Linear.Quaternion:$ctanh from linear-1.19.1.3

Percentage Accurate: 96.2% → 99.6%

Time: 4.1s

Alternatives: 10

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 96.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.55 \cdot 10^{-45}:\\ \;\;\;\;\frac{\frac{\sin y}{z\_m}}{y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{\sin y}{y}}{z\_m}\\ \end{array} \end{array} \]

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 1.55e-45)
    (* (/ (/ (sin y) z_m) y) x)
    (/ (* 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 <= 1.55e-45) {
		tmp = ((sin(y) / z_m) / y) * x;
	} 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 <= 1.55d-45) then
        tmp = ((sin(y) / z_m) / y) * x
    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 <= 1.55e-45) {
		tmp = ((Math.sin(y) / z_m) / y) * x;
	} 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 <= 1.55e-45:
		tmp = ((math.sin(y) / z_m) / y) * x
	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 <= 1.55e-45)
		tmp = Float64(Float64(Float64(sin(y) / z_m) / y) * x);
	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 <= 1.55e-45)
		tmp = ((sin(y) / z_m) / y) * x;
	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, 1.55e-45], N[(N[(N[(N[Sin[y], $MachinePrecision] / z$95$m), $MachinePrecision] / y), $MachinePrecision] * x), $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 < 1.55e-45

   1. Initial program 94.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. distribute-frac-neg2 N/A

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

      6. distribute-rgt-neg-out N/A

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

      7. distribute-frac-neg N/A

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

      8. associate-/l* N/A

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

      9. distribute-lft-neg-in N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. distribute-frac-neg N/A

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

      13. distribute-frac-neg2 N/A

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

      14. associate-/r* N/A

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

      15. *-commutative N/A

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

      16. associate-/r* N/A

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

      17. lower-/.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. lower-neg.f64 N/A

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

      20. lower-neg.f64 90.8

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

   4. Applied rewrites 90.8%

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

   if 1.55e-45 < z

   1. Initial program 99.8%

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

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.55 \cdot 10^{-45}:\\ \;\;\;\;\frac{\frac{\sin y}{z}}{y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{\sin y}{y}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 47.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_0 := \frac{x \cdot \frac{\sin y}{y}}{z\_m}\\ z\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-175}:\\ \;\;\;\;\left(\frac{y \cdot y}{z\_m} \cdot -0.16666666666666666\right) \cdot x\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-233}:\\ \;\;\;\;y \cdot \frac{x}{z\_m \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z\_m}\\ \end{array} \end{array} \end{array} \]

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 -5e-175)
      (* (* (/ (* y y) z_m) -0.16666666666666666) x)
      (if (<= t_0 2e-233) (* y (/ x (* z_m 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 t_0 = (x * (sin(y) / y)) / z_m;
	double tmp;
	if (t_0 <= -5e-175) {
		tmp = (((y * y) / z_m) * -0.16666666666666666) * x;
	} else if (t_0 <= 2e-233) {
		tmp = y * (x / (z_m * y));
	} else {
		tmp = 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) :: t_0
    real(8) :: tmp
    t_0 = (x * (sin(y) / y)) / z_m
    if (t_0 <= (-5d-175)) then
        tmp = (((y * y) / z_m) * (-0.16666666666666666d0)) * x
    else if (t_0 <= 2d-233) then
        tmp = y * (x / (z_m * y))
    else
        tmp = 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 t_0 = (x * (Math.sin(y) / y)) / z_m;
	double tmp;
	if (t_0 <= -5e-175) {
		tmp = (((y * y) / z_m) * -0.16666666666666666) * x;
	} else if (t_0 <= 2e-233) {
		tmp = y * (x / (z_m * y));
	} else {
		tmp = 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):
	t_0 = (x * (math.sin(y) / y)) / z_m
	tmp = 0
	if t_0 <= -5e-175:
		tmp = (((y * y) / z_m) * -0.16666666666666666) * x
	elif t_0 <= 2e-233:
		tmp = y * (x / (z_m * y))
	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 <= -5e-175)
		tmp = Float64(Float64(Float64(Float64(y * y) / z_m) * -0.16666666666666666) * x);
	elseif (t_0 <= 2e-233)
		tmp = Float64(y * Float64(x / Float64(z_m * y)));
	else
		tmp = 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)
	t_0 = (x * (sin(y) / y)) / z_m;
	tmp = 0.0;
	if (t_0 <= -5e-175)
		tmp = (((y * y) / z_m) * -0.16666666666666666) * x;
	elseif (t_0 <= 2e-233)
		tmp = y * (x / (z_m * y));
	else
		tmp = 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_] := 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, -5e-175], N[(N[(N[(N[(y * y), $MachinePrecision] / z$95$m), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t$95$0, 2e-233], N[(y * N[(x / N[(z$95$m * y), $MachinePrecision]), $MachinePrecision]), $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) < -5e-175

