Data.Histogram.Bin.BinF:$cfromIndex from histogram-fill-0.8.4.1

Percentage Accurate: 100.0% → 100.0%

Time: 4.8s

Alternatives: 7

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \left(\frac{x}{2} + y \cdot x\right) + z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ (+ (/ x 2.0) (* y x)) z))

C

double code(double x, double y, double z) {
	return ((x / 2.0) + (y * 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 / 2.0d0) + (y * x)) + z
end function

Java

public static double code(double x, double y, double z) {
	return ((x / 2.0) + (y * x)) + z;
}

Python

def code(x, y, z):
	return ((x / 2.0) + (y * x)) + z

Julia

function code(x, y, z)
	return Float64(Float64(Float64(x / 2.0) + Float64(y * x)) + z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((x / 2.0) + (y * x)) + z;
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(x / 2.0), $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\frac{x}{2} + y \cdot x\right) + z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ (+ (/ x 2.0) (* y x)) z))

C

double code(double x, double y, double z) {
	return ((x / 2.0) + (y * 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 / 2.0d0) + (y * x)) + z
end function

Java

public static double code(double x, double y, double z) {
	return ((x / 2.0) + (y * x)) + z;
}

Python

def code(x, y, z):
	return ((x / 2.0) + (y * x)) + z

Julia

function code(x, y, z)
	return Float64(Float64(Float64(x / 2.0) + Float64(y * x)) + z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((x / 2.0) + (y * x)) + z;
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(x / 2.0), $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\frac{x}{2} + x \cdot y\right) + z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ (+ (/ x 2.0) (* x y)) z))

C

double code(double x, double y, double z) {
	return ((x / 2.0) + (x * 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 / 2.0d0) + (x * y)) + z
end function

Java

public static double code(double x, double y, double z) {
	return ((x / 2.0) + (x * y)) + z;
}

Python

def code(x, y, z):
	return ((x / 2.0) + (x * y)) + z

Julia

function code(x, y, z)
	return Float64(Float64(Float64(x / 2.0) + Float64(x * y)) + z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((x / 2.0) + (x * y)) + z;
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(x / 2.0), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(\frac{x}{2} + y \cdot x\right) + z \]
2. Add Preprocessing

3. Final simplification 100.0%

   \[\leadsto \left(\frac{x}{2} + x \cdot y\right) + z \]
4. Add Preprocessing

Alternative 2: 60.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{+24}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{-111}:\\ \;\;\;\;z\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{-44}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{+18}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -8.5e+24)
   (* x y)
   (if (<= y 1.2e-111)
     z
     (if (<= y 8.5e-44) (* x 0.5) (if (<= y 1.4e+18) z (* x y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -8.5e+24) {
		tmp = x * y;
	} else if (y <= 1.2e-111) {
		tmp = z;
	} else if (y <= 8.5e-44) {
		tmp = x * 0.5;
	} else if (y <= 1.4e+18) {
		tmp = z;
	} else {
		tmp = x * y;
	}
	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 <= (-8.5d+24)) then
        tmp = x * y
    else if (y <= 1.2d-111) then
        tmp = z
    else if (y <= 8.5d-44) then
        tmp = x * 0.5d0
    else if (y <= 1.4d+18) then
        tmp = z
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -8.5e+24) {
		tmp = x * y;
	} else if (y <= 1.2e-111) {
		tmp = z;
	} else if (y <= 8.5e-44) {
		tmp = x * 0.5;
	} else if (y <= 1.4e+18) {
		tmp = z;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -8.5e+24:
		tmp = x * y
	elif y <= 1.2e-111:
		tmp = z
	elif y <= 8.5e-44:
		tmp = x * 0.5
	elif y <= 1.4e+18:
		tmp = z
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -8.5e+24)
		tmp = Float64(x * y);
	elseif (y <= 1.2e-111)
		tmp = z;
	elseif (y <= 8.5e-44)
		tmp = Float64(x * 0.5);
	elseif (y <= 1.4e+18)
		tmp = z;
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -8.5e+24)
		tmp = x * y;
	elseif (y <= 1.2e-111)
		tmp = z;
	elseif (y <= 8.5e-44)
		tmp = x * 0.5;
	elseif (y <= 1.4e+18)
		tmp = z;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -8.5e+24], N[(x * y), $MachinePrecision], If[LessEqual[y, 1.2e-111], z, If[LessEqual[y, 8.5e-44], N[(x * 0.5), $MachinePrecision], If[LessEqual[y, 1.4e+18], z, N[(x * y), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.49999999999999959e24 or 1.4e18 < y

