Linear.Quaternion:$ctanh from linear-1.19.1.3

Percentage Accurate: 96.1% → 99.3%

Time: 10.2s

Alternatives: 12

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: 96.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 5.2 \cdot 10^{+102}:\\ \;\;\;\;\frac{\frac{x\_m}{z}}{\frac{y}{\sin y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m \cdot \sin y}{y}}{z}\\ \end{array} \end{array} \]

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 5.2e+102)
    (/ (/ 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 <= 5.2e+102) {
		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 <= 5.2d+102) 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 <= 5.2e+102) {
		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 <= 5.2e+102:
		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 <= 5.2e+102)
		tmp = Float64(Float64(x_m / z) / Float64(y / sin(y)));
	else
		tmp = Float64(Float64(Float64(x_m * 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 <= 5.2e+102)
		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, 5.2e+102], N[(N[(x$95$m / z), $MachinePrecision] / N[(y / N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x$95$m * N[Sin[y], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 5.20000000000000013e102

   1. Initial program 95.8%

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

      1. associate-*l/ N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{x}{z}\right), \color{blue}{\left(\frac{y}{\sin y}\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, z\right), \left(\frac{\color{blue}{y}}{\sin y}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, z\right), \mathsf{/.f64}\left(y, \color{blue}{\sin y}\right)\right) \]

      7. sin-lowering-sin.f64 97.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, z\right), \mathsf{/.f64}\left(y, \mathsf{sin.f64}\left(y\right)\right)\right) \]

   4. Applied egg-rr 97.6%

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

   if 5.20000000000000013e102 < x

   1. Initial program 99.8%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot \frac{\sin y}{y}\right), \color{blue}{z}\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{x \cdot \sin y}{y}\right), z\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(x \cdot \sin y\right), y\right), z\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \sin y\right), y\right), z\right) \]

      5. sin-lowering-sin.f64 99.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{sin.f64}\left(y\right)\right), y\right), z\right) \]

   3. Simplified 99.8%

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

Alternative 2: 97.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := x\_m \cdot \frac{\sin y}{y}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 2 \cdot 10^{-318}:\\ \;\;\;\;\sin y \cdot \frac{x\_m}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{z}\\ \end{array} \end{array} \end{array} \]

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 (* x_m (/ (sin y) y))))
   (* x_s (if (<= t_0 2e-318) (* (sin y) (/ x_m (* z y))) (/ 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 = x_m * (sin(y) / y);
	double tmp;
	if (t_0 <= 2e-318) {
		tmp = sin(y) * (x_m / (z * y));
	} else {
		tmp = 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 = x_m * (sin(y) / y)
    if (t_0 <= 2d-318) then
        tmp = sin(y) * (x_m / (z * y))
    else
        tmp = 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 = x_m * (Math.sin(y) / y);
	double tmp;
	if (t_0 <= 2e-318) {
		tmp = Math.sin(y) * (x_m / (z * y));
	} else {
		tmp = 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 = x_m * (math.sin(y) / y)
	tmp = 0
	if t_0 <= 2e-318:
		tmp = math.sin(y) * (x_m / (z * y))
	else:
		tmp = 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(x_m * Float64(sin(y) / y))
	tmp = 0.0
	if (t_0 <= 2e-318)
		tmp = Float64(sin(y) * Float64(x_m / Float64(z * y)));
	else
		tmp = Float64(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 = x_m * (sin(y) / y);
	tmp = 0.0;
	if (t_0 <= 2e-318)
		tmp = sin(y) * (x_m / (z * y));
	else
		tmp = 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[(x$95$m * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$0, 2e-318], N[(N[Sin[y], $MachinePrecision] * N[(x$95$m / N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / z), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x (/.f64 (sin.f64 y) y)) < 2.0000024e-318

   1. Initial program 93.3%

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

      1. associate-*r/ N/A

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

      2. associate-/l/ N/A

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

      3. associate-*l/ N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{x}{z \cdot y}\right), \color{blue}{\sin y}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \left(z \cdot y\right)\right), \sin \color{blue}{y}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \left(y \cdot z\right)\right), \sin y\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \sin y\right) \]