   1. Initial program 98.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 97.0

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

   4. Applied rewrites 97.0%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r/ N/A

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

      4. associate-/r* N/A

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

      5. times-frac N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. associate-/r* N/A

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

      10. associate-*r/ N/A

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

      11. associate-*l/ N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. associate-*r* N/A

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

      15. *-commutative N/A

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

      16. distribute-lft1-in N/A

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

      17. +-commutative N/A

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

      18. lower-*.f64 N/A

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

   7. Applied rewrites 53.5%

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

   8. Taylor expanded in y around inf

      \[\leadsto \frac{-1}{6} \cdot \color{blue}{\frac{x \cdot {y}^{2}}{z}} \]
   9. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 3.9

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

   10. Applied rewrites 3.9%

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

   if -5e-175 < (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z) < 1.99999999999999992e-233

   1. Initial program 93.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 95.3

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

   4. Applied rewrites 95.3%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. remove-double-neg N/A

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

      7. lower-/.f64 N/A

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

      8. remove-double-neg 94.4

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

   6. Applied rewrites 94.4%

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

   7. Taylor expanded in y around 0

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

   9. Applied rewrites 66.7%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-*l/ N/A

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

       4. associate-/l* N/A

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

       5. lower-*.f64 N/A

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

       6. lower-/.f64 82.4

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

   11. Applied rewrites 82.4%

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

   if 1.99999999999999992e-233 < (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z)

   1. Initial program 99.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 55.3%

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

Alternative 3: 91.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ z\_s \cdot \begin{array}{l} \mathbf{if}\;x \cdot t\_0 \leq -2 \cdot 10^{-291}:\\ \;\;\;\;\frac{\sin y}{z\_m} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{z\_m} \cdot x\\ \end{array} \end{array} \end{array} \]

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 (/ (sin y) y)))
   (*
    z_s
    (if (<= (* x t_0) -2e-291)
      (* (/ (sin y) z_m) (/ x y))
      (* (/ t_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) {
	double t_0 = sin(y) / y;
	double tmp;
	if ((x * t_0) <= -2e-291) {
		tmp = (sin(y) / z_m) * (x / y);
	} else {
		tmp = (t_0 / z_m) * x;
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = sin(y) / y
    if ((x * t_0) <= (-2d-291)) then
        tmp = (sin(y) / z_m) * (x / y)
    else
        tmp = (t_0 / z_m) * x
    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 t_0 = Math.sin(y) / y;
	double tmp;
	if ((x * t_0) <= -2e-291) {
		tmp = (Math.sin(y) / z_m) * (x / y);
	} else {
		tmp = (t_0 / z_m) * x;
	}
	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):
	t_0 = math.sin(y) / y
	tmp = 0
	if (x * t_0) <= -2e-291:
		tmp = (math.sin(y) / z_m) * (x / y)
	else:
		tmp = (t_0 / z_m) * x
	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(sin(y) / y)
	tmp = 0.0
	if (Float64(x * t_0) <= -2e-291)
		tmp = Float64(Float64(sin(y) / z_m) * Float64(x / y));
	else
		tmp = Float64(Float64(t_0 / z_m) * x);
	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)
	t_0 = sin(y) / y;
	tmp = 0.0;
	if ((x * t_0) <= -2e-291)
		tmp = (sin(y) / z_m) * (x / y);
	else
		tmp = (t_0 / z_m) * x;
	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_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, N[(z$95$s * If[LessEqual[N[(x * t$95$0), $MachinePrecision], -2e-291], N[(N[(N[Sin[y], $MachinePrecision] / z$95$m), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 / z$95$m), $MachinePrecision] * x), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x (/.f64 (sin.f64 y) y)) < -1.99999999999999992e-291

