Linear.Quaternion:$ctanh from linear-1.19.1.3

Percentage Accurate: 96.2% → 94.9%

Time: 7.7s

Alternatives: 11

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 96.2% 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: 94.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{x}{z}}{y} \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (/ (sin y) y) 2e-13) (* (/ (/ x z) y) (sin y)) (/ x z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((sin(y) / y) <= 2e-13) {
		tmp = ((x / z) / y) * sin(y);
	} else {
		tmp = x / z;
	}
	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) :: tmp
    if ((sin(y) / y) <= 2d-13) then
        tmp = ((x / z) / y) * sin(y)
    else
        tmp = x / 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) <= 2e-13) {
		tmp = ((x / z) / y) * Math.sin(y);
	} else {
		tmp = x / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (math.sin(y) / y) <= 2e-13:
		tmp = ((x / z) / y) * math.sin(y)
	else:
		tmp = x / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(sin(y) / y) <= 2e-13)
		tmp = Float64(Float64(Float64(x / z) / y) * sin(y));
	else
		tmp = Float64(x / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((sin(y) / y) <= 2e-13)
		tmp = ((x / z) / y) * sin(y);
	else
		tmp = x / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], 2e-13], N[(N[(N[(x / z), $MachinePrecision] / y), $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision], N[(x / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 y) y) < 2.0000000000000001e-13

   1. Initial program 95.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. div-inv N/A

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

      6. associate-*l* N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. div-inv N/A

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

      11. associate-*l/ N/A

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

      12. *-lft-identity N/A

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

      13. associate-*l/ N/A

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

      14. div-inv N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 95.5

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

   4. Applied rewrites 95.5%

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

   if 2.0000000000000001e-13 < (/.f64 (sin.f64 y) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

Alternative 2: 95.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{x}{y}}{z} \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (/ (sin y) y) 2e-13) (* (/ (/ x y) z) (sin y)) (/ x z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((sin(y) / y) <= 2e-13) {
		tmp = ((x / y) / z) * sin(y);
	} else {
		tmp = x / z;
	}
	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) :: tmp
    if ((sin(y) / y) <= 2d-13) then
        tmp = ((x / y) / z) * sin(y)
    else
        tmp = x / 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) <= 2e-13) {
		tmp = ((x / y) / z) * Math.sin(y);
	} else {
		tmp = x / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (math.sin(y) / y) <= 2e-13:
		tmp = ((x / y) / z) * math.sin(y)
	else:
		tmp = x / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(sin(y) / y) <= 2e-13)
		tmp = Float64(Float64(Float64(x / y) / z) * sin(y));
	else
		tmp = Float64(x / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((sin(y) / y) <= 2e-13)
		tmp = ((x / y) / z) * sin(y);
	else
		tmp = x / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], 2e-13], N[(N[(N[(x / y), $MachinePrecision] / z), $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision], N[(x / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 y) y) < 2.0000000000000001e-13

   1. Initial program 95.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. div-inv N/A

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

      6. associate-*l* N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. div-inv N/A

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

      11. associate-*l/ N/A

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

      12. *-lft-identity N/A

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

      13. associate-*l/ N/A

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

      14. div-inv N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 95.5

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

   4. Applied rewrites 95.5%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 95.0

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

   6. Applied rewrites 95.0%

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

   if 2.0000000000000001e-13 < (/.f64 (sin.f64 y) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

Alternative 3: 95.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\frac{\sin y}{z} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (/ (sin y) y) 2e-13) (* (/ (sin y) z) (/ x y)) (/ x z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((sin(y) / y) <= 2e-13) {
		tmp = (sin(y) / z) * (x / y);
	} else {
		tmp = x / z;
	}
	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) :: tmp
    if ((sin(y) / y) <= 2d-13) then
        tmp = (sin(y) / z) * (x / y)
    else
        tmp = x / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((Math.sin(y) / y) <= 2e-13) {
		tmp = (Math.sin(y) / z) * (x / y);
	} else {
		tmp = x / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (math.sin(y) / y) <= 2e-13:
		tmp = (math.sin(y) / z) * (x / y)
	else:
		tmp = x / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(sin(y) / y) <= 2e-13)
		tmp = Float64(Float64(sin(y) / z) * Float64(x / y));
	else
		tmp = Float64(x / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((sin(y) / y) <= 2e-13)
		tmp = (sin(y) / z) * (x / y);
	else
		tmp = x / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], 2e-13], N[(N[(N[Sin[y], $MachinePrecision] / z), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision], N[(x / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 y) y) < 2.0000000000000001e-13

