Linear.Quaternion:$ctanh from linear-1.19.1.3

Percentage Accurate: 96.4% → 99.7%

Time: 2.4s

Alternatives: 10

Speedup: 0.8×

Specification

Math

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

FPCore

(FPCore (x y z)
  :precision binary64
  :pre TRUE
  (/ (* 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)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x * (sin(y) / y)) / z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

ReFlow

f(x, y, z):
	x in [-inf, +inf],
	y in [-inf, +inf],
	z in [-inf, +inf]
code: THEORY
BEGIN
f(x, y, z: real): real =
	(x * ((sin(y)) / y)) / z
END code

Initial Program: 96.4% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z)
  :precision binary64
  :pre TRUE
  (/ (* 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)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x * (sin(y) / y)) / z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

ReFlow

f(x, y, z):
	x in [-inf, +inf],
	y in [-inf, +inf],
	z in [-inf, +inf]
code: THEORY
BEGIN
f(x, y, z: real): real =
	(x * ((sin(y)) / y)) / z
END code

Alternative 1: 99.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \frac{\sin y}{y}\\ \mathsf{copysign}\left(1, x\right) \cdot \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 2.0043961067331517 \cdot 10^{-110}:\\ \;\;\;\;\frac{\left|x\right|}{z} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|x\right| \cdot t\_0}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  :pre TRUE
  (let* ((t_0 (/ (sin y) y)))
  (*
   (copysign 1.0 x)
   (if (<= (fabs x) 2.0043961067331517e-110)
     (* (/ (fabs x) z) t_0)
     (/ (* (fabs x) t_0) z)))))

C

double code(double x, double y, double z) {
	double t_0 = sin(y) / y;
	double tmp;
	if (fabs(x) <= 2.0043961067331517e-110) {
		tmp = (fabs(x) / z) * t_0;
	} else {
		tmp = (fabs(x) * t_0) / z;
	}
	return copysign(1.0, x) * tmp;
}

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.sin(y) / y;
	double tmp;
	if (Math.abs(x) <= 2.0043961067331517e-110) {
		tmp = (Math.abs(x) / z) * t_0;
	} else {
		tmp = (Math.abs(x) * t_0) / z;
	}
	return Math.copySign(1.0, x) * tmp;
}

Python

def code(x, y, z):
	t_0 = math.sin(y) / y
	tmp = 0
	if math.fabs(x) <= 2.0043961067331517e-110:
		tmp = (math.fabs(x) / z) * t_0
	else:
		tmp = (math.fabs(x) * t_0) / z
	return math.copysign(1.0, x) * tmp

Julia

function code(x, y, z)
	t_0 = Float64(sin(y) / y)
	tmp = 0.0
	if (abs(x) <= 2.0043961067331517e-110)
		tmp = Float64(Float64(abs(x) / z) * t_0);
	else
		tmp = Float64(Float64(abs(x) * t_0) / z);
	end
	return Float64(copysign(1.0, x) * tmp)
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = sin(y) / y;
	tmp = 0.0;
	if (abs(x) <= 2.0043961067331517e-110)
		tmp = (abs(x) / z) * t_0;
	else
		tmp = (abs(x) * t_0) / z;
	end
	tmp_2 = (sign(x) * abs(1.0)) * tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[x], $MachinePrecision], 2.0043961067331517e-110], N[(N[(N[Abs[x], $MachinePrecision] / z), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(N[Abs[x], $MachinePrecision] * t$95$0), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.0043961067331517e-110

   1. Initial program 96.4%

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

   3. Applied rewrites 96.0%

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

   if 2.0043961067331517e-110 < x

   1. Initial program 96.4%

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

Alternative 2: 99.3% accurate, 0.8× speedup

Math

\[\mathsf{copysign}\left(1, x\right) \cdot \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 2.0273008920664322 \cdot 10^{+40}:\\ \;\;\;\;\frac{\left|x\right|}{z} \cdot \frac{\sin y}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left|x\right| \cdot \sin y}{y}}{z}\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  :pre TRUE
  (*
 (copysign 1.0 x)
 (if (<= (fabs x) 2.0273008920664322e+40)
   (* (/ (fabs x) z) (/ (sin y) y))
   (/ (/ (* (fabs x) (sin y)) y) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (fabs(x) <= 2.0273008920664322e+40) {
		tmp = (fabs(x) / z) * (sin(y) / y);
	} else {
		tmp = ((fabs(x) * sin(y)) / y) / z;
	}
	return copysign(1.0, x) * tmp;
}

