Linear.Quaternion:$ctanh from linear-1.19.1.3

Percentage Accurate: 96.0% → 99.7%

Time: 4.1s

Alternatives: 11

Speedup: 0.9×

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: 96.0% 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: 99.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 2 \cdot 10^{-9}:\\ \;\;\;\;\frac{x}{\frac{y}{\sin y} \cdot z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{\sin y}{y}}{z\_m}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m)
 :precision binary64
 (*
  z_s
  (if (<= z_m 2e-9) (/ x (* (/ y (sin y)) z_m)) (/ (* x (/ (sin y) y)) z_m))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (z_m <= 2e-9) {
		tmp = x / ((y / sin(y)) * z_m);
	} else {
		tmp = (x * (sin(y) / y)) / z_m;
	}
	return z_s * tmp;
}

Fortran

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, y, z_m)
use fmin_fmax_functions
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (z_m <= 2d-9) then
        tmp = x / ((y / sin(y)) * z_m)
    else
        tmp = (x * (sin(y) / y)) / z_m
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (z_m <= 2e-9) {
		tmp = x / ((y / Math.sin(y)) * z_m);
	} else {
		tmp = (x * (Math.sin(y) / y)) / z_m;
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m):
	tmp = 0
	if z_m <= 2e-9:
		tmp = x / ((y / math.sin(y)) * z_m)
	else:
		tmp = (x * (math.sin(y) / y)) / z_m
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	tmp = 0.0
	if (z_m <= 2e-9)
		tmp = Float64(x / Float64(Float64(y / sin(y)) * z_m));
	else
		tmp = Float64(Float64(x * Float64(sin(y) / y)) / z_m);
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m)
	tmp = 0.0;
	if (z_m <= 2e-9)
		tmp = x / ((y / sin(y)) * z_m);
	else
		tmp = (x * (sin(y) / y)) / z_m;
	end
	tmp_2 = z_s * tmp;
end

Wolfram

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_, y_, z$95$m_] := N[(z$95$s * If[LessEqual[z$95$m, 2e-9], N[(x / N[(N[(y / N[Sin[y], $MachinePrecision]), $MachinePrecision] * z$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 2.00000000000000012e-9

   1. Initial program 96.0%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. mult-flip-rev N/A

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

      8. lower-/.f64 95.8

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

   3. Applied rewrites 95.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l/ N/A

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

      7. div-flip-rev N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. div-flip-rev N/A

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

      11. associate-/l/ N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 95.8

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

   5. Applied rewrites 95.8%

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

   if 2.00000000000000012e-9 < z

   1. Initial program 96.0%

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

Alternative 2: 99.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 2 \cdot 10^{-9}:\\ \;\;\;\;\frac{x}{\frac{y}{\sin y} \cdot z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin y}{y} \cdot \frac{x}{z\_m}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m)
 :precision binary64
 (*
  z_s
  (if (<= z_m 2e-9) (/ x (* (/ y (sin y)) z_m)) (* (/ (sin y) y) (/ x z_m)))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (z_m <= 2e-9) {
		tmp = x / ((y / sin(y)) * z_m);
	} else {
		tmp = (sin(y) / y) * (x / z_m);
	}
	return z_s * tmp;
}

Fortran

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, y, z_m)
use fmin_fmax_functions
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (z_m <= 2d-9) then
        tmp = x / ((y / sin(y)) * z_m)
    else
        tmp = (sin(y) / y) * (x / z_m)
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (z_m <= 2e-9) {
		tmp = x / ((y / Math.sin(y)) * z_m);
	} else {
		tmp = (Math.sin(y) / y) * (x / z_m);
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m):
	tmp = 0
	if z_m <= 2e-9:
		tmp = x / ((y / math.sin(y)) * z_m)
	else:
		tmp = (math.sin(y) / y) * (x / z_m)
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	tmp = 0.0
	if (z_m <= 2e-9)
		tmp = Float64(x / Float64(Float64(y / sin(y)) * z_m));
	else
		tmp = Float64(Float64(sin(y) / y) * Float64(x / z_m));
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m)
	tmp = 0.0;
	if (z_m <= 2e-9)
		tmp = x / ((y / sin(y)) * z_m);
	else
		tmp = (sin(y) / y) * (x / z_m);
	end
	tmp_2 = z_s * tmp;
end

