Linear.Quaternion:$ctanh from linear-1.19.1.3

Percentage Accurate: 95.9% → 98.3%

Time: 12.8s

Alternatives: 14

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: 95.9% 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: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 6.5 \cdot 10^{-61}:\\ \;\;\;\;\frac{\sin y}{y \cdot \frac{z}{x\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot \frac{\sin y}{y}}{z}\\ \end{array} \end{array} \]

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 6.5e-61)
    (/ (sin y) (* y (/ z x_m)))
    (/ (* 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 <= 6.5e-61) {
		tmp = sin(y) / (y * (z / x_m));
	} 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 <= 6.5d-61) then
        tmp = sin(y) / (y * (z / x_m))
    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 <= 6.5e-61) {
		tmp = Math.sin(y) / (y * (z / x_m));
	} 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 <= 6.5e-61:
		tmp = math.sin(y) / (y * (z / x_m))
	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 <= 6.5e-61)
		tmp = Float64(sin(y) / Float64(y * Float64(z / x_m)));
	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 <= 6.5e-61)
		tmp = sin(y) / (y * (z / x_m));
	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, 6.5e-61], N[(N[Sin[y], $MachinePrecision] / N[(y * N[(z / x$95$m), $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 < 6.4999999999999994e-61

   1. Initial program 92.3%

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

      1. clear-num N/A

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

      2. associate-/r* N/A

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

      3. clear-num N/A

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

      4. associate-/l/ N/A

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

      5. remove-double-div N/A

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

      6. div-inv N/A

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

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

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

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

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

      9. div-inv N/A

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

      10. remove-double-div N/A

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

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

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

      12. /-lowering-/.f64 93.2

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

   4. Applied egg-rr 93.2%

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

   if 6.4999999999999994e-61 < 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 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6.5 \cdot 10^{-61}:\\ \;\;\;\;\frac{\sin y}{y \cdot \frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{\sin y}{y}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 0.9999999995:\\ \;\;\;\;\frac{\sin y}{y \cdot \frac{z}{x\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z}\\ \end{array} \end{array} \]

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) 0.9999999995) (/ (sin y) (* y (/ z x_m))) (/ 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) <= 0.9999999995) {
		tmp = sin(y) / (y * (z / x_m));
	} 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) <= 0.9999999995d0) then
        tmp = sin(y) / (y * (z / x_m))
    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) <= 0.9999999995) {
		tmp = Math.sin(y) / (y * (z / x_m));
	} 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) <= 0.9999999995:
		tmp = math.sin(y) / (y * (z / x_m))
	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) <= 0.9999999995)
		tmp = Float64(sin(y) / Float64(y * Float64(z / x_m)));
	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) <= 0.9999999995)
		tmp = sin(y) / (y * (z / x_m));
	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], 0.9999999995], N[(N[Sin[y], $MachinePrecision] / N[(y * N[(z / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 y) y) < 0.99999999949999996

   1. Initial program 90.0%

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

      1. clear-num N/A

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

      2. associate-/r* N/A

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

      3. clear-num N/A

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

      4. associate-/l/ N/A

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

      5. remove-double-div N/A

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

      6. div-inv N/A

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

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

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

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

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

      9. div-inv N/A

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

      10. remove-double-div N/A

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

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

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

      12. /-lowering-/.f64 94.5

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

   4. Applied egg-rr 94.5%

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

   if 0.99999999949999996 < (/.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. /-lowering-/.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 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 0.9999999995:\\ \;\;\;\;\frac{\sin y}{y \cdot \frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 0.9999999995:\\ \;\;\;\;\frac{\sin y}{z} \cdot \frac{x\_m}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z}\\ \end{array} \end{array} \]