   1. Initial program 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 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 92.4

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

   4. Applied rewrites 92.4%

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

   if -1.99999999999999992e-291 < (*.f64 x (/.f64 (sin.f64 y) y))

   1. Initial program 94.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 97.8

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

   4. Applied rewrites 97.8%

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

Alternative 4: 96.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 0.9999999999998048:\\ \;\;\;\;\frac{\sin y}{z\_m} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z\_m}\\ \end{array} \end{array} \]

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 (<= (/ (sin y) y) 0.9999999999998048)
    (* (/ (sin y) z_m) (/ x 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 ((sin(y) / y) <= 0.9999999999998048) {
		tmp = (sin(y) / z_m) * (x / y);
	} else {
		tmp = 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 ((sin(y) / y) <= 0.9999999999998048d0) then
        tmp = (sin(y) / z_m) * (x / y)
    else
        tmp = 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 ((Math.sin(y) / y) <= 0.9999999999998048) {
		tmp = (Math.sin(y) / z_m) * (x / y);
	} else {
		tmp = 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 (math.sin(y) / y) <= 0.9999999999998048:
		tmp = (math.sin(y) / z_m) * (x / y)
	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)
	tmp = 0.0
	if (Float64(sin(y) / y) <= 0.9999999999998048)
		tmp = Float64(Float64(sin(y) / z_m) * Float64(x / y));
	else
		tmp = 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 ((sin(y) / y) <= 0.9999999999998048)
		tmp = (sin(y) / z_m) * (x / y);
	else
		tmp = 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[N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], 0.9999999999998048], N[(N[(N[Sin[y], $MachinePrecision] / z$95$m), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision], N[(x / z$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 y) y) < 0.99999999999980482

   1. Initial program 93.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 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 93.7

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

   4. Applied rewrites 93.7%

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

   if 0.99999999999980482 < (/.f64 (sin.f64 y) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 100.0%

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

Alternative 5: 95.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 0.9999999999998048:\\ \;\;\;\;\frac{\sin y}{z\_m \cdot y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z\_m}\\ \end{array} \end{array} \]

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 (<= (/ (sin y) y) 0.9999999999998048)
    (* (/ (sin y) (* z_m y)) 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 tmp;
	if ((sin(y) / y) <= 0.9999999999998048) {
		tmp = (sin(y) / (z_m * y)) * x;
	} else {
		tmp = 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 ((sin(y) / y) <= 0.9999999999998048d0) then
        tmp = (sin(y) / (z_m * y)) * x
    else
        tmp = 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 ((Math.sin(y) / y) <= 0.9999999999998048) {
		tmp = (Math.sin(y) / (z_m * y)) * x;
	} else {
		tmp = 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 (math.sin(y) / y) <= 0.9999999999998048:
		tmp = (math.sin(y) / (z_m * y)) * 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)
	tmp = 0.0
	if (Float64(sin(y) / y) <= 0.9999999999998048)
		tmp = Float64(Float64(sin(y) / Float64(z_m * y)) * x);
	else
		tmp = 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 ((sin(y) / y) <= 0.9999999999998048)
		tmp = (sin(y) / (z_m * y)) * x;
	else
		tmp = 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[N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], 0.9999999999998048], N[(N[(N[Sin[y], $MachinePrecision] / N[(z$95$m * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], N[(x / z$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 y) y) < 0.99999999999980482

   1. Initial program 93.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 92.5

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

   4. Applied rewrites 92.5%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. remove-double-neg N/A

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

      7. lower-/.f64 N/A

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

      8. remove-double-neg 91.8

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

   6. Applied rewrites 91.8%

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

   if 0.99999999999980482 < (/.f64 (sin.f64 y) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 100.0%