Wolfram

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_, y_, z$95$m_] := N[(z$95$s * If[LessEqual[z$95$m, 2e-9], N[(x / N[(N[(y / N[Sin[y], $MachinePrecision]), $MachinePrecision] * z$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision] * N[(x / z$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 2.00000000000000012e-9

   1. Initial program 96.0%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. mult-flip-rev N/A

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

      8. lower-/.f64 95.8

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

   3. Applied rewrites 95.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l/ N/A

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

      7. div-flip-rev N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. div-flip-rev N/A

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

      11. associate-/l/ N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 95.8

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

   5. Applied rewrites 95.8%

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

   if 2.00000000000000012e-9 < z

   1. Initial program 96.0%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. mult-flip-rev N/A

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

      8. lower-/.f64 95.8

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

   3. Applied rewrites 95.8%

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

Alternative 3: 95.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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_, y_, z$95$m_] := N[(z$95$s * N[(N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision] * N[(x / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.0%

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

   1. lift-/.f64 N/A

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

   2. mult-flip N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. lower-*.f64 N/A

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

   7. mult-flip-rev N/A

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

   8. lower-/.f64 95.8

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

3. Applied rewrites 95.8%

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

Alternative 4: 77.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 7.5 \cdot 10^{-30}:\\ \;\;\;\;\frac{x}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin y \cdot x}{z\_m \cdot y}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m)
 :precision binary64
 (* z_s (if (<= y 7.5e-30) (/ x z_m) (/ (* (sin y) x) (* z_m y)))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (y <= 7.5e-30) {
		tmp = x / z_m;
	} else {
		tmp = (sin(y) * x) / (z_m * y);
	}
	return z_s * tmp;
}

Fortran

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, y, z_m)
use fmin_fmax_functions
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (y <= 7.5d-30) then
        tmp = x / z_m
    else
        tmp = (sin(y) * x) / (z_m * y)
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (y <= 7.5e-30) {
		tmp = x / z_m;
	} else {
		tmp = (Math.sin(y) * x) / (z_m * y);
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m):
	tmp = 0
	if y <= 7.5e-30:
		tmp = x / z_m
	else:
		tmp = (math.sin(y) * x) / (z_m * y)
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	tmp = 0.0
	if (y <= 7.5e-30)
		tmp = Float64(x / z_m);
	else
		tmp = Float64(Float64(sin(y) * x) / Float64(z_m * y));
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m)
	tmp = 0.0;
	if (y <= 7.5e-30)
		tmp = x / z_m;
	else
		tmp = (sin(y) * x) / (z_m * y);
	end
	tmp_2 = z_s * tmp;
end

Wolfram

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_, y_, z$95$m_] := N[(z$95$s * If[LessEqual[y, 7.5e-30], N[(x / z$95$m), $MachinePrecision], N[(N[(N[Sin[y], $MachinePrecision] * x), $MachinePrecision] / N[(z$95$m * y), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 7.5000000000000006e-30

   1. Initial program 96.0%

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

   2. Taylor expanded in y around 0

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

      1. lower-/.f64 58.3

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

   4. Applied rewrites 58.3%

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

   if 7.5000000000000006e-30 < y

   1. Initial program 96.0%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   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. 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-neg-frac2 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. *-commutative N/A

