Linear.Quaternion:$ctanh from linear-1.19.1.3

Percentage Accurate: 96.0% → 97.4%

Time: 4.0s

Alternatives: 9

Speedup: 0.4×

Specification

Math

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

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

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.4% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (x y z)
  :precision binary64
  (let* ((t_0 (/ (* (fabs x) (/ (sin y) y)) (fabs z))))
  (*
   (copysign 1.0 x)
   (*
    (copysign 1.0 z)
    (if (<= t_0 0.0) (* (/ (fabs x) (* (fabs z) y)) (sin y)) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = (fabs(x) * (sin(y) / y)) / fabs(z);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = (fabs(x) / (fabs(z) * y)) * sin(y);
	} else {
		tmp = t_0;
	}
	return copysign(1.0, x) * (copysign(1.0, z) * tmp);
}

Java

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

Python

def code(x, y, z):
	t_0 = (math.fabs(x) * (math.sin(y) / y)) / math.fabs(z)
	tmp = 0
	if t_0 <= 0.0:
		tmp = (math.fabs(x) / (math.fabs(z) * y)) * math.sin(y)
	else:
		tmp = t_0
	return math.copysign(1.0, x) * (math.copysign(1.0, z) * tmp)

Julia

function code(x, y, z)
	t_0 = Float64(Float64(abs(x) * Float64(sin(y) / y)) / abs(z))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(Float64(abs(x) / Float64(abs(z) * y)) * sin(y));
	else
		tmp = t_0;
	end
	return Float64(copysign(1.0, x) * Float64(copysign(1.0, z) * tmp))
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (abs(x) * (sin(y) / y)) / abs(z);
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = (abs(x) / (abs(z) * y)) * sin(y);
	else
		tmp = t_0;
	end
	tmp_2 = (sign(x) * abs(1.0)) * ((sign(z) * abs(1.0)) * tmp);
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[Abs[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / N[Abs[z], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$0, 0.0], N[(N[(N[Abs[x], $MachinePrecision] / N[(N[Abs[z], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision], t$95$0]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 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 83.9%

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

   3. Applied rewrites 83.9%

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

   if -0.0 < (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) 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: 95.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \frac{\sin y}{y}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-295}:\\ \;\;\;\;\frac{x}{z \cdot y} \cdot \sin y\\ \mathbf{elif}\;t\_0 \leq 0.99999999998:\\ \;\;\;\;\frac{\sin y}{z} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  (let* ((t_0 (/ (sin y) y)))
  (if (<= t_0 -2e-295)
    (* (/ x (* z y)) (sin y))
    (if (<= t_0 0.99999999998) (* (/ (sin y) z) (/ x y)) (/ x z)))))

C

double code(double x, double y, double z) {
	double t_0 = sin(y) / y;
	double tmp;
	if (t_0 <= -2e-295) {
		tmp = (x / (z * y)) * sin(y);
	} else if (t_0 <= 0.99999999998) {
		tmp = (sin(y) / z) * (x / y);
	} else {
		tmp = x / z;
	}
	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) :: tmp
    t_0 = sin(y) / y
    if (t_0 <= (-2d-295)) then
        tmp = (x / (z * y)) * sin(y)
    else if (t_0 <= 0.99999999998d0) then
        tmp = (sin(y) / z) * (x / y)
    else
        tmp = x / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.sin(y) / y;
	double tmp;
	if (t_0 <= -2e-295) {
		tmp = (x / (z * y)) * Math.sin(y);
	} else if (t_0 <= 0.99999999998) {
		tmp = (Math.sin(y) / z) * (x / y);
	} else {
		tmp = x / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.sin(y) / y
	tmp = 0
	if t_0 <= -2e-295:
		tmp = (x / (z * y)) * math.sin(y)
	elif t_0 <= 0.99999999998:
		tmp = (math.sin(y) / z) * (x / y)
	else:
		tmp = x / z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(sin(y) / y)
	tmp = 0.0
	if (t_0 <= -2e-295)
		tmp = Float64(Float64(x / Float64(z * y)) * sin(y));
	elseif (t_0 <= 0.99999999998)
		tmp = Float64(Float64(sin(y) / z) * Float64(x / y));
	else
		tmp = Float64(x / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = sin(y) / y;
	tmp = 0.0;
	if (t_0 <= -2e-295)
		tmp = (x / (z * y)) * sin(y);
	elseif (t_0 <= 0.99999999998)
		tmp = (sin(y) / z) * (x / y);
	else
		tmp = x / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-295], N[(N[(x / N[(z * y), $MachinePrecision]), $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.99999999998], N[(N[(N[Sin[y], $MachinePrecision] / z), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision], N[(x / z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 y) y) < -2.0000000000000001e-295

