Linear.Quaternion:$ctanh from linear-1.19.1.3

Percentage Accurate: 95.8% → 99.8%

Time: 10.6s

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

real(8) function code(x, y, z)
    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: 95.8% 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

real(8) function code(x, y, z)
    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: 99.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 5 \cdot 10^{-35}:\\ \;\;\;\;\frac{\frac{x\_m}{z}}{\frac{y}{\sin y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot \frac{\sin y}{y}}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= x_m 5e-35) (/ (/ x_m z) (/ y (sin y))) (/ (* x_m (/ (sin y) y)) z))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (x_m <= 5e-35) {
		tmp = (x_m / z) / (y / sin(y));
	} else {
		tmp = (x_m * (sin(y) / y)) / z;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x_m <= 5d-35) then
        tmp = (x_m / z) / (y / sin(y))
    else
        tmp = (x_m * (sin(y) / y)) / z
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (x_m <= 5e-35) {
		tmp = (x_m / z) / (y / Math.sin(y));
	} else {
		tmp = (x_m * (Math.sin(y) / y)) / z;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if x_m <= 5e-35:
		tmp = (x_m / z) / (y / math.sin(y))
	else:
		tmp = (x_m * (math.sin(y) / y)) / z
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (x_m <= 5e-35)
		tmp = Float64(Float64(x_m / z) / Float64(y / sin(y)));
	else
		tmp = Float64(Float64(x_m * Float64(sin(y) / y)) / z);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if (x_m <= 5e-35)
		tmp = (x_m / z) / (y / sin(y));
	else
		tmp = (x_m * (sin(y) / y)) / z;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[x$95$m, 5e-35], N[(N[(x$95$m / z), $MachinePrecision] / N[(y / N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 4.99999999999999964e-35

   1. Initial program 93.4%

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

      1. lift-sin.f64 N/A

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

      2. associate-*r/ N/A

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

      3. associate-/l/ N/A

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

      4. times-frac N/A

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

      5. clear-num N/A

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

      6. div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 97.9

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

   4. Applied egg-rr 97.9%

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

   if 4.99999999999999964e-35 < x

   1. Initial program 99.7%

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

Alternative 2: 94.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{x\_m \cdot t\_0}{z} \leq -2 \cdot 10^{-16}:\\ \;\;\;\;\frac{\sin y}{z} \cdot \frac{x\_m}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z} \cdot t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)))
   (*
    x_s
    (if (<= (/ (* x_m t_0) z) -2e-16)
      (* (/ (sin y) z) (/ x_m y))
      (* (/ x_m z) t_0)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double t_0 = sin(y) / y;
	double tmp;
	if (((x_m * t_0) / z) <= -2e-16) {
		tmp = (sin(y) / z) * (x_m / y);
	} else {
		tmp = (x_m / z) * t_0;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin(y) / y
    if (((x_m * t_0) / z) <= (-2d-16)) then
        tmp = (sin(y) / z) * (x_m / y)
    else
        tmp = (x_m / z) * t_0
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double t_0 = Math.sin(y) / y;
	double tmp;
	if (((x_m * t_0) / z) <= -2e-16) {
		tmp = (Math.sin(y) / z) * (x_m / y);
	} else {
		tmp = (x_m / z) * t_0;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	t_0 = math.sin(y) / y
	tmp = 0
	if ((x_m * t_0) / z) <= -2e-16:
		tmp = (math.sin(y) / z) * (x_m / y)
	else:
		tmp = (x_m / z) * t_0
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	t_0 = Float64(sin(y) / y)
	tmp = 0.0
	if (Float64(Float64(x_m * t_0) / z) <= -2e-16)
		tmp = Float64(Float64(sin(y) / z) * Float64(x_m / y));
	else
		tmp = Float64(Float64(x_m / z) * t_0);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	t_0 = sin(y) / y;
	tmp = 0.0;
	if (((x_m * t_0) / z) <= -2e-16)
		tmp = (sin(y) / z) * (x_m / y);
	else
		tmp = (x_m / z) * t_0;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, N[(x$95$s * If[LessEqual[N[(N[(x$95$m * t$95$0), $MachinePrecision] / z), $MachinePrecision], -2e-16], N[(N[(N[Sin[y], $MachinePrecision] / z), $MachinePrecision] * N[(x$95$m / y), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / z), $MachinePrecision] * t$95$0), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z) < -2e-16