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

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

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

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

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

      13. remove-double-neg N/A

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

      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 84.3

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

   3. Applied rewrites 84.3%

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

Alternative 5: 75.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 0.000216:\\ \;\;\;\;\frac{x}{z\_m} \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z\_m \cdot y} \cdot \sin y\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m)
 :precision binary64
 (*
  z_s
  (if (<= y 0.000216)
    (* (/ x z_m) (fma -0.16666666666666666 (* y y) 1.0))
    (* (/ x (* z_m y)) (sin y)))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (y <= 0.000216) {
		tmp = (x / z_m) * fma(-0.16666666666666666, (y * y), 1.0);
	} else {
		tmp = (x / (z_m * y)) * sin(y);
	}
	return z_s * tmp;
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	tmp = 0.0
	if (y <= 0.000216)
		tmp = Float64(Float64(x / z_m) * fma(-0.16666666666666666, Float64(y * y), 1.0));
	else
		tmp = Float64(Float64(x / Float64(z_m * y)) * sin(y));
	end
	return Float64(z_s * tmp)
end

Wolfram

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_, y_, z$95$m_] := N[(z$95$s * If[LessEqual[y, 0.000216], N[(N[(x / z$95$m), $MachinePrecision] * N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(z$95$m * y), $MachinePrecision]), $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 2.1599999999999999e-4

   1. Initial program 96.0%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. mult-flip-rev N/A

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

      8. lower-/.f64 95.8

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

   3. Applied rewrites 95.8%

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

   4. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 55.0

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

   6. Applied rewrites 55.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 55.0

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 55.0

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

   8. Applied rewrites 55.0%

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

   if 2.1599999999999999e-4 < y

   1. Initial program 96.0%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. mult-flip N/A

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

      7. associate-*l* N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. mult-flip-rev N/A

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

      12. *-commutative N/A

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

      13. mult-flip-rev N/A

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

      14. associate-/l/ N/A

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

      15. lower-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 84.2

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

   3. Applied rewrites 84.2%

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

Alternative 6: 66.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{x \cdot \frac{\sin y}{y}}{z\_m} \leq -1 \cdot 10^{-68}:\\ \;\;\;\;\frac{x}{z\_m} \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z\_m + 0.16666666666666666 \cdot \left({y}^{2} \cdot z\_m\right)}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m)
 :precision binary64
 (*
  z_s
  (if (<= (/ (* x (/ (sin y) y)) z_m) -1e-68)
    (* (/ x z_m) (fma -0.16666666666666666 (* y y) 1.0))
    (/ x (+ z_m (* 0.16666666666666666 (* (pow y 2.0) z_m)))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (((x * (sin(y) / y)) / z_m) <= -1e-68) {
		tmp = (x / z_m) * fma(-0.16666666666666666, (y * y), 1.0);
	} else {
		tmp = x / (z_m + (0.16666666666666666 * (pow(y, 2.0) * z_m)));
	}
	return z_s * tmp;
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	tmp = 0.0
	if (Float64(Float64(x * Float64(sin(y) / y)) / z_m) <= -1e-68)
		tmp = Float64(Float64(x / z_m) * fma(-0.16666666666666666, Float64(y * y), 1.0));
	else
		tmp = Float64(x / Float64(z_m + Float64(0.16666666666666666 * Float64((y ^ 2.0) * z_m))));
	end
	return Float64(z_s * tmp)
end

Wolfram

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_, y_, z$95$m_] := N[(z$95$s * If[LessEqual[N[(N[(x * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision], -1e-68], N[(N[(x / z$95$m), $MachinePrecision] * N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(x / N[(z$95$m + N[(0.16666666666666666 * N[(N[Power[y, 2.0], $MachinePrecision] * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z) < -1.00000000000000007e-68

   1. Initial program 96.0%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. mult-flip-rev N/A

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

      8. lower-/.f64 95.8

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

   3. Applied rewrites 95.8%

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

   4. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 55.0

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

   6. Applied rewrites 55.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 55.0

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 55.0

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

   8. Applied rewrites 55.0%

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

   if -1.00000000000000007e-68 < (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z)