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) 0.9999999995) (* (/ (sin y) z) (/ x_m 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) <= 0.9999999995) {
		tmp = (sin(y) / z) * (x_m / 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) <= 0.9999999995d0) then
        tmp = (sin(y) / z) * (x_m / 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) <= 0.9999999995) {
		tmp = (Math.sin(y) / z) * (x_m / 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) <= 0.9999999995:
		tmp = (math.sin(y) / z) * (x_m / 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) <= 0.9999999995)
		tmp = Float64(Float64(sin(y) / z) * Float64(x_m / 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) <= 0.9999999995)
		tmp = (sin(y) / z) * (x_m / 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], 0.9999999995], N[(N[(N[Sin[y], $MachinePrecision] / z), $MachinePrecision] * N[(x$95$m / y), $MachinePrecision]), $MachinePrecision], N[(x$95$m / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 y) y) < 0.99999999949999996

   1. Initial program 90.0%

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

      1. *-commutative N/A

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

      2. div-inv N/A

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

      3. associate-*l* N/A

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

      4. associate-*l/ N/A

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

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

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

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

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

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

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

      8. associate-*l/ N/A

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

      9. *-lft-identity N/A

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

      10. /-lowering-/.f64 90.1

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

   4. Applied egg-rr 90.1%

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

   if 0.99999999949999996 < (/.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. /-lowering-/.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. Add Preprocessing

Alternative 4: 95.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 0.9999999995:\\ \;\;\;\;\frac{x\_m \cdot \sin y}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z}\\ \end{array} \end{array} \]

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) 0.9999999995) (/ (* x_m (sin y)) (* y z)) (/ 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) <= 0.9999999995) {
		tmp = (x_m * sin(y)) / (y * z);
	} 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) <= 0.9999999995d0) then
        tmp = (x_m * sin(y)) / (y * z)
    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) <= 0.9999999995) {
		tmp = (x_m * Math.sin(y)) / (y * z);
	} 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) <= 0.9999999995:
		tmp = (x_m * math.sin(y)) / (y * z)
	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) <= 0.9999999995)
		tmp = Float64(Float64(x_m * sin(y)) / Float64(y * z));
	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) <= 0.9999999995)
		tmp = (x_m * sin(y)) / (y * z);
	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], 0.9999999995], N[(N[(x$95$m * N[Sin[y], $MachinePrecision]), $MachinePrecision] / N[(y * z), $MachinePrecision]), $MachinePrecision], N[(x$95$m / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 y) y) < 0.99999999949999996

   1. Initial program 90.0%

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

      1. associate-*r/ N/A

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

      2. associate-/l/ N/A

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

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

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

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

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 88.5

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

   4. Applied egg-rr 88.5%

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

   if 0.99999999949999996 < (/.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. /-lowering-/.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. Add Preprocessing

Alternative 5: 95.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 0.9999999995:\\ \;\;\;\;\sin y \cdot \frac{x\_m}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z}\\ \end{array} \end{array} \]

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) 0.9999999995) (* (sin y) (/ x_m (* y z))) (/ 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) <= 0.9999999995) {
		tmp = sin(y) * (x_m / (y * z));
	} 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) <= 0.9999999995d0) then
        tmp = sin(y) * (x_m / (y * z))
    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) <= 0.9999999995) {
		tmp = Math.sin(y) * (x_m / (y * z));
	} 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) <= 0.9999999995:
		tmp = math.sin(y) * (x_m / (y * z))
	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) <= 0.9999999995)
		tmp = Float64(sin(y) * Float64(x_m / Float64(y * z)));
	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) <= 0.9999999995)
		tmp = sin(y) * (x_m / (y * z));
	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], 0.9999999995], N[(N[Sin[y], $MachinePrecision] * N[(x$95$m / N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 y) y) < 0.99999999949999996