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

Alternative 6: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.55 \cdot 10^{-45}:\\ \;\;\;\;\frac{t\_0}{z\_m} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot t\_0}{z\_m}\\ \end{array} \end{array} \end{array} \]

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 (/ (sin y) y)))
   (* z_s (if (<= z_m 1.55e-45) (* (/ t_0 z_m) x) (/ (* x t_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 t_0 = sin(y) / y;
	double tmp;
	if (z_m <= 1.55e-45) {
		tmp = (t_0 / z_m) * x;
	} else {
		tmp = (x * t_0) / 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) :: t_0
    real(8) :: tmp
    t_0 = sin(y) / y
    if (z_m <= 1.55d-45) then
        tmp = (t_0 / z_m) * x
    else
        tmp = (x * t_0) / 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 t_0 = Math.sin(y) / y;
	double tmp;
	if (z_m <= 1.55e-45) {
		tmp = (t_0 / z_m) * x;
	} else {
		tmp = (x * t_0) / 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):
	t_0 = math.sin(y) / y
	tmp = 0
	if z_m <= 1.55e-45:
		tmp = (t_0 / z_m) * x
	else:
		tmp = (x * t_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)
	t_0 = Float64(sin(y) / y)
	tmp = 0.0
	if (z_m <= 1.55e-45)
		tmp = Float64(Float64(t_0 / z_m) * x);
	else
		tmp = Float64(Float64(x * t_0) / 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)
	t_0 = sin(y) / y;
	tmp = 0.0;
	if (z_m <= 1.55e-45)
		tmp = (t_0 / z_m) * x;
	else
		tmp = (x * t_0) / 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_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, N[(z$95$s * If[LessEqual[z$95$m, 1.55e-45], N[(N[(t$95$0 / z$95$m), $MachinePrecision] * x), $MachinePrecision], N[(N[(x * t$95$0), $MachinePrecision] / z$95$m), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.55e-45

   1. Initial program 94.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 96.9

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

   4. Applied rewrites 96.9%

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

   if 1.55e-45 < z

   1. Initial program 99.8%

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

Alternative 7: 75.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 6.4 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666 \cdot y, y, 1\right) \cdot \frac{x}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \frac{x}{z\_m \cdot y}\\ \end{array} \end{array} \]

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 6.4e-6)
    (* (fma (* -0.16666666666666666 y) y 1.0) (/ 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 <= 6.4e-6) {
		tmp = fma((-0.16666666666666666 * y), y, 1.0) * (x / z_m);
	} else {
		tmp = 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 <= 6.4e-6)
		tmp = Float64(fma(Float64(-0.16666666666666666 * y), y, 1.0) * Float64(x / z_m));
	else
		tmp = Float64(sin(y) * Float64(x / Float64(z_m * 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, 6.4e-6], N[(N[(N[(-0.16666666666666666 * y), $MachinePrecision] * y + 1.0), $MachinePrecision] * N[(x / z$95$m), $MachinePrecision]), $MachinePrecision], N[(N[Sin[y], $MachinePrecision] * N[(x / N[(z$95$m * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 6.3999999999999997e-6

   1. Initial program 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 97.0

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

   4. Applied rewrites 97.0%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r/ N/A

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

      4. associate-/r* N/A

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

      5. times-frac N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. associate-/r* N/A

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

      10. associate-*r/ N/A

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

      11. associate-*l/ N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. associate-*r* N/A

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

      15. *-commutative N/A

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

      16. distribute-lft1-in N/A

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

      17. +-commutative N/A

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

      18. lower-*.f64 N/A

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

   7. Applied rewrites 66.9%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 66.9

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

   9. Applied rewrites 66.9%

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

   if 6.3999999999999997e-6 < y

   1. Initial program 88.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 92.5

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

   4. Applied rewrites 92.5%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. remove-double-neg N/A

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

      7. lower-/.f64 N/A

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

      8. remove-double-neg 92.2

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

   6. Applied rewrites 92.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 92.2

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

   8. Applied rewrites 92.2%

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

Alternative 8: 60.6% accurate, 3.8× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 7.5 \cdot 10^{+92}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666 \cdot y, y, 1\right) \cdot \frac{x}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z\_m \cdot y}\\ \end{array} \end{array} \]