      8. sin-lowering-sin.f64 86.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{sin.f64}\left(y\right)\right) \]

   4. Applied egg-rr 86.0%

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

   if 2.0000024e-318 < (*.f64 x (/.f64 (sin.f64 y) y))

   1. Initial program 99.7%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \frac{\sin y}{y} \leq 2 \cdot 10^{-318}:\\ \;\;\;\;\sin y \cdot \frac{x}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{\sin y}{y}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 5.2 \cdot 10^{+102}:\\ \;\;\;\;\frac{\frac{x\_m}{z}}{\frac{y}{\sin y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot \frac{\sin y}{y}}{z}\\ \end{array} \end{array} \]

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 5.2e+102)
    (/ (/ 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 <= 5.2e+102) {
		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 <= 5.2d+102) 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 <= 5.2e+102) {
		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 <= 5.2e+102:
		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 <= 5.2e+102)
		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 <= 5.2e+102)
		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, 5.2e+102], 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 < 5.20000000000000013e102

   1. Initial program 95.8%

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

      1. associate-*l/ N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{x}{z}\right), \color{blue}{\left(\frac{y}{\sin y}\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, z\right), \left(\frac{\color{blue}{y}}{\sin y}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, z\right), \mathsf{/.f64}\left(y, \color{blue}{\sin y}\right)\right) \]

      7. sin-lowering-sin.f64 97.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, z\right), \mathsf{/.f64}\left(y, \mathsf{sin.f64}\left(y\right)\right)\right) \]

   4. Applied egg-rr 97.6%

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

   if 5.20000000000000013e102 < 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. Add Preprocessing

Alternative 4: 74.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 1.95 \cdot 10^{-7}:\\ \;\;\;\;\frac{x\_m \cdot \left(1 + y \cdot \left(y \cdot -0.16666666666666666\right)\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \frac{x\_m}{z \cdot y}\\ \end{array} \end{array} \]

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 1.95e-7)
    (/ (* x_m (+ 1.0 (* y (* y -0.16666666666666666)))) z)
    (* (sin y) (/ x_m (* z 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 <= 1.95e-7) {
		tmp = (x_m * (1.0 + (y * (y * -0.16666666666666666)))) / z;
	} else {
		tmp = sin(y) * (x_m / (z * y));
	}
	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 (y <= 1.95d-7) then
        tmp = (x_m * (1.0d0 + (y * (y * (-0.16666666666666666d0))))) / z
    else
        tmp = sin(y) * (x_m / (z * y))
    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 (y <= 1.95e-7) {
		tmp = (x_m * (1.0 + (y * (y * -0.16666666666666666)))) / z;
	} else {
		tmp = Math.sin(y) * (x_m / (z * y));
	}
	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 y <= 1.95e-7:
		tmp = (x_m * (1.0 + (y * (y * -0.16666666666666666)))) / z
	else:
		tmp = math.sin(y) * (x_m / (z * 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 <= 1.95e-7)
		tmp = Float64(Float64(x_m * Float64(1.0 + Float64(y * Float64(y * -0.16666666666666666)))) / z);
	else
		tmp = Float64(sin(y) * Float64(x_m / Float64(z * y)));
	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 (y <= 1.95e-7)
		tmp = (x_m * (1.0 + (y * (y * -0.16666666666666666)))) / z;
	else
		tmp = sin(y) * (x_m / (z * y));
	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[y, 1.95e-7], N[(N[(x$95$m * N[(1.0 + N[(y * N[(y * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[Sin[y], $MachinePrecision] * N[(x$95$m / N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 1.95000000000000012e-7

   1. Initial program 96.9%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. distribute-lft1-in N/A

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

      5. +-commutative N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-out N/A

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

      9. *-rgt-identity N/A

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

      10. *-commutative N/A

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

      11. associate-*l* N/A

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

      12. *-commutative N/A

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

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(x + \frac{-1}{6} \cdot \left(x \cdot {y}^{2}\right)\right), \color{blue}{z}\right) \]