   1. Initial program 99.8%

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

      1. lift-sin.f64 N/A

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

      2. associate-*r/ N/A

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

      3. associate-/l/ N/A

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

      4. *-commutative N/A

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

      5. times-frac N/A

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

      6. *-lft-identity N/A

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

      7. associate-*l/ N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. associate-*l/ N/A

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

      11. *-lft-identity N/A

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

      12. lower-/.f64 83.1

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

   4. Applied egg-rr 83.1%

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

   if -2e-16 < (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z)

   1. Initial program 93.8%

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

      1. lift-sin.f64 N/A

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

      2. frac-2neg N/A

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

      3. div-inv N/A

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

      4. div-inv N/A

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

      5. frac-2neg N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 97.5

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

   4. Applied egg-rr 97.5%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \frac{\sin y}{y}}{z} \leq -2 \cdot 10^{-16}:\\ \;\;\;\;\frac{\sin y}{z} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{\sin y}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{x\_m \cdot t\_0}{z} \leq -4 \cdot 10^{+150}:\\ \;\;\;\;x\_m \cdot \frac{\sin y}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z} \cdot t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)))
   (*
    x_s
    (if (<= (/ (* x_m t_0) z) -4e+150)
      (* x_m (/ (sin y) (* z y)))
      (* (/ x_m z) t_0)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double t_0 = sin(y) / y;
	double tmp;
	if (((x_m * t_0) / z) <= -4e+150) {
		tmp = x_m * (sin(y) / (z * y));
	} else {
		tmp = (x_m / z) * t_0;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin(y) / y
    if (((x_m * t_0) / z) <= (-4d+150)) then
        tmp = x_m * (sin(y) / (z * y))
    else
        tmp = (x_m / z) * t_0
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double t_0 = Math.sin(y) / y;
	double tmp;
	if (((x_m * t_0) / z) <= -4e+150) {
		tmp = x_m * (Math.sin(y) / (z * y));
	} else {
		tmp = (x_m / z) * t_0;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	t_0 = math.sin(y) / y
	tmp = 0
	if ((x_m * t_0) / z) <= -4e+150:
		tmp = x_m * (math.sin(y) / (z * y))
	else:
		tmp = (x_m / z) * t_0
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	t_0 = Float64(sin(y) / y)
	tmp = 0.0
	if (Float64(Float64(x_m * t_0) / z) <= -4e+150)
		tmp = Float64(x_m * Float64(sin(y) / Float64(z * y)));
	else
		tmp = Float64(Float64(x_m / z) * t_0);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	t_0 = sin(y) / y;
	tmp = 0.0;
	if (((x_m * t_0) / z) <= -4e+150)
		tmp = x_m * (sin(y) / (z * y));
	else
		tmp = (x_m / z) * t_0;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, N[(x$95$s * If[LessEqual[N[(N[(x$95$m * t$95$0), $MachinePrecision] / z), $MachinePrecision], -4e+150], N[(x$95$m * N[(N[Sin[y], $MachinePrecision] / N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / z), $MachinePrecision] * t$95$0), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z) < -3.99999999999999992e150

   1. Initial program 99.8%

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

      1. lift-sin.f64 N/A

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

      2. frac-2neg N/A

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

      3. div-inv N/A

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

      4. div-inv N/A

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

      5. frac-2neg N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-/l/ N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 95.4

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

   4. Applied egg-rr 95.4%

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

   if -3.99999999999999992e150 < (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z)