   1. Initial program 90.0%

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

      1. *-commutative N/A

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

      2. div-inv N/A

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

      3. associate-*l* N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

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

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

      7. associate-*l/ N/A

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

      8. *-lft-identity N/A

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

      9. associate-/l/ N/A

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

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

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

      11. *-commutative N/A

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

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

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

      13. sin-lowering-sin.f64 88.4

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

   4. Applied egg-rr 88.4%

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

   if 0.99999999949999996 < (/.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. /-lowering-/.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 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 0.9999999995:\\ \;\;\;\;\sin y \cdot \frac{x}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 66.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 10^{-5}:\\ \;\;\;\;y \cdot \frac{\frac{1}{z}}{\frac{y}{x\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m + y \cdot \left(\mathsf{fma}\left(y, y \cdot 0.008333333333333333, -0.16666666666666666\right) \cdot \left(x\_m \cdot y\right)\right)}{z}\\ \end{array} \end{array} \]

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) 1e-5)
    (* y (/ (/ 1.0 z) (/ y x_m)))
    (/
     (+
      x_m
      (*
       y
       (* (fma y (* y 0.008333333333333333) -0.16666666666666666) (* x_m 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 ((sin(y) / y) <= 1e-5) {
		tmp = y * ((1.0 / z) / (y / x_m));
	} else {
		tmp = (x_m + (y * (fma(y, (y * 0.008333333333333333), -0.16666666666666666) * (x_m * 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 (Float64(sin(y) / y) <= 1e-5)
		tmp = Float64(y * Float64(Float64(1.0 / z) / Float64(y / x_m)));
	else
		tmp = Float64(Float64(x_m + Float64(y * Float64(fma(y, Float64(y * 0.008333333333333333), -0.16666666666666666) * Float64(x_m * y)))) / z);
	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[N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], 1e-5], N[(y * N[(N[(1.0 / z), $MachinePrecision] / N[(y / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m + N[(y * N[(N[(y * N[(y * 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(x$95$m * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 y) y) < 1.00000000000000008e-5

   1. Initial program 89.9%

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

      1. *-commutative N/A

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

      2. div-inv N/A

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

      3. associate-*l* N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

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

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

      7. associate-*l/ N/A

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

      8. *-lft-identity N/A

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

      9. associate-/l/ N/A

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

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

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

      11. *-commutative N/A

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

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

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

      13. sin-lowering-sin.f64 88.2

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

   4. Applied egg-rr 88.2%

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

   5. Taylor expanded in y around 0

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

   7. Simplified 22.2%

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

      1. associate-/r* N/A

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

      2. div-inv N/A

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

      3. clear-num N/A

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

      4. associate-*l/ N/A

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

      5. div-inv N/A

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

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

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

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

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

      8. /-lowering-/.f64 22.6

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

   9. Applied egg-rr 22.6%

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

   if 1.00000000000000008e-5 < (/.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

      \[\leadsto \color{blue}{{y}^{2} \cdot \left(\frac{-1}{6} \cdot \frac{x}{z} + \frac{1}{120} \cdot \frac{x \cdot {y}^{2}}{z}\right) + \frac{x}{z}} \]

   5. Simplified 99.3%

      \[\leadsto \color{blue}{\frac{x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.008333333333333333, -0.16666666666666666\right), 1\right)}{z}} \]
   6. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \frac{\color{blue}{\left(\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{120} + \frac{-1}{6}\right)\right) \cdot x + 1 \cdot x}}{z} \]

      2. *-lft-identity N/A

         \[\leadsto \frac{\left(\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{120} + \frac{-1}{6}\right)\right) \cdot x + \color{blue}{x}}{z} \]

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

         \[\leadsto \frac{\color{blue}{\left(\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{120} + \frac{-1}{6}\right)\right) \cdot x + x}}{z} \]

      4. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(y \cdot \left(y \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{120} + \frac{-1}{6}\right)\right)\right)} \cdot x + x}{z} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(\left(y \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{120} + \frac{-1}{6}\right)\right) \cdot x\right)} + x}{z} \]

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

         \[\leadsto \frac{\color{blue}{y \cdot \left(\left(y \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{120} + \frac{-1}{6}\right)\right) \cdot x\right)} + x}{z} \]