   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 83.9%

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

   3. Applied rewrites 83.9%

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

   if -2.0000000000000001e-295 < (/.f64 (sin.f64 y) y) < 0.99999999998

   1. Initial program 96.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. mult-flip-rev N/A

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

      10. lower-/.f64 88.2%

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

   3. Applied rewrites 88.2%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. lift-sin.f64 N/A

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

      8. mult-flip N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. associate-/l/ N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. lower-/.f64 88.0%

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

   5. Applied rewrites 88.0%

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

      1. lift-*.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 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. lift-sin.f64 N/A

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

      10. lower-/.f64 84.7%

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

   7. Applied rewrites 84.7%

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

   if 0.99999999998 < (/.f64 (sin.f64 y) y)

   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 57.4%

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

   4. Applied rewrites 57.4%

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

Alternative 3: 95.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|y\right| \leq 1.3 \cdot 10^{-5}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z \cdot \left|y\right|} \cdot \sin \left(\left|y\right|\right)\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  (if (<= (fabs y) 1.3e-5)
  (/ x z)
  (* (/ x (* z (fabs y))) (sin (fabs y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (fabs(y) <= 1.3e-5) {
		tmp = x / z;
	} else {
		tmp = (x / (z * fabs(y))) * sin(fabs(y));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (abs(y) <= 1.3d-5) then
        tmp = x / z
    else
        tmp = (x / (z * abs(y))) * sin(abs(y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (Math.abs(y) <= 1.3e-5) {
		tmp = x / z;
	} else {
		tmp = (x / (z * Math.abs(y))) * Math.sin(Math.abs(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if math.fabs(y) <= 1.3e-5:
		tmp = x / z
	else:
		tmp = (x / (z * math.fabs(y))) * math.sin(math.fabs(y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (abs(y) <= 1.3e-5)
		tmp = Float64(x / z);
	else
		tmp = Float64(Float64(x / Float64(z * abs(y))) * sin(abs(y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (abs(y) <= 1.3e-5)
		tmp = x / z;
	else
		tmp = (x / (z * abs(y))) * sin(abs(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[Abs[y], $MachinePrecision], 1.3e-5], N[(x / z), $MachinePrecision], N[(N[(x / N[(z * N[Abs[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[N[Abs[y], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.2999999999999999e-5

   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 57.4%

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

   4. Applied rewrites 57.4%

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

   if 1.2999999999999999e-5 < 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 83.9%

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

   3. Applied rewrites 83.9%

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

Alternative 4: 64.9% accurate, 1.2× speedup

Math

\[\frac{1}{\mathsf{fma}\left(0.16666666666666666, \frac{{y}^{2} \cdot z}{x}, \frac{z}{x}\right)} \]

FPCore

(FPCore (x y z)
  :precision binary64
  (/ 1.0 (fma 0.16666666666666666 (/ (* (pow y 2.0) z) x) (/ z x))))

C

double code(double x, double y, double z) {
	return 1.0 / fma(0.16666666666666666, ((pow(y, 2.0) * z) / x), (z / x));
}

Julia

function code(x, y, z)
	return Float64(1.0 / fma(0.16666666666666666, Float64(Float64((y ^ 2.0) * z) / x), Float64(z / x)))
end

Wolfram

code[x_, y_, z_] := N[(1.0 / N[(0.16666666666666666 * N[(N[(N[Power[y, 2.0], $MachinePrecision] * z), $MachinePrecision] / x), $MachinePrecision] + N[(z / 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

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

   1. lower-+.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-pow.f64 52.0%

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

4. Applied rewrites 52.0%

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

   1. lift-/.f64 N/A

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

   2. div-flip N/A

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

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

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

   4. lower-unsound-/.f64 51.7%

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 51.7%

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 51.7%

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

   13. lift-pow.f64 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f64 51.7%

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

6. Applied rewrites 51.7%

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

7. Taylor expanded in y around 0

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

   1. lower-fma.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-pow.f64 N/A

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

   5. lower-/.f64 64.9%

      \[\leadsto \frac{1}{\mathsf{fma}\left(0.16666666666666666, \frac{{y}^{2} \cdot z}{x}, \frac{z}{x}\right)} \]