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (Math.abs(x) <= 2.0273008920664322e+40) {
		tmp = (Math.abs(x) / z) * (Math.sin(y) / y);
	} else {
		tmp = ((Math.abs(x) * Math.sin(y)) / y) / z;
	}
	return Math.copySign(1.0, x) * tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if math.fabs(x) <= 2.0273008920664322e+40:
		tmp = (math.fabs(x) / z) * (math.sin(y) / y)
	else:
		tmp = ((math.fabs(x) * math.sin(y)) / y) / z
	return math.copysign(1.0, x) * tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (abs(x) <= 2.0273008920664322e+40)
		tmp = Float64(Float64(abs(x) / z) * Float64(sin(y) / y));
	else
		tmp = Float64(Float64(Float64(abs(x) * sin(y)) / y) / z);
	end
	return Float64(copysign(1.0, x) * tmp)
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (abs(x) <= 2.0273008920664322e+40)
		tmp = (abs(x) / z) * (sin(y) / y);
	else
		tmp = ((abs(x) * sin(y)) / y) / z;
	end
	tmp_2 = (sign(x) * abs(1.0)) * tmp;
end

Wolfram

code[x_, y_, z_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[x], $MachinePrecision], 2.0273008920664322e+40], N[(N[(N[Abs[x], $MachinePrecision] / z), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Abs[x], $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2.0273008920664322e40

   1. Initial program 96.4%

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

   3. Applied rewrites 96.0%

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

   if 2.0273008920664322e40 < x

   1. Initial program 96.4%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 88.7%

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

Alternative 3: 99.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \frac{\sin y}{y}\\ \mathsf{copysign}\left(1, z\right) \cdot \begin{array}{l} \mathbf{if}\;\left|z\right| \leq 2.2705743560222577 \cdot 10^{-89}:\\ \;\;\;\;x \cdot \frac{t\_0}{\left|z\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left|z\right|} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  :pre TRUE
  (let* ((t_0 (/ (sin y) y)))
  (*
   (copysign 1.0 z)
   (if (<= (fabs z) 2.2705743560222577e-89)
     (* x (/ t_0 (fabs z)))
     (* (/ x (fabs z)) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = sin(y) / y;
	double tmp;
	if (fabs(z) <= 2.2705743560222577e-89) {
		tmp = x * (t_0 / fabs(z));
	} else {
		tmp = (x / fabs(z)) * t_0;
	}
	return copysign(1.0, z) * tmp;
}

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.sin(y) / y;
	double tmp;
	if (Math.abs(z) <= 2.2705743560222577e-89) {
		tmp = x * (t_0 / Math.abs(z));
	} else {
		tmp = (x / Math.abs(z)) * t_0;
	}
	return Math.copySign(1.0, z) * tmp;
}

Python

def code(x, y, z):
	t_0 = math.sin(y) / y
	tmp = 0
	if math.fabs(z) <= 2.2705743560222577e-89:
		tmp = x * (t_0 / math.fabs(z))
	else:
		tmp = (x / math.fabs(z)) * t_0
	return math.copysign(1.0, z) * tmp

Julia

function code(x, y, z)
	t_0 = Float64(sin(y) / y)
	tmp = 0.0
	if (abs(z) <= 2.2705743560222577e-89)
		tmp = Float64(x * Float64(t_0 / abs(z)));
	else
		tmp = Float64(Float64(x / abs(z)) * t_0);
	end
	return Float64(copysign(1.0, z) * tmp)
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = sin(y) / y;
	tmp = 0.0;
	if (abs(z) <= 2.2705743560222577e-89)
		tmp = x * (t_0 / abs(z));
	else
		tmp = (x / abs(z)) * t_0;
	end
	tmp_2 = (sign(z) * abs(1.0)) * tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[z], $MachinePrecision], 2.2705743560222577e-89], N[(x * N[(t$95$0 / N[Abs[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[Abs[z], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 2.2705743560222577e-89

   1. Initial program 96.4%

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

   3. Applied rewrites 88.3%

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

   5. Applied rewrites 96.2%

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

   if 2.2705743560222577e-89 < z

   1. Initial program 96.4%

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

   3. Applied rewrites 96.0%

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

Alternative 4: 96.2% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z)
  :precision binary64
  :pre TRUE
  (* 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)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x * ((sin(y) / y) / z)
end function