   5. Simplified 67.9%

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

   if 1.95000000000000012e-7 < y

   1. Initial program 94.6%

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

      1. associate-*r/ N/A

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

      2. associate-/l/ N/A

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

      3. associate-*l/ N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{x}{z \cdot y}\right), \color{blue}{\sin y}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \left(z \cdot y\right)\right), \sin \color{blue}{y}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \left(y \cdot z\right)\right), \sin y\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \sin y\right) \]

      8. sin-lowering-sin.f64 94.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{sin.f64}\left(y\right)\right) \]

   4. Applied egg-rr 94.1%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.95 \cdot 10^{-7}:\\ \;\;\;\;\frac{x \cdot \left(1 + y \cdot \left(y \cdot -0.16666666666666666\right)\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \frac{x}{z \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 62.6% accurate, 7.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 2.5:\\ \;\;\;\;\frac{x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{6 \cdot \frac{\frac{x\_m}{z}}{y}}{y}\\ \end{array} \end{array} \]

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 2.5) (/ x_m z) (/ (* 6.0 (/ (/ x_m 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 <= 2.5) {
		tmp = x_m / z;
	} else {
		tmp = (6.0 * ((x_m / z) / y)) / y;
	}
	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 (y <= 2.5d0) then
        tmp = x_m / z
    else
        tmp = (6.0d0 * ((x_m / z) / y)) / y
    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 (y <= 2.5) {
		tmp = x_m / z;
	} else {
		tmp = (6.0 * ((x_m / z) / y)) / y;
	}
	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 y <= 2.5:
		tmp = x_m / z
	else:
		tmp = (6.0 * ((x_m / 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 <= 2.5)
		tmp = Float64(x_m / z);
	else
		tmp = Float64(Float64(6.0 * Float64(Float64(x_m / z) / y)) / y);
	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 (y <= 2.5)
		tmp = x_m / z;
	else
		tmp = (6.0 * ((x_m / z) / y)) / y;
	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[y, 2.5], N[(x$95$m / z), $MachinePrecision], N[(N[(6.0 * N[(N[(x$95$m / z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 2.5

   1. Initial program 96.9%

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

   3. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 70.9%

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{z}\right) \]

   5. Simplified 70.9%

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

   if 2.5 < y

   1. Initial program 94.4%

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

      1. associate-*l/ N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{x}{z}\right), \color{blue}{\left(\frac{y}{\sin y}\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, z\right), \left(\frac{\color{blue}{y}}{\sin y}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, z\right), \mathsf{/.f64}\left(y, \color{blue}{\sin y}\right)\right) \]

      7. sin-lowering-sin.f64 89.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, z\right), \mathsf{/.f64}\left(y, \mathsf{sin.f64}\left(y\right)\right)\right) \]

   4. Applied egg-rr 89.4%

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

   5. Taylor expanded in y around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, z\right), \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)}\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, z\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, z\right), \mathsf{+.f64}\left(1, \left({y}^{2} \cdot \color{blue}{\frac{1}{6}}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, z\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\frac{1}{6}}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, z\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(y \cdot y\right), \frac{1}{6}\right)\right)\right) \]

      5. *-lowering-*.f64 32.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, z\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{6}\right)\right)\right) \]

   7. Simplified 32.7%

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

   8. Taylor expanded in y around inf

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

      1. associate-*r/ N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. times-frac N/A

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

      5. associate-*l/ N/A

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

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(6 \cdot \frac{x}{y \cdot z}\right), \color{blue}{y}\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(6, \left(\frac{x}{y \cdot z}\right)\right), y\right) \]

      8. associate-/l/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(6, \left(\frac{\frac{x}{z}}{y}\right)\right), y\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(6, \mathsf{/.f64}\left(\left(\frac{x}{z}\right), y\right)\right), y\right) \]

      10. /-lowering-/.f64 33.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(6, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, z\right), y\right)\right), y\right) \]

   10. Simplified 33.0%

       \[\leadsto \color{blue}{\frac{6 \cdot \frac{\frac{x}{z}}{y}}{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 62.5% accurate, 8.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 0.2:\\ \;\;\;\;\frac{x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{\frac{x\_m}{y}}}\\ \end{array} \end{array} \]