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+92)
    (* (fma (* -0.16666666666666666 y) y 1.0) (/ x z_m))
    (* 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+92) {
		tmp = fma((-0.16666666666666666 * y), y, 1.0) * (x / z_m);
	} else {
		tmp = 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+92)
		tmp = Float64(fma(Float64(-0.16666666666666666 * y), y, 1.0) * Float64(x / z_m));
	else
		tmp = Float64(y * Float64(x / Float64(z_m * 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, 7.5e+92], N[(N[(N[(-0.16666666666666666 * y), $MachinePrecision] * y + 1.0), $MachinePrecision] * N[(x / z$95$m), $MachinePrecision]), $MachinePrecision], N[(y * N[(x / N[(z$95$m * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 7.49999999999999946e92

   1. Initial program 99.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 97.3

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

   4. Applied rewrites 97.3%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r/ N/A

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

      4. associate-/r* N/A

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

      5. times-frac N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. associate-/r* N/A

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

      10. associate-*r/ N/A

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

      11. associate-*l/ N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. associate-*r* N/A

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

      15. *-commutative N/A

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

      16. distribute-lft1-in N/A

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

      17. +-commutative N/A

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

      18. lower-*.f64 N/A

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

   7. Applied rewrites 62.6%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 62.6

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

   9. Applied rewrites 62.6%

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

   if 7.49999999999999946e92 < y

   1. Initial program 84.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 88.9

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

   4. Applied rewrites 88.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. remove-double-neg N/A

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

      7. lower-/.f64 N/A

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

      8. remove-double-neg 88.7

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

   6. Applied rewrites 88.7%

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

   7. Taylor expanded in y around 0

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

   9. Applied rewrites 30.4%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-*l/ N/A

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

       4. associate-/l* N/A

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

       5. lower-*.f64 N/A

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

       6. lower-/.f64 44.9

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

   11. Applied rewrites 44.9%

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

Alternative 9: 62.5% accurate, 4.6× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 1.5 \cdot 10^{+41}:\\ \;\;\;\;\frac{x}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z\_m \cdot y}\\ \end{array} \end{array} \]

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 1.5e+41) (/ x z_m) (* 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 <= 1.5e+41) {
		tmp = x / z_m;
	} else {
		tmp = 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 <= 1.5d+41) then
        tmp = x / z_m
    else
        tmp = 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 <= 1.5e+41) {
		tmp = x / z_m;
	} else {
		tmp = 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 <= 1.5e+41:
		tmp = x / z_m
	else:
		tmp = 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 <= 1.5e+41)
		tmp = Float64(x / z_m);
	else
		tmp = Float64(y * Float64(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 <= 1.5e+41)
		tmp = x / z_m;
	else
		tmp = 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, 1.5e+41], N[(x / z$95$m), $MachinePrecision], N[(y * N[(x / N[(z$95$m * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 1.4999999999999999e41

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 64.8%

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

   if 1.4999999999999999e41 < y

   1. Initial program 85.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 91.0

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

   4. Applied rewrites 91.0%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. remove-double-neg N/A

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

      7. lower-/.f64 N/A

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

      8. remove-double-neg 90.7

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

   6. Applied rewrites 90.7%

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

   7. Taylor expanded in y around 0

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

   9. Applied rewrites 28.4%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-*l/ N/A

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

       4. associate-/l* N/A

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

       5. lower-*.f64 N/A

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

       6. lower-/.f64 40.2

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

   11. Applied rewrites 40.2%

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

Alternative 10: 58.9% accurate, 10.7× 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 96.4%

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

3. Taylor expanded in y around 0

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

5. Applied rewrites 53.6%

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

Developer Target 1: 99.5% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2025026 
(FPCore (x y z)
  :name "Linear.Quaternion:$ctanh from linear-1.19.1.3"
  :precision binary64

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

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