate-*r/ N/A

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

      6. associate-/r* N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 91.5

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

   3. Applied rewrites 91.5%

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

   4. Taylor expanded in y around 0

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

   6. Applied rewrites 33.3%

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

   if 0.999998999999999971 < (/.f64 (sin.f64 y) y)

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 99.7%

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

Alternative 11: 58.8% accurate, 9.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp = code(z_s, x_s, x_m, y, z_m)
	tmp = z_s * (x_s * (x_m / z_m));
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]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x$95$s_, x$95$m_, y_, z$95$m_] := N[(z$95$s * N[(x$95$s * N[(x$95$m / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.1%

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

2. Taylor expanded in y around 0

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

4. Applied rewrites 58.8%

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

Reproduce

herbie shell --seed 2025112 
(FPCore (x y z)
  :name "Linear.Quaternion:$ctanh from linear-1.19.1.3"
  :precision binary64
  (/ (* x (/ (sin y) y)) z))