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 0.2) (/ x_m z) (/ y (/ z (/ x_m 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 <= 0.2) {
		tmp = x_m / z;
	} else {
		tmp = y / (z / (x_m / y));
	}
	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 (y <= 0.2d0) then
        tmp = x_m / z
    else
        tmp = y / (z / (x_m / y))
    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 (y <= 0.2) {
		tmp = x_m / z;
	} else {
		tmp = y / (z / (x_m / y));
	}
	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 y <= 0.2:
		tmp = x_m / z
	else:
		tmp = y / (z / (x_m / 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 <= 0.2)
		tmp = Float64(x_m / z);
	else
		tmp = Float64(y / Float64(z / Float64(x_m / y)));
	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 (y <= 0.2)
		tmp = x_m / z;
	else
		tmp = y / (z / (x_m / y));
	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[y, 0.2], N[(x$95$m / z), $MachinePrecision], N[(y / N[(z / N[(x$95$m / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 0.20000000000000001

   1. Initial program 96.9%

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

   3. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 70.9%

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{z}\right) \]

   5. Simplified 70.9%

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

   if 0.20000000000000001 < y

   1. Initial program 94.4%

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

   3. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 15.3%

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{z}\right) \]

   5. Simplified 15.3%

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

      1. clear-num N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{z}{x}\right)}\right) \]

      3. /-lowering-/.f64 16.3%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(z, \color{blue}{x}\right)\right) \]

   7. Applied egg-rr 16.3%

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

      1. clear-num N/A

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

      2. *-rgt-identity N/A

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

      3. lft-mult-inverse N/A

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

      4. associate-*l* N/A

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

      5. div-inv N/A

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

      6. associate-/r* N/A

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

      7. *-commutative N/A

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

      8. clear-num N/A

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

      9. un-div-inv N/A

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

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(y, \color{blue}{\left(\frac{z \cdot y}{x}\right)}\right) \]

      11. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(y, \left(z \cdot \color{blue}{\frac{y}{x}}\right)\right) \]

      12. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(y, \left(z \cdot \frac{1}{\color{blue}{\frac{x}{y}}}\right)\right) \]

      13. div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(y, \left(\frac{z}{\color{blue}{\frac{x}{y}}}\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(z, \color{blue}{\left(\frac{x}{y}\right)}\right)\right) \]

      15. /-lowering-/.f64 32.6%

          \[\leadsto \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(z, \mathsf{/.f64}\left(x, \color{blue}{y}\right)\right)\right) \]

   9. Applied egg-rr 32.6%

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

Alternative 7: 62.2% accurate, 8.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 0.002:\\ \;\;\;\;\frac{x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x\_m}{z \cdot y}\\ \end{array} \end{array} \]

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 0.002) (/ x_m z) (* y (/ x_m (* z 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 <= 0.002) {
		tmp = x_m / z;
	} else {
		tmp = y * (x_m / (z * y));
	}
	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 (y <= 0.002d0) then
        tmp = x_m / z
    else
        tmp = y * (x_m / (z * y))
    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 (y <= 0.002) {
		tmp = x_m / z;
	} else {
		tmp = y * (x_m / (z * y));
	}
	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 y <= 0.002:
		tmp = x_m / z
	else:
		tmp = y * (x_m / (z * 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 <= 0.002)
		tmp = Float64(x_m / z);
	else
		tmp = Float64(y * Float64(x_m / Float64(z * y)));
	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 (y <= 0.002)
		tmp = x_m / z;
	else
		tmp = y * (x_m / (z * y));
	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[y, 0.002], N[(x$95$m / z), $MachinePrecision], N[(y * N[(x$95$m / N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 2e-3

   1. Initial program 96.9%

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

   3. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 70.9%

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{z}\right) \]

   5. Simplified 70.9%

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

   if 2e-3 < y

   1. Initial program 94.4%

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

   3. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 15.3%

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{z}\right) \]