Java

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

Python

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

Julia

function code(x, y, z)
	return Float64(x * Float64(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[(x * N[(N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]

ReFlow

f(x, y, z):
	x in [-inf, +inf],
	y in [-inf, +inf],
	z in [-inf, +inf]
code: THEORY
BEGIN
f(x, y, z: real): real =
	x * (((sin(y)) / y) / z)
END code

Derivation

1. Initial program 96.4%

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

3. Applied rewrites 88.3%

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

5. Applied rewrites 96.2%

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

Alternative 5: 96.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 0.999998:\\ \;\;\;\;x \cdot \frac{\sin y}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(1 + -0.16666666666666666 \cdot {y}^{2}\right)}{z}\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  :pre TRUE
  (if (<= (/ (sin y) y) 0.999998)
  (* x (/ (sin y) (* z y)))
  (/ (* x (+ 1.0 (* -0.16666666666666666 (pow y 2.0)))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((sin(y) / y) <= 0.999998) {
		tmp = x * (sin(y) / (z * y));
	} else {
		tmp = (x * (1.0 + (-0.16666666666666666 * pow(y, 2.0)))) / z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((sin(y) / y) <= 0.999998d0) then
        tmp = x * (sin(y) / (z * y))
    else
        tmp = (x * (1.0d0 + ((-0.16666666666666666d0) * (y ** 2.0d0)))) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((Math.sin(y) / y) <= 0.999998) {
		tmp = x * (Math.sin(y) / (z * y));
	} else {
		tmp = (x * (1.0 + (-0.16666666666666666 * Math.pow(y, 2.0)))) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (math.sin(y) / y) <= 0.999998:
		tmp = x * (math.sin(y) / (z * y))
	else:
		tmp = (x * (1.0 + (-0.16666666666666666 * math.pow(y, 2.0)))) / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(sin(y) / y) <= 0.999998)
		tmp = Float64(x * Float64(sin(y) / Float64(z * y)));
	else
		tmp = Float64(Float64(x * Float64(1.0 + Float64(-0.16666666666666666 * (y ^ 2.0)))) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((sin(y) / y) <= 0.999998)
		tmp = x * (sin(y) / (z * y));
	else
		tmp = (x * (1.0 + (-0.16666666666666666 * (y ^ 2.0)))) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], 0.999998], N[(x * N[(N[Sin[y], $MachinePrecision] / N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(1.0 + N[(-0.16666666666666666 * N[Power[y, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]

ReFlow

f(x, y, z):
	x in [-inf, +inf],
	y in [-inf, +inf],
	z in [-inf, +inf]
code: THEORY
BEGIN
f(x, y, z: real): real =
	LET tmp = IF (((sin(y)) / y) <= (99999800000000005351097343009314499795436859130859375e-53)) THEN (x * ((sin(y)) / (z * y))) ELSE ((x * ((1) + ((-1666666666666666574148081281236954964697360992431640625e-55) * (y ^ (2))))) / z) ENDIF IN
	tmp
END code

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 y) y) < 0.99999800000000005

   1. Initial program 96.4%

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

   3. Applied rewrites 88.3%

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

   if 0.99999800000000005 < (/.f64 (sin.f64 y) y)