   1. Initial program 94.6%

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

      1. lift-sin.f64 N/A

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

      2. frac-2neg N/A

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

      3. div-inv N/A

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

      4. div-inv N/A

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

      5. frac-2neg N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 97.8

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

   4. Applied egg-rr 97.8%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \frac{\sin y}{y}}{z} \leq -4 \cdot 10^{+150}:\\ \;\;\;\;x \cdot \frac{\sin y}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{\sin y}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 0.999998582761237:\\ \;\;\;\;\sin y \cdot \frac{x\_m}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= (/ (sin y) y) 0.999998582761237)
    (* (sin y) (/ x_m (* z y)))
    (/ x_m z))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((sin(y) / y) <= 0.999998582761237) {
		tmp = sin(y) * (x_m / (z * y));
	} else {
		tmp = x_m / z;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((sin(y) / y) <= 0.999998582761237d0) then
        tmp = sin(y) * (x_m / (z * y))
    else
        tmp = x_m / z
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((Math.sin(y) / y) <= 0.999998582761237) {
		tmp = Math.sin(y) * (x_m / (z * y));
	} else {
		tmp = x_m / z;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if (math.sin(y) / y) <= 0.999998582761237:
		tmp = math.sin(y) * (x_m / (z * y))
	else:
		tmp = x_m / z
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (Float64(sin(y) / y) <= 0.999998582761237)
		tmp = Float64(sin(y) * Float64(x_m / Float64(z * y)));
	else
		tmp = Float64(x_m / z);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if ((sin(y) / y) <= 0.999998582761237)
		tmp = sin(y) * (x_m / (z * y));
	else
		tmp = x_m / z;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], 0.999998582761237], N[(N[Sin[y], $MachinePrecision] * N[(x$95$m / N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 y) y) < 0.99999858276123699

   1. Initial program 90.1%

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

      1. lift-sin.f64 N/A

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

      2. associate-*r/ N/A

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

      3. associate-/l/ N/A

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

      4. *-commutative N/A

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

      5. times-frac N/A

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

      6. *-lft-identity N/A

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

      7. associate-*l/ N/A

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

      8. associate-*l/ N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. associate-*l/ N/A

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

      13. *-lft-identity N/A

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

      14. associate-/l/ N/A

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

      15. lower-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 91.5

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

   4. Applied egg-rr 91.5%

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

   if 0.99999858276123699 < (/.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

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

      1. lower-/.f64 100.0

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

   5. Simplified 100.0%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 0.999998582761237:\\ \;\;\;\;\sin y \cdot \frac{x}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 54.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{x\_m \cdot \frac{\sin y}{y}}{z} \leq 10^{-295}:\\ \;\;\;\;\frac{x\_m \cdot y}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= (/ (* x_m (/ (sin y) y)) z) 1e-295)
    (/ (* x_m y) (* z y))
    (/ x_m z))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (((x_m * (sin(y) / y)) / z) <= 1e-295) {
		tmp = (x_m * y) / (z * y);
	} else {
		tmp = x_m / z;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (((x_m * (sin(y) / y)) / z) <= 1d-295) then
        tmp = (x_m * y) / (z * y)
    else
        tmp = x_m / z
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (((x_m * (Math.sin(y) / y)) / z) <= 1e-295) {
		tmp = (x_m * y) / (z * y);
	} else {
		tmp = x_m / z;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if ((x_m * (math.sin(y) / y)) / z) <= 1e-295:
		tmp = (x_m * y) / (z * y)
	else:
		tmp = x_m / z
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (Float64(Float64(x_m * Float64(sin(y) / y)) / z) <= 1e-295)
		tmp = Float64(Float64(x_m * y) / Float64(z * y));
	else
		tmp = Float64(x_m / z);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if (((x_m * (sin(y) / y)) / z) <= 1e-295)
		tmp = (x_m * y) / (z * y);
	else
		tmp = x_m / z;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[N[(N[(x$95$m * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], 1e-295], N[(N[(x$95$m * y), $MachinePrecision] / N[(z * y), $MachinePrecision]), $MachinePrecision], N[(x$95$m / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z) < 1.00000000000000006e-295

   1. Initial program 92.0%

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

      1. lift-sin.f64 N/A

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

      2. associate-*r/ N/A

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

      3. associate-/l/ N/A

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

      4. times-frac N/A

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

      5. clear-num N/A

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

      6. div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 97.6

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

   4. Applied egg-rr 97.6%

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

      1. lift-/.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. frac-2neg N/A

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

      4. frac-2neg N/A

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

      5. associate-/r/ N/A

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

      6. *-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/l/ N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 89.5