   1. Initial program 96.0%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. mult-flip-rev N/A

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

      8. lower-/.f64 95.8

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

   3. Applied rewrites 95.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l/ N/A

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

      7. div-flip-rev N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. div-flip-rev N/A

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

      11. associate-/l/ N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 95.8

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

   5. Applied rewrites 95.8%

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

   6. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 66.2

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

   8. Applied rewrites 66.2%

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

Alternative 7: 60.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_0 := \frac{x \cdot \frac{\sin y}{y}}{z\_m}\\ z\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-123}:\\ \;\;\;\;\frac{x}{z\_m} \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-309}:\\ \;\;\;\;\frac{\left(\frac{1}{z\_m} - z\_m\right) + \left(\frac{1}{z\_m} + z\_m\right)}{2} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m)
 :precision binary64
 (let* ((t_0 (/ (* x (/ (sin y) y)) z_m)))
   (*
    z_s
    (if (<= t_0 -2e-123)
      (* (/ x z_m) (fma -0.16666666666666666 (* y y) 1.0))
      (if (<= t_0 5e-309)
        (* (/ (+ (- (/ 1.0 z_m) z_m) (+ (/ 1.0 z_m) z_m)) 2.0) x)
        (/ x z_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double t_0 = (x * (sin(y) / y)) / z_m;
	double tmp;
	if (t_0 <= -2e-123) {
		tmp = (x / z_m) * fma(-0.16666666666666666, (y * y), 1.0);
	} else if (t_0 <= 5e-309) {
		tmp = ((((1.0 / z_m) - z_m) + ((1.0 / z_m) + z_m)) / 2.0) * x;
	} else {
		tmp = x / z_m;
	}
	return z_s * tmp;
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	t_0 = Float64(Float64(x * Float64(sin(y) / y)) / z_m)
	tmp = 0.0
	if (t_0 <= -2e-123)
		tmp = Float64(Float64(x / z_m) * fma(-0.16666666666666666, Float64(y * y), 1.0));
	elseif (t_0 <= 5e-309)
		tmp = Float64(Float64(Float64(Float64(Float64(1.0 / z_m) - z_m) + Float64(Float64(1.0 / z_m) + z_m)) / 2.0) * x);
	else
		tmp = Float64(x / z_m);
	end
	return Float64(z_s * tmp)
end

Wolfram

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_, y_, z$95$m_] := Block[{t$95$0 = N[(N[(x * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision]}, N[(z$95$s * If[LessEqual[t$95$0, -2e-123], N[(N[(x / z$95$m), $MachinePrecision] * N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e-309], N[(N[(N[(N[(N[(1.0 / z$95$m), $MachinePrecision] - z$95$m), $MachinePrecision] + N[(N[(1.0 / z$95$m), $MachinePrecision] + z$95$m), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] * x), $MachinePrecision], N[(x / z$95$m), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z) < -2.0000000000000001e-123

   1. Initial program 96.0%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. mult-flip-rev N/A

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

      8. lower-/.f64 95.8

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

   3. Applied rewrites 95.8%

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

   4. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 55.0

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

   6. Applied rewrites 55.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 55.0

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 55.0

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

   8. Applied rewrites 55.0%

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

   if -2.0000000000000001e-123 < (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z) < 4.9999999999999995e-309

   1. Initial program 96.0%

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

   2. Taylor expanded in y around 0

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

      1. lower-/.f64 58.3

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

   4. Applied rewrites 58.3%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 58.2

         \[\leadsto \frac{1}{z} \cdot x \]

   6. Applied rewrites 58.2%

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

      1. lift-/.f64 N/A

         \[\leadsto \frac{1}{z} \cdot x \]

      2. inv-pow N/A

         \[\leadsto {z}^{-1} \cdot x \]

      3. pow-to-exp N/A

         \[\leadsto e^{\log z \cdot -1} \cdot x \]

      4. lower-unsound-exp.f64 N/A

         \[\leadsto e^{\log z \cdot -1} \cdot x \]