      7. *-commutative N/A

         \[\leadsto \frac{y \cdot \left(\color{blue}{\left(\left(\left(y \cdot y\right) \cdot \frac{1}{120} + \frac{-1}{6}\right) \cdot y\right)} \cdot x\right) + x}{z} \]

      8. associate-*l* N/A

         \[\leadsto \frac{y \cdot \color{blue}{\left(\left(\left(y \cdot y\right) \cdot \frac{1}{120} + \frac{-1}{6}\right) \cdot \left(y \cdot x\right)\right)} + x}{z} \]

      9. *-commutative N/A

         \[\leadsto \frac{y \cdot \left(\left(\left(y \cdot y\right) \cdot \frac{1}{120} + \frac{-1}{6}\right) \cdot \color{blue}{\left(x \cdot y\right)}\right) + x}{z} \]

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

          \[\leadsto \frac{y \cdot \color{blue}{\left(\left(\left(y \cdot y\right) \cdot \frac{1}{120} + \frac{-1}{6}\right) \cdot \left(x \cdot y\right)\right)} + x}{z} \]

      11. associate-*l* N/A

          \[\leadsto \frac{y \cdot \left(\left(\color{blue}{y \cdot \left(y \cdot \frac{1}{120}\right)} + \frac{-1}{6}\right) \cdot \left(x \cdot y\right)\right) + x}{z} \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{y \cdot \left(\color{blue}{\mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{-1}{6}\right)} \cdot \left(x \cdot y\right)\right) + x}{z} \]

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

          \[\leadsto \frac{y \cdot \left(\mathsf{fma}\left(y, \color{blue}{y \cdot \frac{1}{120}}, \frac{-1}{6}\right) \cdot \left(x \cdot y\right)\right) + x}{z} \]

      14. *-commutative N/A

          \[\leadsto \frac{y \cdot \left(\mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{-1}{6}\right) \cdot \color{blue}{\left(y \cdot x\right)}\right) + x}{z} \]

      15. *-lowering-*.f64 99.3

          \[\leadsto \frac{y \cdot \left(\mathsf{fma}\left(y, y \cdot 0.008333333333333333, -0.16666666666666666\right) \cdot \color{blue}{\left(y \cdot x\right)}\right) + x}{z} \]

   7. Applied egg-rr 99.3%

      \[\leadsto \frac{\color{blue}{y \cdot \left(\mathsf{fma}\left(y, y \cdot 0.008333333333333333, -0.16666666666666666\right) \cdot \left(y \cdot x\right)\right) + x}}{z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 10^{-5}:\\ \;\;\;\;y \cdot \frac{\frac{1}{z}}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y \cdot \left(\mathsf{fma}\left(y, y \cdot 0.008333333333333333, -0.16666666666666666\right) \cdot \left(x \cdot y\right)\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 66.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 10^{-5}:\\ \;\;\;\;y \cdot \frac{\frac{1}{z}}{\frac{y}{x\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.008333333333333333, -0.16666666666666666\right), 1\right)}{z}\\ \end{array} \end{array} \]

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) 1e-5)
    (* y (/ (/ 1.0 z) (/ y x_m)))
    (/
     (*
      x_m
      (fma
       (* y y)
       (fma (* y y) 0.008333333333333333 -0.16666666666666666)
       1.0))
     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 tmp;
	if ((sin(y) / y) <= 1e-5) {
		tmp = y * ((1.0 / z) / (y / x_m));
	} else {
		tmp = (x_m * fma((y * y), fma((y * y), 0.008333333333333333, -0.16666666666666666), 1.0)) / z;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (Float64(sin(y) / y) <= 1e-5)
		tmp = Float64(y * Float64(Float64(1.0 / z) / Float64(y / x_m)));
	else
		tmp = Float64(Float64(x_m * fma(Float64(y * y), fma(Float64(y * y), 0.008333333333333333, -0.16666666666666666), 1.0)) / z);
	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[N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], 1e-5], N[(y * N[(N[(1.0 / z), $MachinePrecision] / N[(y / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 y) y) < 1.00000000000000008e-5