   5. Simplified 15.3%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{z}\right), \color{blue}{x}\right) \]

      4. /-lowering-/.f64 15.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, z\right), x\right) \]

   7. Applied egg-rr 15.3%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

      3. *-inverses N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{x}{z \cdot y}\right)}\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(x, \color{blue}{\left(z \cdot y\right)}\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(x, \left(y \cdot \color{blue}{z}\right)\right)\right) \]

      10. *-lowering-*.f64 32.1%

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{z}\right)\right)\right) \]

   9. Applied egg-rr 32.1%

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

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.002:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 60.4% accurate, 8.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 9 \cdot 10^{-6}:\\ \;\;\;\;\frac{x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \frac{y}{z \cdot y}\\ \end{array} \end{array} \]

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 9e-6) (/ x_m z) (* x_m (/ y (* z 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 <= 9e-6) {
		tmp = x_m / z;
	} else {
		tmp = x_m * (y / (z * y));
	}
	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 (y <= 9d-6) then
        tmp = x_m / z
    else
        tmp = x_m * (y / (z * y))
    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 (y <= 9e-6) {
		tmp = x_m / z;
	} else {
		tmp = x_m * (y / (z * y));
	}
	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 y <= 9e-6:
		tmp = x_m / z
	else:
		tmp = x_m * (y / (z * 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 <= 9e-6)
		tmp = Float64(x_m / z);
	else
		tmp = Float64(x_m * Float64(y / Float64(z * y)));
	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 (y <= 9e-6)
		tmp = x_m / z;
	else
		tmp = x_m * (y / (z * y));
	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[y, 9e-6], N[(x$95$m / z), $MachinePrecision], N[(x$95$m * N[(y / N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 9.00000000000000023e-6

   1. Initial program 96.9%

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

   3. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 71.0%

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{z}\right) \]

   5. Simplified 71.0%

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

   if 9.00000000000000023e-6 < y

   1. Initial program 94.5%

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

   3. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 15.9%

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{z}\right) \]

   5. Simplified 15.9%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{z}\right), \color{blue}{x}\right) \]

      4. /-lowering-/.f64 15.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, z\right), x\right) \]

   7. Applied egg-rr 15.9%

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

      1. associate-*l/ N/A

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

      2. *-lft-identity N/A

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

      3. *-rgt-identity N/A

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

      4. *-inverses N/A

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

      5. times-frac N/A

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

      6. associate-/l* N/A

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

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{y}{z \cdot y}\right)}\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \color{blue}{\left(z \cdot y\right)}\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \left(y \cdot \color{blue}{z}\right)\right)\right) \]

      10. *-lowering-*.f64 24.1%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{z}\right)\right)\right) \]

   9. Applied egg-rr 24.1%

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

4. Final simplification 58.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9 \cdot 10^{-6}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 66.9% accurate, 9.7× speedup

Math

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

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) (+ 1.0 (* (* y y) 0.16666666666666666)))))

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) / (1.0 + ((y * y) * 0.16666666666666666)));
}

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) / (1.0d0 + ((y * y) * 0.16666666666666666d0)))
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) / (1.0 + ((y * y) * 0.16666666666666666)));
}

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) / (1.0 + ((y * y) * 0.16666666666666666)))

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) / Float64(1.0 + Float64(Float64(y * y) * 0.16666666666666666))))
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) / (1.0 + ((y * y) * 0.16666666666666666)));
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[(1.0 + N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.3%

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

   1. associate-*l/ N/A

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

   2. clear-num N/A

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

   3. un-div-inv N/A

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

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{x}{z}\right), \color{blue}{\left(\frac{y}{\sin y}\right)}\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, z\right), \left(\frac{\color{blue}{y}}{\sin y}\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, z\right), \mathsf{/.f64}\left(y, \color{blue}{\sin y}\right)\right) \]

   7. sin-lowering-sin.f64 96.1%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, z\right), \mathsf{/.f64}\left(y, \mathsf{sin.f64}\left(y\right)\right)\right) \]

4. Applied egg-rr 96.1%

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

5. Taylor expanded in y around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, z\right), \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)}\right) \]
6. Step-by-step derivation