   1. Initial program 95.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. div-inv N/A

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

      6. associate-*l* N/A

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

      7. associate-*l/ N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. associate-*l/ N/A

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

      11. *-lft-identity N/A

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

      12. lower-/.f64 94.9

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

   4. Applied rewrites 94.9%

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

   if 2.0000000000000001e-13 < (/.f64 (sin.f64 y) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

Alternative 4: 94.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\frac{x}{y \cdot z} \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (/ (sin y) y) 2e-13) (* (/ x (* y z)) (sin y)) (/ x z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((sin(y) / y) <= 2e-13) {
		tmp = (x / (y * z)) * sin(y);
	} else {
		tmp = x / z;
	}
	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) :: tmp
    if ((sin(y) / y) <= 2d-13) then
        tmp = (x / (y * z)) * sin(y)
    else
        tmp = x / 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) <= 2e-13) {
		tmp = (x / (y * z)) * Math.sin(y);
	} else {
		tmp = x / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (math.sin(y) / y) <= 2e-13:
		tmp = (x / (y * z)) * math.sin(y)
	else:
		tmp = x / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(sin(y) / y) <= 2e-13)
		tmp = Float64(Float64(x / Float64(y * z)) * sin(y));
	else
		tmp = Float64(x / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((sin(y) / y) <= 2e-13)
		tmp = (x / (y * z)) * sin(y);
	else
		tmp = x / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], 2e-13], N[(N[(x / N[(y * z), $MachinePrecision]), $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision], N[(x / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 y) y) < 2.0000000000000001e-13

   1. Initial program 95.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. div-inv N/A

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

      6. associate-*l* N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. div-inv N/A

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

      11. associate-*l/ N/A

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

      12. *-lft-identity N/A

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

      13. associate-*l/ N/A

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

      14. div-inv N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 95.5

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

   4. Applied rewrites 95.5%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lift-*.f64 N/A

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

      5. lower-/.f64 90.2

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 90.2

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

   6. Applied rewrites 90.2%

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

   if 2.0000000000000001e-13 < (/.f64 (sin.f64 y) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

Alternative 5: 55.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x \cdot \frac{\sin y}{y}}{z} \leq 0:\\ \;\;\;\;\frac{y \cdot x}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \end{array} \]

FPCore

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

C

double code(double x, double y, double z) {
	double tmp;
	if (((x * (sin(y) / y)) / z) <= 0.0) {
		tmp = (y * x) / (z * y);
	} else {
		tmp = x / z;
	}
	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) :: tmp
    if (((x * (sin(y) / y)) / z) <= 0.0d0) then
        tmp = (y * x) / (z * y)
    else
        tmp = x / z
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[(x * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], 0.0], N[(N[(y * x), $MachinePrecision] / N[(z * y), $MachinePrecision]), $MachinePrecision], N[(x / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 97.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 86.6

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

   4. Applied rewrites 86.6%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 57.9

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

   7. Applied rewrites 57.9%

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

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

   1. Initial program 99.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 58.1

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

   5. Applied rewrites 58.1%

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

Alternative 6: 62.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 10^{-32}:\\ \;\;\;\;\frac{\frac{x \cdot x}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (/ (sin y) y) 1e-32) (/ (/ (* x x) z) x) (/ x z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((sin(y) / y) <= 1e-32) {
		tmp = ((x * x) / z) / x;
	} else {
		tmp = x / z;
	}
	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) :: tmp
    if ((sin(y) / y) <= 1d-32) then
        tmp = ((x * x) / z) / x
    else
        tmp = x / 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) <= 1e-32) {
		tmp = ((x * x) / z) / x;
	} else {
		tmp = x / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (math.sin(y) / y) <= 1e-32:
		tmp = ((x * x) / z) / x
	else:
		tmp = x / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(sin(y) / y) <= 1e-32)
		tmp = Float64(Float64(Float64(x * x) / z) / x);
	else
		tmp = Float64(x / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((sin(y) / y) <= 1e-32)
		tmp = ((x * x) / z) / x;
	else
		tmp = x / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], 1e-32], N[(N[(N[(x * x), $MachinePrecision] / z), $MachinePrecision] / x), $MachinePrecision], N[(x / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 y) y) < 1.00000000000000006e-32