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 89.5

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

   6. Applied egg-rr 89.5%

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

   7. Taylor expanded in y around 0

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

      1. lower-*.f64 57.8

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

   9. Simplified 57.8%

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

   if 1.00000000000000006e-295 < (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z)

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 58.6

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

   5. Simplified 58.6%

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

Alternative 6: 65.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 2 \cdot 10^{-76}:\\ \;\;\;\;\frac{x\_m \cdot 6}{z \cdot \left(y \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (* x_s (if (<= (/ (sin y) y) 2e-76) (/ (* x_m 6.0) (* z (* y y))) (/ x_m z))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((sin(y) / y) <= 2e-76) {
		tmp = (x_m * 6.0) / (z * (y * y));
	} else {
		tmp = x_m / z;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((sin(y) / y) <= 2d-76) then
        tmp = (x_m * 6.0d0) / (z * (y * y))
    else
        tmp = x_m / z
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((Math.sin(y) / y) <= 2e-76) {
		tmp = (x_m * 6.0) / (z * (y * y));
	} else {
		tmp = x_m / z;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if (math.sin(y) / y) <= 2e-76:
		tmp = (x_m * 6.0) / (z * (y * y))
	else:
		tmp = x_m / z
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (Float64(sin(y) / y) <= 2e-76)
		tmp = Float64(Float64(x_m * 6.0) / Float64(z * Float64(y * y)));
	else
		tmp = Float64(x_m / z);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if ((sin(y) / y) <= 2e-76)
		tmp = (x_m * 6.0) / (z * (y * y));
	else
		tmp = x_m / z;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], 2e-76], N[(N[(x$95$m * 6.0), $MachinePrecision] / N[(z * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 y) y) < 1.99999999999999985e-76

   1. Initial program 88.5%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-rgt-identity N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 4.4

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

   5. Simplified 4.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. +-commutative N/A

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

      4. flip-+ N/A

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

      5. div-inv N/A

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

      6. lower-*.f64 N/A

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

   7. Applied egg-rr 2.8%

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

   8. Taylor expanded in y around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 11.8

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

   10. Simplified 11.8%

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

   11. Taylor expanded in y around inf

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

       1. associate-*r/ N/A

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

       2. lower-/.f64 N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 33.2

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

   13. Simplified 33.2%

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

   if 1.99999999999999985e-76 < (/.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

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

      1. lower-/.f64 93.4

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

   5. Simplified 93.4%

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

Alternative 7: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 6 \cdot 10^{-64}:\\ \;\;\;\;\frac{x\_m}{z} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot t\_0}{z}\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)))
   (* x_s (if (<= x_m 6e-64) (* (/ x_m z) t_0) (/ (* x_m t_0) z)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double t_0 = sin(y) / y;
	double tmp;
	if (x_m <= 6e-64) {
		tmp = (x_m / z) * t_0;
	} else {
		tmp = (x_m * t_0) / z;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin(y) / y
    if (x_m <= 6d-64) then
        tmp = (x_m / z) * t_0
    else
        tmp = (x_m * t_0) / z
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double t_0 = Math.sin(y) / y;
	double tmp;
	if (x_m <= 6e-64) {
		tmp = (x_m / z) * t_0;
	} else {
		tmp = (x_m * t_0) / z;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	t_0 = math.sin(y) / y
	tmp = 0
	if x_m <= 6e-64:
		tmp = (x_m / z) * t_0
	else:
		tmp = (x_m * t_0) / z
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	t_0 = Float64(sin(y) / y)
	tmp = 0.0
	if (x_m <= 6e-64)
		tmp = Float64(Float64(x_m / z) * t_0);
	else
		tmp = Float64(Float64(x_m * t_0) / z);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	t_0 = sin(y) / y;
	tmp = 0.0;
	if (x_m <= 6e-64)
		tmp = (x_m / z) * t_0;
	else
		tmp = (x_m * t_0) / z;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, N[(x$95$s * If[LessEqual[x$95$m, 6e-64], N[(N[(x$95$m / z), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(x$95$m * t$95$0), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 6.0000000000000001e-64