   1. Initial program 89.9%

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

      1. *-commutative N/A

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

      2. div-inv N/A

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

      3. associate-*l* N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

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

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

      7. associate-*l/ N/A

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

      8. *-lft-identity N/A

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

      9. associate-/l/ N/A

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

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

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

      11. *-commutative N/A

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

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

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

      13. sin-lowering-sin.f64 88.2

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

   4. Applied egg-rr 88.2%

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

   5. Taylor expanded in y around 0

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

   7. Simplified 22.2%

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

      1. associate-/r* N/A

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

      2. div-inv N/A

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

      3. clear-num N/A

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

      4. associate-*l/ N/A

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

      5. div-inv N/A

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

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

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

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

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

      8. /-lowering-/.f64 22.6

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

   9. Applied egg-rr 22.6%

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

   if 1.00000000000000008e-5 < (/.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

      \[\leadsto \color{blue}{{y}^{2} \cdot \left(\frac{-1}{6} \cdot \frac{x}{z} + \frac{1}{120} \cdot \frac{x \cdot {y}^{2}}{z}\right) + \frac{x}{z}} \]

   5. Simplified 99.3%

      \[\leadsto \color{blue}{\frac{x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.008333333333333333, -0.16666666666666666\right), 1\right)}{z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 10^{-5}:\\ \;\;\;\;y \cdot \frac{\frac{1}{z}}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.008333333333333333, -0.16666666666666666\right), 1\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 66.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 0.02:\\ \;\;\;\;y \cdot \frac{\frac{1}{z}}{\frac{y}{x\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right)\\ \end{array} \end{array} \]

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 (<= (/ (sin y) y) 0.02)
    (* y (/ (/ 1.0 z) (/ y x_m)))
    (* (/ 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) {
	double tmp;
	if ((sin(y) / y) <= 0.02) {
		tmp = y * ((1.0 / z) / (y / x_m));
	} else {
		tmp = (x_m / z) * fma(y, (y * -0.16666666666666666), 1.0);
	}
	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) <= 0.02)
		tmp = Float64(y * Float64(Float64(1.0 / z) / Float64(y / x_m)));
	else
		tmp = Float64(Float64(x_m / z) * fma(y, Float64(y * -0.16666666666666666), 1.0));
	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[N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], 0.02], N[(y * N[(N[(1.0 / z), $MachinePrecision] / N[(y / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / z), $MachinePrecision] * N[(y * N[(y * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 y) y) < 0.0200000000000000004

   1. Initial program 90.0%

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

      1. *-commutative N/A

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

      2. div-inv N/A

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

      3. associate-*l* N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

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

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

      7. associate-*l/ N/A

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

      8. *-lft-identity N/A

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

      9. associate-/l/ N/A

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

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

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

      11. *-commutative N/A

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

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

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

      13. sin-lowering-sin.f64 88.3

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

   4. Applied egg-rr 88.3%

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

   5. Taylor expanded in y around 0

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

   7. Simplified 22.2%

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

      1. associate-/r* N/A

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

      2. div-inv N/A

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

      3. clear-num N/A

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

      4. associate-*l/ N/A

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

      5. div-inv N/A

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

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

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

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

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

      8. /-lowering-/.f64 22.5

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

   9. Applied egg-rr 22.5%

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

   if 0.0200000000000000004 < (/.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

      \[\leadsto \color{blue}{{y}^{2} \cdot \left(\frac{-1}{6} \cdot \frac{x}{z} + \frac{1}{120} \cdot \frac{x \cdot {y}^{2}}{z}\right) + \frac{x}{z}} \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.008333333333333333, -0.16666666666666666\right), 1\right)}{z}} \]