9. Applied rewrites 64.9%

   \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(0.16666666666666666, \frac{{y}^{2} \cdot z}{x}, \frac{z}{x}\right)}} \]
10. Add Preprocessing

Alternative 5: 63.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|y\right| \leq 1.35:\\ \;\;\;\;\frac{x}{\frac{z}{\mathsf{fma}\left(-0.16666666666666666, \left|y\right| \cdot \left|y\right|, 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{1}{z} - z\right) + \left(\frac{1}{z} + z\right)}{2} \cdot x\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  (if (<= (fabs y) 1.35)
  (/ x (/ z (fma -0.16666666666666666 (* (fabs y) (fabs y)) 1.0)))
  (* (/ (+ (- (/ 1.0 z) z) (+ (/ 1.0 z) z)) 2.0) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (fabs(y) <= 1.35) {
		tmp = x / (z / fma(-0.16666666666666666, (fabs(y) * fabs(y)), 1.0));
	} else {
		tmp = ((((1.0 / z) - z) + ((1.0 / z) + z)) / 2.0) * x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (abs(y) <= 1.35)
		tmp = Float64(x / Float64(z / fma(-0.16666666666666666, Float64(abs(y) * abs(y)), 1.0)));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(1.0 / z) - z) + Float64(Float64(1.0 / z) + z)) / 2.0) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[Abs[y], $MachinePrecision], 1.35], N[(x / N[(z / N[(-0.16666666666666666 * N[(N[Abs[y], $MachinePrecision] * N[Abs[y], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(1.0 / z), $MachinePrecision] - z), $MachinePrecision] + N[(N[(1.0 / z), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.3500000000000001

   1. Initial program 96.0%

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

   2. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 52.0%

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

   4. Applied rewrites 52.0%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

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

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

      4. lower-unsound-/.f64 51.7%

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 51.7%

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 51.7%

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 51.7%

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

   6. Applied rewrites 51.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. div-flip-rev N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 52.0%

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

      8. lift-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. rem-exp-log N/A

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

      11. lift-log.f64 N/A

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

      12. lift-exp.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-+.f64 52.0%

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

   8. Applied rewrites 52.0%

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

   if 1.3500000000000001 < y

   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 57.4%

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

   4. Applied rewrites 57.4%

      \[\leadsto \color{blue}{\frac{x}{z}} \]
   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 57.2%

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

   6. Applied rewrites 57.2%

      \[\leadsto \frac{1}{z} \cdot \color{blue}{x} \]
   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 26.9%

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

   8. Applied rewrites 26.9%

      \[\leadsto e^{\log z \cdot -1} \cdot x \]
   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 46.2%

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

Alternative 6: 60.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|y\right| \leq 1.35:\\ \;\;\;\;\frac{x}{\frac{z}{\mathsf{fma}\left(-0.16666666666666666, \left|y\right| \cdot \left|y\right|, 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left|y\right|}{z \cdot \left|y\right|}\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  (if (<= (fabs y) 1.35)
  (/ x (/ z (fma -0.16666666666666666 (* (fabs y) (fabs y)) 1.0)))
  (/ (* x (fabs y)) (* z (fabs y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (fabs(y) <= 1.35) {
		tmp = x / (z / fma(-0.16666666666666666, (fabs(y) * fabs(y)), 1.0));
	} else {
		tmp = (x * fabs(y)) / (z * fabs(y));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (abs(y) <= 1.35)
		tmp = Float64(x / Float64(z / fma(-0.16666666666666666, Float64(abs(y) * abs(y)), 1.0)));
	else
		tmp = Float64(Float64(x * abs(y)) / Float64(z * abs(y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[Abs[y], $MachinePrecision], 1.35], N[(x / N[(z / N[(-0.16666666666666666 * N[(N[Abs[y], $MachinePrecision] * N[Abs[y], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[Abs[y], $MachinePrecision]), $MachinePrecision] / N[(z * N[Abs[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.3500000000000001