      5. lower-unsound-*.f64 N/A

         \[\leadsto e^{\log z \cdot -1} \cdot x \]

      6. lower-unsound-log.f64 55.1

         \[\leadsto e^{\log z \cdot -1} \cdot x \]

   8. Applied rewrites 55.1%

      \[\leadsto e^{\log z \cdot -1} \cdot x \]
   9. Step-by-step derivation

      1. lift-exp.f64 N/A

         \[\leadsto e^{\log z \cdot -1} \cdot x \]

      2. sinh-+-cosh-rev N/A

         \[\leadsto \left(\cosh \left(\log z \cdot -1\right) + \sinh \left(\log z \cdot -1\right)\right) \cdot x \]

      3. +-commutative N/A

         \[\leadsto \left(\sinh \left(\log z \cdot -1\right) + \cosh \left(\log z \cdot -1\right)\right) \cdot x \]

      4. sinh-def N/A

         \[\leadsto \left(\frac{e^{\log z \cdot -1} - e^{\mathsf{neg}\left(\log z \cdot -1\right)}}{2} + \cosh \left(\log z \cdot -1\right)\right) \cdot x \]

      5. cosh-def N/A

         \[\leadsto \left(\frac{e^{\log z \cdot -1} - e^{\mathsf{neg}\left(\log z \cdot -1\right)}}{2} + \frac{e^{\log z \cdot -1} + e^{\mathsf{neg}\left(\log z \cdot -1\right)}}{2}\right) \cdot x \]

      6. div-add-rev N/A

         \[\leadsto \frac{\left(e^{\log z \cdot -1} - e^{\mathsf{neg}\left(\log z \cdot -1\right)}\right) + \left(e^{\log z \cdot -1} + e^{\mathsf{neg}\left(\log z \cdot -1\right)}\right)}{2} \cdot x \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\left(e^{\log z \cdot -1} - e^{\mathsf{neg}\left(\log z \cdot -1\right)}\right) + \left(e^{\log z \cdot -1} + e^{\mathsf{neg}\left(\log z \cdot -1\right)}\right)}{2} \cdot x \]

   10. Applied rewrites 45.8%

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

   if 4.9999999999999995e-309 < (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z)

   1. Initial program 96.0%

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

   2. Taylor expanded in y around 0

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

      1. lower-/.f64 58.3

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

   4. Applied rewrites 58.3%

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

Alternative 8: 58.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{x \cdot \frac{\sin y}{y}}{z\_m} \leq -4 \cdot 10^{-226}:\\ \;\;\;\;\frac{x}{z\_m} \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{x}{z\_m}}}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m)
 :precision binary64
 (*
  z_s
  (if (<= (/ (* x (/ (sin y) y)) z_m) -4e-226)
    (* (/ x z_m) (fma -0.16666666666666666 (* y y) 1.0))
    (/ 1.0 (/ 1.0 (/ x z_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (((x * (sin(y) / y)) / z_m) <= -4e-226) {
		tmp = (x / z_m) * fma(-0.16666666666666666, (y * y), 1.0);
	} else {
		tmp = 1.0 / (1.0 / (x / z_m));
	}
	return z_s * tmp;
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	tmp = 0.0
	if (Float64(Float64(x * Float64(sin(y) / y)) / z_m) <= -4e-226)
		tmp = Float64(Float64(x / z_m) * fma(-0.16666666666666666, Float64(y * y), 1.0));
	else
		tmp = Float64(1.0 / Float64(1.0 / Float64(x / z_m)));
	end
	return Float64(z_s * tmp)
end

Wolfram

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_, y_, z$95$m_] := N[(z$95$s * If[LessEqual[N[(N[(x * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision], -4e-226], N[(N[(x / z$95$m), $MachinePrecision] * N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 / N[(x / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z) < -3.99999999999999969e-226

   1. Initial program 96.0%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. mult-flip-rev N/A

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

      8. lower-/.f64 95.8

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

   3. Applied rewrites 95.8%

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

   4. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 55.0

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

   6. Applied rewrites 55.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 55.0

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 55.0

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

   8. Applied rewrites 55.0%

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

   if -3.99999999999999969e-226 < (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z)