   1. Initial program 96.4%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 53.3%

      \[\leadsto \frac{x \cdot \left(1 + -0.16666666666666666 \cdot {y}^{2}\right)}{z} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 96.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|y\right| \leq 0.00023416457863042044:\\ \;\;\;\;\frac{x \cdot \left(1 + -0.16666666666666666 \cdot {\left(\left|y\right|\right)}^{2}\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \sin \left(\left|y\right|\right)}{\left|y\right| \cdot z}\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  :pre TRUE
  (if (<= (fabs y) 0.00023416457863042044)
  (/ (* x (+ 1.0 (* -0.16666666666666666 (pow (fabs y) 2.0)))) z)
  (/ (* x (sin (fabs y))) (* (fabs y) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (fabs(y) <= 0.00023416457863042044) {
		tmp = (x * (1.0 + (-0.16666666666666666 * pow(fabs(y), 2.0)))) / z;
	} else {
		tmp = (x * sin(fabs(y))) / (fabs(y) * z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (abs(y) <= 0.00023416457863042044d0) then
        tmp = (x * (1.0d0 + ((-0.16666666666666666d0) * (abs(y) ** 2.0d0)))) / z
    else
        tmp = (x * sin(abs(y))) / (abs(y) * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (Math.abs(y) <= 0.00023416457863042044) {
		tmp = (x * (1.0 + (-0.16666666666666666 * Math.pow(Math.abs(y), 2.0)))) / z;
	} else {
		tmp = (x * Math.sin(Math.abs(y))) / (Math.abs(y) * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if math.fabs(y) <= 0.00023416457863042044:
		tmp = (x * (1.0 + (-0.16666666666666666 * math.pow(math.fabs(y), 2.0)))) / z
	else:
		tmp = (x * math.sin(math.fabs(y))) / (math.fabs(y) * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (abs(y) <= 0.00023416457863042044)
		tmp = Float64(Float64(x * Float64(1.0 + Float64(-0.16666666666666666 * (abs(y) ^ 2.0)))) / z);
	else
		tmp = Float64(Float64(x * sin(abs(y))) / Float64(abs(y) * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (abs(y) <= 0.00023416457863042044)
		tmp = (x * (1.0 + (-0.16666666666666666 * (abs(y) ^ 2.0)))) / z;
	else
		tmp = (x * sin(abs(y))) / (abs(y) * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[Abs[y], $MachinePrecision], 0.00023416457863042044], N[(N[(x * N[(1.0 + N[(-0.16666666666666666 * N[Power[N[Abs[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(x * N[Sin[N[Abs[y], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[Abs[y], $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]]

ReFlow

f(x, y, z):
	x in [-inf, +inf],
	y in [-inf, +inf],
	z in [-inf, +inf]
code: THEORY
BEGIN
f(x, y, z: real): real =
	LET tmp = IF ((abs(y)) <= (23416457863042043651725931940887903692782856523990631103515625e-65)) THEN ((x * ((1) + ((-1666666666666666574148081281236954964697360992431640625e-55) * ((abs(y)) ^ (2))))) / z) ELSE ((x * (sin((abs(y))))) / ((abs(y)) * z)) ENDIF IN
	tmp
END code

Derivation

1. Split input into 2 regimes

2. if y < 2.3416457863042044e-4

   1. Initial program 96.4%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 53.3%

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

   if 2.3416457863042044e-4 < y

   1. Initial program 96.4%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 84.3%

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

Alternative 7: 68.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \frac{\left|x\right|}{\left|z\right|}\\ t_1 := \frac{\left|x\right| \cdot \frac{\sin y}{y}}{\left|z\right|}\\ \mathsf{copysign}\left(1, x\right) \cdot \left(\mathsf{copysign}\left(1, z\right) \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-96}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right) \cdot t\_0\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-319}:\\ \;\;\;\;\frac{0}{\left|z\right|}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\right) \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  :pre TRUE
  (let* ((t_0 (/ (fabs x) (fabs z)))
       (t_1 (/ (* (fabs x) (/ (sin y) y)) (fabs z))))
  (*
   (copysign 1.0 x)
   (*
    (copysign 1.0 z)
    (if (<= t_1 -1e-96)
      (* (fma (* y y) -0.16666666666666666 1.0) t_0)
      (if (<= t_1 5e-319) (/ 0.0 (fabs z)) t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = fabs(x) / fabs(z);
	double t_1 = (fabs(x) * (sin(y) / y)) / fabs(z);
	double tmp;
	if (t_1 <= -1e-96) {
		tmp = fma((y * y), -0.16666666666666666, 1.0) * t_0;
	} else if (t_1 <= 5e-319) {
		tmp = 0.0 / fabs(z);
	} else {
		tmp = t_0;
	}
	return copysign(1.0, x) * (copysign(1.0, z) * tmp);
}

Julia

function code(x, y, z)
	t_0 = Float64(abs(x) / abs(z))
	t_1 = Float64(Float64(abs(x) * Float64(sin(y) / y)) / abs(z))
	tmp = 0.0
	if (t_1 <= -1e-96)
		tmp = Float64(fma(Float64(y * y), -0.16666666666666666, 1.0) * t_0);
	elseif (t_1 <= 5e-319)
		tmp = Float64(0.0 / abs(z));
	else
		tmp = t_0;
	end
	return Float64(copysign(1.0, x) * Float64(copysign(1.0, z) * tmp))
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Abs[x], $MachinePrecision] / N[Abs[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Abs[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / N[Abs[z], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$1, -1e-96], N[(N[(N[(y * y), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[t$95$1, 5e-319], N[(0.0 / N[Abs[z], $MachinePrecision]), $MachinePrecision], t$95$0]]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z) < -9.9999999999999991e-97