   6. Taylor expanded in y around 0

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

   8. Simplified 100.0%

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

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

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

      9. /-lowering-/.f64 100.0

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

   10. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right) \cdot \frac{x}{z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 0.02:\\ \;\;\;\;y \cdot \frac{\frac{1}{z}}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 60.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{x\_m \cdot \frac{\sin y}{y}}{z} \leq 2 \cdot 10^{-273}:\\ \;\;\;\;y \cdot \frac{x\_m}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z}\\ \end{array} \end{array} \]

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) 2e-273)
    (* y (/ x_m (* y z)))
    (/ 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) <= 2e-273) {
		tmp = y * (x_m / (y * z));
	} 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) <= 2d-273) then
        tmp = y * (x_m / (y * z))
    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) <= 2e-273) {
		tmp = y * (x_m / (y * z));
	} 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) <= 2e-273:
		tmp = y * (x_m / (y * z))
	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) <= 2e-273)
		tmp = Float64(y * Float64(x_m / Float64(y * z)));
	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) <= 2e-273)
		tmp = y * (x_m / (y * z));
	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], 2e-273], N[(y * N[(x$95$m / N[(y * z), $MachinePrecision]), $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) < 2e-273

   1. Initial program 92.2%

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

      1. *-commutative N/A

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

      2. div-inv N/A

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

      3. associate-*l* N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

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

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

      7. associate-*l/ N/A

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

      8. *-lft-identity N/A

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

      9. associate-/l/ N/A

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

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

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

      11. *-commutative N/A

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

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

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

      13. sin-lowering-sin.f64 85.0

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

   4. Applied egg-rr 85.0%

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

   5. Taylor expanded in y around 0

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

   7. Simplified 53.1%

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

   if 2e-273 < (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z)

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 60.7

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

   5. Simplified 60.7%

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

4. Final simplification 56.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \frac{\sin y}{y}}{z} \leq 2 \cdot 10^{-273}:\\ \;\;\;\;y \cdot \frac{x}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 66.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 0.02:\\ \;\;\;\;\frac{y}{y \cdot \frac{z}{x\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right)\\ \end{array} \end{array} \]

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 (<= (/ (sin y) y) 0.02)
    (/ y (* y (/ z x_m)))
    (* (/ 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) {
	double tmp;
	if ((sin(y) / y) <= 0.02) {
		tmp = y / (y * (z / x_m));
	} else {
		tmp = (x_m / z) * fma(y, (y * -0.16666666666666666), 1.0);
	}
	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) <= 0.02)
		tmp = Float64(y / Float64(y * Float64(z / x_m)));
	else
		tmp = Float64(Float64(x_m / z) * fma(y, Float64(y * -0.16666666666666666), 1.0));
	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[N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], 0.02], N[(y / N[(y * N[(z / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / z), $MachinePrecision] * N[(y * N[(y * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 y) y) < 0.0200000000000000004

   1. Initial program 90.0%

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

      1. clear-num N/A

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

      2. associate-/r* N/A

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

      3. clear-num N/A

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

      4. associate-/l/ N/A

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

      5. remove-double-div N/A

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

      6. div-inv N/A

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

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

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

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

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

      9. div-inv N/A

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

      10. remove-double-div N/A

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

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

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

      12. /-lowering-/.f64 94.4

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

   4. Applied egg-rr 94.4%

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

   5. Taylor expanded in y around 0

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

   7. Simplified 22.2%

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

   if 0.0200000000000000004 < (/.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

      \[\leadsto \color{blue}{{y}^{2} \cdot \left(\frac{-1}{6} \cdot \frac{x}{z} + \frac{1}{120} \cdot \frac{x \cdot {y}^{2}}{z}\right) + \frac{x}{z}} \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.008333333333333333, -0.16666666666666666\right), 1\right)}{z}} \]

   6. Taylor expanded in y around 0

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

   8. Simplified 100.0%

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

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

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

      9. /-lowering-/.f64 100.0

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

   10. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right) \cdot \frac{x}{z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 0.02:\\ \;\;\;\;\frac{y}{y \cdot \frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 66.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 0.02:\\ \;\;\;\;y \cdot \frac{\frac{x\_m}{y}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right)\\ \end{array} \end{array} \]