   1. Initial program 96.0%

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

   2. Taylor expanded in y around 0

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

      1. lower-/.f64 58.3

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

   4. Applied rewrites 58.3%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. associate-*r/ N/A

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

      4. div-flip N/A

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

      5. *-rgt-identity N/A

         \[\leadsto \frac{1}{\frac{z}{x}} \]

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

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

      7. lower-/.f64 N/A

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

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

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

      9. lower-/.f64 58.2

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

   6. Applied rewrites 58.2%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. div-flip N/A

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

      4. lower-unsound-/.f32 N/A

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

      5. lower-/.f32 N/A

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

      6. frac-2neg N/A

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

      7. div-flip-rev N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lower-unsound-/.f64 58.2

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

      11. lift-/.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. div-flip-rev N/A

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

      14. lower-/.f64 58.3

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

   8. Applied rewrites 58.3%

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

Alternative 9: 58.3% accurate, 3.8× speedup

Math

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

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m) :precision binary64 (* z_s (/ 1.0 (/ 1.0 (/ x z_m)))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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_, y_, z$95$m_] := N[(z$95$s * N[(1.0 / N[(1.0 / N[(x / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.0%

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

2. Taylor expanded in y around 0

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

   1. lower-/.f64 58.3

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

4. Applied rewrites 58.3%

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

   1. lift-/.f64 N/A

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

   2. mult-flip N/A

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

   3. associate-*r/ N/A

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

   4. div-flip N/A

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

   5. *-rgt-identity N/A

      \[\leadsto \frac{1}{\frac{z}{x}} \]

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

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

   7. lower-/.f64 N/A

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

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

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

   9. lower-/.f64 58.2

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

6. Applied rewrites 58.2%

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

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. div-flip N/A

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

   4. lower-unsound-/.f32 N/A

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

   5. lower-/.f32 N/A

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

   6. frac-2neg N/A

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

   7. div-flip-rev N/A

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

   8. lift-/.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. lower-unsound-/.f64 58.2

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

   11. lift-/.f64 N/A

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

   12. lift-/.f64 N/A

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

   13. div-flip-rev N/A

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

   14. lower-/.f64 58.3

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

8. Applied rewrites 58.3%

   \[\leadsto \frac{1}{\frac{1}{\color{blue}{\frac{x}{z}}}} \]
9. Add Preprocessing

Alternative 10: 58.3% accurate, 5.5× speedup

Math

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

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m) :precision binary64 (* z_s (/ 1.0 (/ z_m x))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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_, y_, z$95$m_] := N[(z$95$s * N[(1.0 / N[(z$95$m / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.0%

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

2. Taylor expanded in y around 0

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

   1. lower-/.f64 58.3

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

4. Applied rewrites 58.3%

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

   1. lift-/.f64 N/A

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

   2. mult-flip N/A

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

   3. associate-*r/ N/A

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

   4. div-flip N/A

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

   5. *-rgt-identity N/A

      \[\leadsto \frac{1}{\frac{z}{x}} \]

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

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

   7. lower-/.f64 N/A

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

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

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

   9. lower-/.f64 58.2

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

6. Applied rewrites 58.2%

   \[\leadsto \frac{1}{\color{blue}{\frac{z}{x}}} \]
7. Add Preprocessing

Alternative 11: 58.2% accurate, 9.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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_, y_, z$95$m_] := N[(z$95$s * N[(x / z$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.0%

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

2. Taylor expanded in y around 0

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

   1. lower-/.f64 58.3

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

4. Applied rewrites 58.3%

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

Reproduce

herbie shell --seed 2025155 
(FPCore (x y z)
  :name "Linear.Quaternion:$ctanh from linear-1.19.1.3"
  :precision binary64
  (/ (* x (/ (sin y) y)) z))