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, z\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)}\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, z\right), \mathsf{+.f64}\left(1, \left({y}^{2} \cdot \color{blue}{\frac{1}{6}}\right)\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, z\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\frac{1}{6}}\right)\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, z\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(y \cdot y\right), \frac{1}{6}\right)\right)\right) \]

   5. *-lowering-*.f64 66.2%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, z\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{6}\right)\right)\right) \]

7. Simplified 66.2%

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

Alternative 10: 66.6% accurate, 9.7× speedup

Math

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

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 (+ 1.0 (* y (* y 0.16666666666666666)))))))

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 * (1.0 + (y * (y * 0.16666666666666666)))));
}

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 * (1.0d0 + (y * (y * 0.16666666666666666d0)))))
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 * (1.0 + (y * (y * 0.16666666666666666)))));
}

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 * (1.0 + (y * (y * 0.16666666666666666)))))

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 / Float64(z * Float64(1.0 + Float64(y * Float64(y * 0.16666666666666666))))))
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 * (1.0 + (y * (y * 0.16666666666666666)))));
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 / N[(z * N[(1.0 + N[(y * N[(y * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.3%

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

   1. associate-*l/ N/A

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

   2. clear-num N/A

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

   3. un-div-inv N/A

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

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{x}{z}\right), \color{blue}{\left(\frac{y}{\sin y}\right)}\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, z\right), \left(\frac{\color{blue}{y}}{\sin y}\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, z\right), \mathsf{/.f64}\left(y, \color{blue}{\sin y}\right)\right) \]

   7. sin-lowering-sin.f64 96.1%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, z\right), \mathsf{/.f64}\left(y, \mathsf{sin.f64}\left(y\right)\right)\right) \]

4. Applied egg-rr 96.1%

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

5. Taylor expanded in y around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, z\right), \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)}\right) \]
6. Step-by-step derivation

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, z\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)}\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, z\right), \mathsf{+.f64}\left(1, \left({y}^{2} \cdot \color{blue}{\frac{1}{6}}\right)\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, z\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\frac{1}{6}}\right)\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, z\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(y \cdot y\right), \frac{1}{6}\right)\right)\right) \]

   5. *-lowering-*.f64 66.2%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, z\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{6}\right)\right)\right) \]

7. Simplified 66.2%

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

   1. associate-/l/ N/A

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

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(\left(1 + \left(y \cdot y\right) \cdot \frac{1}{6}\right) \cdot z\right)}\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(x, \left(z \cdot \color{blue}{\left(1 + \left(y \cdot y\right) \cdot \frac{1}{6}\right)}\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(z, \color{blue}{\left(1 + \left(y \cdot y\right) \cdot \frac{1}{6}\right)}\right)\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \color{blue}{\left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)}\right)\right)\right) \]

   6. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \left(y \cdot \color{blue}{\left(y \cdot \frac{1}{6}\right)}\right)\right)\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(y \cdot \frac{1}{6}\right)}\right)\right)\right)\right) \]

   8. *-lowering-*.f64 65.5%

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right) \]

9. Applied egg-rr 65.5%

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

Alternative 11: 58.3% accurate, 21.4× speedup

Math

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

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 (/ 1.0 (/ z x_m))))

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 * (1.0 / (z / x_m));
}

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 * (1.0d0 / (z / x_m))
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 * (1.0 / (z / x_m));
}

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 96.3%

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

3. Taylor expanded in y around 0

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

   1. /-lowering-/.f64 56.8%

      \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{z}\right) \]

5. Simplified 56.8%

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

   1. clear-num N/A

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

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{z}{x}\right)}\right) \]

   3. /-lowering-/.f64 57.0%

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(z, \color{blue}{x}\right)\right) \]

7. Applied egg-rr 57.0%

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

Alternative 12: 58.4% accurate, 35.7× 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 96.3%

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

3. Taylor expanded in y around 0

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

   1. /-lowering-/.f64 56.8%

      \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{z}\right) \]

5. Simplified 56.8%

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

Developer Target 1: 99.6% accurate, 0.9× 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 2024140 
(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))