   1. Initial program 96.4%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 53.3%

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

   6. Applied rewrites 55.0%

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

   if -9.9999999999999991e-97 < (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z) < 4.9999937424779992e-319

   1. Initial program 96.4%

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

   2. Taylor expanded in y around 0

      \[\leadsto \frac{x}{z} \]
   3. Step-by-step derivation

   4. Applied rewrites 58.9%

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

   5. Taylor expanded in undef-var around zero

      \[\leadsto \frac{0}{z} \]
   6. Step-by-step derivation

   7. Applied rewrites 23.6%

      \[\leadsto \frac{0}{z} \]

   if 4.9999937424779992e-319 < (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z)

   1. Initial program 96.4%

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

   2. Taylor expanded in y around 0

      \[\leadsto \frac{x}{z} \]
   3. Step-by-step derivation

   4. Applied rewrites 58.9%

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

Alternative 8: 68.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \frac{\left|x\right| \cdot \frac{\sin y}{y}}{\left|z\right|}\\ \mathsf{copysign}\left(1, x\right) \cdot \left(\mathsf{copysign}\left(1, z\right) \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-96}:\\ \;\;\;\;\left|x\right| \cdot \frac{\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right)}{\left|z\right|}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-319}:\\ \;\;\;\;\frac{0}{\left|z\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|x\right|}{\left|z\right|}\\ \end{array}\right) \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  :pre TRUE
  (let* ((t_0 (/ (* (fabs x) (/ (sin y) y)) (fabs z))))
  (*
   (copysign 1.0 x)
   (*
    (copysign 1.0 z)
    (if (<= t_0 -1e-96)
      (* (fabs x) (/ (fma (* y y) -0.16666666666666666 1.0) (fabs z)))
      (if (<= t_0 5e-319) (/ 0.0 (fabs z)) (/ (fabs x) (fabs z))))))))

C

double code(double x, double y, double z) {
	double t_0 = (fabs(x) * (sin(y) / y)) / fabs(z);
	double tmp;
	if (t_0 <= -1e-96) {
		tmp = fabs(x) * (fma((y * y), -0.16666666666666666, 1.0) / fabs(z));
	} else if (t_0 <= 5e-319) {
		tmp = 0.0 / fabs(z);
	} else {
		tmp = fabs(x) / fabs(z);
	}
	return copysign(1.0, x) * (copysign(1.0, z) * tmp);
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(abs(x) * Float64(sin(y) / y)) / abs(z))
	tmp = 0.0
	if (t_0 <= -1e-96)
		tmp = Float64(abs(x) * Float64(fma(Float64(y * y), -0.16666666666666666, 1.0) / abs(z)));
	elseif (t_0 <= 5e-319)
		tmp = Float64(0.0 / abs(z));
	else
		tmp = Float64(abs(x) / abs(z));
	end
	return Float64(copysign(1.0, x) * Float64(copysign(1.0, z) * tmp))
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[Abs[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / N[Abs[z], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$0, -1e-96], N[(N[Abs[x], $MachinePrecision] * N[(N[(N[(y * y), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] / N[Abs[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e-319], N[(0.0 / N[Abs[z], $MachinePrecision]), $MachinePrecision], N[(N[Abs[x], $MachinePrecision] / N[Abs[z], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z) < -9.9999999999999991e-97

   1. Initial program 96.4%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 53.3%

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

   6. Applied rewrites 53.2%

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

   if -9.9999999999999991e-97 < (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z) < 4.9999937424779992e-319

   1. Initial program 96.4%

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

   2. Taylor expanded in y around 0

      \[\leadsto \frac{x}{z} \]
   3. Step-by-step derivation

   4. Applied rewrites 58.9%

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

   5. Taylor expanded in undef-var around zero

      \[\leadsto \frac{0}{z} \]
   6. Step-by-step derivation

   7. Applied rewrites 23.6%

      \[\leadsto \frac{0}{z} \]

   if 4.9999937424779992e-319 < (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z)