   1. Initial program 93.0%

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

      1. lift-sin.f64 N/A

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

      2. frac-2neg N/A

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

      3. div-inv N/A

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

      4. div-inv N/A

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

      5. frac-2neg N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 97.7

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

   4. Applied egg-rr 97.7%

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

   if 6.0000000000000001e-64 < x

   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 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6 \cdot 10^{-64}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{\sin y}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{\sin y}{y}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 60.3% accurate, 3.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 230000:\\ \;\;\;\;\frac{x\_m}{z} \cdot \mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot 6}{z \cdot \left(y \cdot y\right)}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= y 230000.0)
    (* (/ x_m z) (fma y (* y -0.16666666666666666) 1.0))
    (/ (* x_m 6.0) (* z (* y y))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (y <= 230000.0) {
		tmp = (x_m / z) * fma(y, (y * -0.16666666666666666), 1.0);
	} else {
		tmp = (x_m * 6.0) / (z * (y * y));
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (y <= 230000.0)
		tmp = Float64(Float64(x_m / z) * fma(y, Float64(y * -0.16666666666666666), 1.0));
	else
		tmp = Float64(Float64(x_m * 6.0) / Float64(z * Float64(y * y)));
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[y, 230000.0], N[(N[(x$95$m / z), $MachinePrecision] * N[(y * N[(y * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * 6.0), $MachinePrecision] / N[(z * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 2.3e5

   1. Initial program 97.1%

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

      1. lift-sin.f64 N/A

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

      2. frac-2neg N/A

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

      3. div-inv N/A

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

      4. div-inv N/A

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

      5. frac-2neg N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 98.5

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

   4. Applied egg-rr 98.5%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 67.8

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

   7. Simplified 67.8%

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

   if 2.3e5 < y

   1. Initial program 87.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-rgt-identity N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 7.3

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

   5. Simplified 7.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. +-commutative N/A

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

      4. flip-+ N/A

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

      5. div-inv N/A

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

      6. lower-*.f64 N/A

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

   7. Applied egg-rr 4.9%

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

   8. Taylor expanded in y around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 13.1

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

   10. Simplified 13.1%

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

   11. Taylor expanded in y around inf

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

       1. associate-*r/ N/A

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

       2. lower-/.f64 N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 29.0

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

   13. Simplified 29.0%

       \[\leadsto \color{blue}{\frac{x \cdot 6}{z \cdot \left(y \cdot y\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 230000:\\ \;\;\;\;\frac{x}{z} \cdot \mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 6}{z \cdot \left(y \cdot y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 66.3% accurate, 3.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \frac{\frac{x\_m}{z}}{\mathsf{fma}\left(y, y \cdot 0.16666666666666666, 1\right)} \end{array} \]

FPCore

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

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	return x_s * ((x_m / z) / fma(y, (y * 0.16666666666666666), 1.0));
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	return Float64(x_s * Float64(Float64(x_m / z) / fma(y, Float64(y * 0.16666666666666666), 1.0)))
end

Wolfram

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

Derivation

1. Initial program 95.1%

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

   1. lift-sin.f64 N/A

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

   2. associate-*r/ N/A

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

   3. associate-/l/ N/A

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

   4. times-frac N/A

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

   5. clear-num N/A

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

   6. div-inv N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 96.6

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

4. Applied egg-rr 96.6%

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

5. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. unpow2 N/A

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

   4. associate-*l* N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-*.f64 67.9

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

7. Simplified 67.9%

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

Alternative 10: 57.9% accurate, 10.7× speedup

Math

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

FPCore

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

C

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

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x_s * (x_m / z)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.1%

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

3. Taylor expanded in y around 0

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

   1. lower-/.f64 59.3

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

5. Simplified 59.3%

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

Developer Target 1: 99.6% 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

real(8) function code(x, y, z)
    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 2024207 
(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))