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 (<= (/ (sin y) y) 0.02)
    (* y (/ (/ x_m y) z))
    (* (/ 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) {
	double tmp;
	if ((sin(y) / y) <= 0.02) {
		tmp = y * ((x_m / y) / z);
	} else {
		tmp = (x_m / z) * fma(y, (y * -0.16666666666666666), 1.0);
	}
	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) <= 0.02)
		tmp = Float64(y * Float64(Float64(x_m / y) / z));
	else
		tmp = Float64(Float64(x_m / z) * fma(y, Float64(y * -0.16666666666666666), 1.0));
	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[N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], 0.02], N[(y * N[(N[(x$95$m / y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / z), $MachinePrecision] * N[(y * N[(y * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 y) y) < 0.0200000000000000004

   1. Initial program 90.0%

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

      1. *-commutative N/A

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

      2. div-inv N/A

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

      3. associate-*l* N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

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

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

      7. associate-*l/ N/A

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

      8. *-lft-identity N/A

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

      9. associate-/l/ N/A

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

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

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

      11. *-commutative N/A

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

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

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

      13. sin-lowering-sin.f64 88.3

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

   4. Applied egg-rr 88.3%

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

   5. Taylor expanded in y around 0

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

   7. Simplified 22.2%

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

      1. associate-/r* N/A

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

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

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

      3. /-lowering-/.f64 22.2

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

   9. Applied egg-rr 22.2%

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

   if 0.0200000000000000004 < (/.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

      \[\leadsto \color{blue}{{y}^{2} \cdot \left(\frac{-1}{6} \cdot \frac{x}{z} + \frac{1}{120} \cdot \frac{x \cdot {y}^{2}}{z}\right) + \frac{x}{z}} \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.008333333333333333, -0.16666666666666666\right), 1\right)}{z}} \]

   6. Taylor expanded in y around 0

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

   8. Simplified 100.0%

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

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

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

      9. /-lowering-/.f64 100.0

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

   10. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right) \cdot \frac{x}{z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 0.02:\\ \;\;\;\;y \cdot \frac{\frac{x}{y}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 66.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 4 \cdot 10^{-73}:\\ \;\;\;\;y \cdot \frac{\frac{x\_m}{y}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z}\\ \end{array} \end{array} \]

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) 4e-73) (* y (/ (/ x_m y) z)) (/ 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) <= 4e-73) {
		tmp = y * ((x_m / y) / z);
	} 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) <= 4d-73) then
        tmp = y * ((x_m / y) / z)
    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) <= 4e-73) {
		tmp = y * ((x_m / y) / z);
	} 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) <= 4e-73:
		tmp = y * ((x_m / y) / z)
	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) <= 4e-73)
		tmp = Float64(y * Float64(Float64(x_m / y) / z));
	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) <= 4e-73)
		tmp = y * ((x_m / y) / z);
	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], 4e-73], N[(y * N[(N[(x$95$m / y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(x$95$m / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 y) y) < 3.99999999999999999e-73

   1. Initial program 88.5%

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

      1. *-commutative N/A

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

      2. div-inv N/A

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

      3. associate-*l* N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

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

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

      7. associate-*l/ N/A

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

      8. *-lft-identity N/A

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

      9. associate-/l/ N/A

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

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

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

      11. *-commutative N/A

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

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

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

      13. sin-lowering-sin.f64 86.6

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

   4. Applied egg-rr 86.6%

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

   5. Taylor expanded in y around 0

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

   7. Simplified 21.7%

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

      1. associate-/r* N/A

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

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

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

      3. /-lowering-/.f64 21.8

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

   9. Applied egg-rr 21.8%

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

   if 3.99999999999999999e-73 < (/.f64 (sin.f64 y) y)

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 91.1

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

   5. Simplified 91.1%

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

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 4 \cdot 10^{-73}:\\ \;\;\;\;y \cdot \frac{\frac{x}{y}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 66.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \frac{-1}{\frac{z}{x\_m} \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, -0.16666666666666666\right), -1\right)} \end{array} \]