   1. Initial program 94.8%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 15.2

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

   5. Applied rewrites 15.2%

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

   7. Applied rewrites 15.2%

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

   9. Applied rewrites 25.8%

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

   11. Applied rewrites 25.8%

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

   if 1.00000000000000006e-32 < (/.f64 (sin.f64 y) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 97.5

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

   5. Applied rewrites 97.5%

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

Alternative 7: 66.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ {\left(\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right) \cdot \frac{z}{x}\right)}^{-1} \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 97.7%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 56.9

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

5. Applied rewrites 56.9%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. lower-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/r* N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 58.4

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

7. Applied rewrites 58.4%

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

8. Taylor expanded in y around 0

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

   1. associate-/l* N/A

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

   2. associate-*r* N/A

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

   3. distribute-lft1-in N/A

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

   4. lower-*.f64 N/A

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

   5. lower-fma.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 70.0

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

10. Applied rewrites 70.0%

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

11. Final simplification 70.0%

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

Alternative 8: 96.2% 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]

Derivation

1. Initial program 97.7%

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

Alternative 9: 60.1% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.2 \cdot 10^{+27}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot x}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 3.2e+27) (/ x z) (/ (* (/ x z) x) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 3.2e+27) {
		tmp = x / z;
	} else {
		tmp = ((x / z) * x) / x;
	}
	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) :: tmp
    if (y <= 3.2d+27) then
        tmp = x / z
    else
        tmp = ((x / z) * x) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 3.2e+27) {
		tmp = x / z;
	} else {
		tmp = ((x / z) * x) / x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 3.2e+27:
		tmp = x / z
	else:
		tmp = ((x / z) * x) / x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 3.2e+27)
		tmp = Float64(x / z);
	else
		tmp = Float64(Float64(Float64(x / z) * x) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 3.2e+27)
		tmp = x / z;
	else
		tmp = ((x / z) * x) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 3.2e+27], N[(x / z), $MachinePrecision], N[(N[(N[(x / z), $MachinePrecision] * x), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.20000000000000015e27

   1. Initial program 98.6%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 77.8

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

   5. Applied rewrites 77.8%

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

   if 3.20000000000000015e27 < y

   1. Initial program 95.2%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 13.0

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

   5. Applied rewrites 13.0%

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

   7. Applied rewrites 13.0%

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

   9. Applied rewrites 22.8%

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

   11. Applied rewrites 21.4%

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

Alternative 10: 59.2% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.5 \cdot 10^{+119}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot x}{z \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 4.5e+119) (/ x z) (/ (* x x) (* z x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 4.5e+119) {
		tmp = x / z;
	} else {
		tmp = (x * x) / (z * x);
	}
	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) :: tmp
    if (y <= 4.5d+119) then
        tmp = x / z
    else
        tmp = (x * x) / (z * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 4.5e+119) {
		tmp = x / z;
	} else {
		tmp = (x * x) / (z * x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 4.5e+119:
		tmp = x / z
	else:
		tmp = (x * x) / (z * x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 4.5e+119)
		tmp = Float64(x / z);
	else
		tmp = Float64(Float64(x * x) / Float64(z * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 4.5e+119)
		tmp = x / z;
	else
		tmp = (x * x) / (z * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 4.5e+119], N[(x / z), $MachinePrecision], N[(N[(x * x), $MachinePrecision] / N[(z * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.5000000000000002e119

   1. Initial program 98.3%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 72.0

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

   5. Applied rewrites 72.0%

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

   if 4.5000000000000002e119 < y

   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. lower-/.f64 10.1

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

   5. Applied rewrites 10.1%

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

   7. Applied rewrites 10.1%

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

   9. Applied rewrites 21.9%

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

   11. Applied rewrites 16.5%

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

Alternative 11: 58.0% accurate, 10.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.7%

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

3. Taylor expanded in y around 0

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

   1. lower-/.f64 61.8

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

5. Applied rewrites 61.8%

   \[\leadsto \color{blue}{\frac{x}{z}} \]
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 2024307 
(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))