   1. Initial program 96.4%

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

   2. Taylor expanded in y around 0

      \[\leadsto \frac{x}{z} \]
   3. Step-by-step derivation

   4. Applied rewrites 58.9%

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

Alternative 9: 67.3% accurate, 0.7× speedup

Math

\[\mathsf{copysign}\left(1, x\right) \cdot \left(\mathsf{copysign}\left(1, z\right) \cdot \begin{array}{l} \mathbf{if}\;\frac{\left|x\right| \cdot \frac{\sin y}{y}}{\left|z\right|} \leq 5 \cdot 10^{-319}:\\ \;\;\;\;\frac{0}{\left|z\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|x\right|}{\left|z\right|}\\ \end{array}\right) \]

FPCore

(FPCore (x y z)
  :precision binary64
  :pre TRUE
  (*
 (copysign 1.0 x)
 (*
  (copysign 1.0 z)
  (if (<= (/ (* (fabs x) (/ (sin y) y)) (fabs z)) 5e-319)
    (/ 0.0 (fabs z))
    (/ (fabs x) (fabs z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (((fabs(x) * (sin(y) / y)) / fabs(z)) <= 5e-319) {
		tmp = 0.0 / fabs(z);
	} else {
		tmp = fabs(x) / fabs(z);
	}
	return copysign(1.0, x) * (copysign(1.0, z) * tmp);
}

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (((Math.abs(x) * (Math.sin(y) / y)) / Math.abs(z)) <= 5e-319) {
		tmp = 0.0 / Math.abs(z);
	} else {
		tmp = Math.abs(x) / Math.abs(z);
	}
	return Math.copySign(1.0, x) * (Math.copySign(1.0, z) * tmp);
}

Python

def code(x, y, z):
	tmp = 0
	if ((math.fabs(x) * (math.sin(y) / y)) / math.fabs(z)) <= 5e-319:
		tmp = 0.0 / math.fabs(z)
	else:
		tmp = math.fabs(x) / math.fabs(z)
	return math.copysign(1.0, x) * (math.copysign(1.0, z) * tmp)

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(abs(x) * Float64(sin(y) / y)) / abs(z)) <= 5e-319)
		tmp = Float64(0.0 / abs(z));
	else
		tmp = Float64(abs(x) / abs(z));
	end
	return Float64(copysign(1.0, x) * Float64(copysign(1.0, z) * tmp))
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (((abs(x) * (sin(y) / y)) / abs(z)) <= 5e-319)
		tmp = 0.0 / abs(z);
	else
		tmp = abs(x) / abs(z);
	end
	tmp_2 = (sign(x) * abs(1.0)) * ((sign(z) * abs(1.0)) * tmp);
end

Wolfram

code[x_, y_, z_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(N[(N[Abs[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / N[Abs[z], $MachinePrecision]), $MachinePrecision], 5e-319], N[(0.0 / N[Abs[z], $MachinePrecision]), $MachinePrecision], N[(N[Abs[x], $MachinePrecision] / N[Abs[z], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z) < 4.9999937424779992e-319

   1. Initial program 96.4%

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

   2. Taylor expanded in y around 0

      \[\leadsto \frac{x}{z} \]
   3. Step-by-step derivation

   4. Applied rewrites 58.9%

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

   5. Taylor expanded in undef-var around zero

      \[\leadsto \frac{0}{z} \]
   6. Step-by-step derivation

   7. Applied rewrites 23.6%

      \[\leadsto \frac{0}{z} \]

   if 4.9999937424779992e-319 < (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z)

   1. Initial program 96.4%

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

   2. Taylor expanded in y around 0

      \[\leadsto \frac{x}{z} \]
   3. Step-by-step derivation

   4. Applied rewrites 58.9%

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

Alternative 10: 58.9% accurate, 9.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

ReFlow

f(x, y, z):
	x in [-inf, +inf],
	y in [-inf, +inf],
	z in [-inf, +inf]
code: THEORY
BEGIN
f(x, y, z: real): real =
	x / z
END code

Derivation

1. Initial program 96.4%

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

2. Taylor expanded in y around 0

   \[\leadsto \frac{x}{z} \]
3. Step-by-step derivation

4. Applied rewrites 58.9%

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

Reproduce

herbie shell --seed 2026092 
(FPCore (x y z)
  :name "Linear.Quaternion:$ctanh from linear-1.19.1.3"
  :precision binary64
  (/ (* x (/ (sin y) y)) z))