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
  (/
   -1.0
   (*
    (/ z x_m)
    (fma
     (* y y)
     (fma y (* y 0.008333333333333333) -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 * (-1.0 / ((z / x_m) * fma((y * y), fma(y, (y * 0.008333333333333333), -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(-1.0 / Float64(Float64(z / x_m) * fma(Float64(y * y), fma(y, Float64(y * 0.008333333333333333), -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[(-1.0 / N[(N[(z / x$95$m), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.0%

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

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{{y}^{2} \cdot \left(\frac{-1}{6} \cdot \frac{x}{z} + \frac{1}{120} \cdot \frac{x \cdot {y}^{2}}{z}\right) + \frac{x}{z}} \]

5. Simplified 53.0%

   \[\leadsto \color{blue}{\frac{x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.008333333333333333, -0.16666666666666666\right), 1\right)}{z}} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\left(\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{120} + \frac{-1}{6}\right) + 1\right) \cdot x}}{z} \]

   2. associate-/l* N/A

      \[\leadsto \color{blue}{\left(\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{120} + \frac{-1}{6}\right) + 1\right) \cdot \frac{x}{z}} \]

   3. flip-+ N/A

      \[\leadsto \color{blue}{\frac{\left(\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{120} + \frac{-1}{6}\right)\right) \cdot \left(\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{120} + \frac{-1}{6}\right)\right) - 1 \cdot 1}{\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{120} + \frac{-1}{6}\right) - 1}} \cdot \frac{x}{z} \]

   4. clear-num N/A

      \[\leadsto \frac{\left(\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{120} + \frac{-1}{6}\right)\right) \cdot \left(\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{120} + \frac{-1}{6}\right)\right) - 1 \cdot 1}{\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{120} + \frac{-1}{6}\right) - 1} \cdot \color{blue}{\frac{1}{\frac{z}{x}}} \]

   5. frac-times N/A

      \[\leadsto \color{blue}{\frac{\left(\left(\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{120} + \frac{-1}{6}\right)\right) \cdot \left(\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{120} + \frac{-1}{6}\right)\right) - 1 \cdot 1\right) \cdot 1}{\left(\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{120} + \frac{-1}{6}\right) - 1\right) \cdot \frac{z}{x}}} \]

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

      \[\leadsto \color{blue}{\frac{\left(\left(\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{120} + \frac{-1}{6}\right)\right) \cdot \left(\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{120} + \frac{-1}{6}\right)\right) - 1 \cdot 1\right) \cdot 1}{\left(\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{120} + \frac{-1}{6}\right) - 1\right) \cdot \frac{z}{x}}} \]

7. Applied egg-rr 52.2%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot 0.008333333333333333, -0.16666666666666666\right), \left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \mathsf{fma}\left(y, y \cdot 0.008333333333333333, -0.16666666666666666\right)\right), -1\right) \cdot 1}{\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, -0.16666666666666666\right), -1\right) \cdot \frac{z}{x}}} \]

8. Taylor expanded in y around 0

   \[\leadsto \frac{\color{blue}{-1}}{\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{-1}{6}\right), -1\right) \cdot \frac{z}{x}} \]
9. Step-by-step derivation

10. Simplified 61.2%

    \[\leadsto \frac{\color{blue}{-1}}{\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, -0.16666666666666666\right), -1\right) \cdot \frac{z}{x}} \]

11. Final simplification 61.2%

    \[\leadsto \frac{-1}{\frac{z}{x} \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, -0.16666666666666666\right), -1\right)} \]
12. Add Preprocessing

Alternative 14: 59.0% 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 95.0%

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

3. Taylor expanded in y around 0

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

   1. /-lowering-/.f64 56.7

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

5. Simplified 56.7%

   \[\leadsto \color{blue}{\frac{x}{z}} \]
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 2024198 
(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))