   1. Initial program 96.0%

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

   2. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 52.0%

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

   4. Applied rewrites 52.0%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

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

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

      4. lower-unsound-/.f64 51.7%

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 51.7%

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 51.7%

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 51.7%

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

   6. Applied rewrites 51.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. div-flip-rev N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 52.0%

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

      8. lift-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. rem-exp-log N/A

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

      11. lift-log.f64 N/A

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

      12. lift-exp.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-+.f64 52.0%

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

   8. Applied rewrites 52.0%

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

   if 1.3500000000000001 < 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 84.2%

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

   3. Applied rewrites 84.2%

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

   4. Taylor expanded in y around 0

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

      1. lower-*.f64 48.9%

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

   6. Applied rewrites 48.9%

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

Alternative 7: 60.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|y\right| \leq 1.35:\\ \;\;\;\;\frac{x}{z} \cdot \mathsf{fma}\left(\left|y\right| \cdot \left|y\right|, -0.16666666666666666, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left|y\right|}{z \cdot \left|y\right|}\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  (if (<= (fabs y) 1.35)
  (* (/ x z) (fma (* (fabs y) (fabs y)) -0.16666666666666666 1.0))
  (/ (* x (fabs y)) (* z (fabs y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (fabs(y) <= 1.35) {
		tmp = (x / z) * fma((fabs(y) * fabs(y)), -0.16666666666666666, 1.0);
	} else {
		tmp = (x * fabs(y)) / (z * fabs(y));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (abs(y) <= 1.35)
		tmp = Float64(Float64(x / z) * fma(Float64(abs(y) * abs(y)), -0.16666666666666666, 1.0));
	else
		tmp = Float64(Float64(x * abs(y)) / Float64(z * abs(y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[Abs[y], $MachinePrecision], 1.35], N[(N[(x / z), $MachinePrecision] * N[(N[(N[Abs[y], $MachinePrecision] * N[Abs[y], $MachinePrecision]), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[Abs[y], $MachinePrecision]), $MachinePrecision] / N[(z * N[Abs[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.3500000000000001

   1. Initial program 96.0%

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

   2. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 52.0%

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

   4. Applied rewrites 52.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lift-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 53.6%

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 53.6%

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 53.6%

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

   6. Applied rewrites 53.6%

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

   if 1.3500000000000001 < 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 84.2%

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

   3. Applied rewrites 84.2%

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

   4. Taylor expanded in y around 0

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

      1. lower-*.f64 48.9%

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

   6. Applied rewrites 48.9%

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

Alternative 8: 60.1% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (x y z)
  :precision binary64
  (*
 (copysign 1.0 x)
 (*
  (copysign 1.0 z)
  (if (<= (/ (* (fabs x) (/ (sin y) y)) (fabs z)) 0.0)
    (/ (* (fabs x) y) (* (fabs z) y))
    (/ (fabs x) (fabs z))))))

C

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

Java

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

Python

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

Julia

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

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (((abs(x) * (sin(y) / y)) / abs(z)) <= 0.0)
		tmp = (abs(x) * y) / (abs(z) * y);
	else
		tmp = abs(x) / abs(z);
	end
	tmp_2 = (sign(x) * abs(1.0)) * ((sign(z) * abs(1.0)) * tmp);
end

Wolfram

code[x_, y_, z_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(N[(N[Abs[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / N[Abs[z], $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(N[Abs[x], $MachinePrecision] * y), $MachinePrecision] / N[(N[Abs[z], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(N[Abs[x], $MachinePrecision] / N[Abs[z], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 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 84.2%

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

   3. Applied rewrites 84.2%

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

   4. Taylor expanded in y around 0

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

      1. lower-*.f64 48.9%

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

   6. Applied rewrites 48.9%

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

   if -0.0 < (/.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 57.4%

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

   4. Applied rewrites 57.4%

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

Alternative 9: 57.4% accurate, 9.4× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return x / 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 / z
end function

Java

public static double code(double x, double y, double z) {
	return x / z;
}

Python

def code(x, y, z):
	return x / z

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(x / z), $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 57.4%

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

4. Applied rewrites 57.4%

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

Reproduce

herbie shell --seed 2025212 
(FPCore (x y z)
  :name "Linear.Quaternion:$ctanh from linear-1.19.1.3"
  :precision binary64
  (/ (* x (/ (sin y) y)) z))