   1. Initial program 100.0%

      \[\left(\frac{x}{2} + y \cdot x\right) + z \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 73.5%

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

   5. Simplified 73.5%

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

   if -8.49999999999999959e24 < y < 1.2e-111 or 8.5000000000000002e-44 < y < 1.4e18

   1. Initial program 100.0%

      \[\left(\frac{x}{2} + y \cdot x\right) + z \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

   5. Simplified 65.5%

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

   if 1.2e-111 < y < 8.5000000000000002e-44

   1. Initial program 100.0%

      \[\left(\frac{x}{2} + y \cdot x\right) + z \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

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

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

      2. +-lowering-+.f64 70.3%

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

   5. Simplified 70.3%

      \[\leadsto \color{blue}{x \cdot \left(0.5 + y\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot x} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto x \cdot \color{blue}{\frac{1}{2}} \]

      2. *-lowering-*.f64 70.3%

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{2}}\right) \]

   8. Simplified 70.3%

      \[\leadsto \color{blue}{x \cdot 0.5} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 97.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z + x \cdot y\\ \mathbf{if}\;y \leq -3.55 \cdot 10^{+21}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.16 \cdot 10^{-18}:\\ \;\;\;\;z + x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ z (* x y))))
   (if (<= y -3.55e+21) t_0 (if (<= y 1.16e-18) (+ z (* x 0.5)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = z + (x * y);
	double tmp;
	if (y <= -3.55e+21) {
		tmp = t_0;
	} else if (y <= 1.16e-18) {
		tmp = z + (x * 0.5);
	} else {
		tmp = t_0;
	}
	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) :: tmp
    t_0 = z + (x * y)
    if (y <= (-3.55d+21)) then
        tmp = t_0
    else if (y <= 1.16d-18) then
        tmp = z + (x * 0.5d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = z + (x * y);
	double tmp;
	if (y <= -3.55e+21) {
		tmp = t_0;
	} else if (y <= 1.16e-18) {
		tmp = z + (x * 0.5);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z + (x * y)
	tmp = 0
	if y <= -3.55e+21:
		tmp = t_0
	elif y <= 1.16e-18:
		tmp = z + (x * 0.5)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z + Float64(x * y))
	tmp = 0.0
	if (y <= -3.55e+21)
		tmp = t_0;
	elseif (y <= 1.16e-18)
		tmp = Float64(z + Float64(x * 0.5));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z + (x * y);
	tmp = 0.0;
	if (y <= -3.55e+21)
		tmp = t_0;
	elseif (y <= 1.16e-18)
		tmp = z + (x * 0.5);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z + N[(x * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -3.55e+21], t$95$0, If[LessEqual[y, 1.16e-18], N[(z + N[(x * 0.5), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.55e21 or 1.16e-18 < y

   1. Initial program 100.0%

      \[\left(\frac{x}{2} + y \cdot x\right) + z \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(x \cdot y\right)}, z\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 100.0%

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

   5. Simplified 100.0%

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

   if -3.55e21 < y < 1.16e-18

   1. Initial program 100.0%

      \[\left(\frac{x}{2} + y \cdot x\right) + z \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(x \cdot \frac{1}{2}\right), z\right) \]

      2. *-lowering-*.f64 99.6%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), z\right) \]

   5. Simplified 99.6%

      \[\leadsto \color{blue}{x \cdot 0.5} + z \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.55 \cdot 10^{+21}:\\ \;\;\;\;z + x \cdot y\\ \mathbf{elif}\;y \leq 1.16 \cdot 10^{-18}:\\ \;\;\;\;z + x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;z + x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.55 \cdot 10^{+22}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{+17}:\\ \;\;\;\;z + x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.55e+22)
   (* x y)
   (if (<= y 4.2e+17) (+ z (* x 0.5)) (* x (+ y 0.5)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.55e+22) {
		tmp = x * y;
	} else if (y <= 4.2e+17) {
		tmp = z + (x * 0.5);
	} else {
		tmp = x * (y + 0.5);
	}
	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 <= (-1.55d+22)) then
        tmp = x * y
    else if (y <= 4.2d+17) then
        tmp = z + (x * 0.5d0)
    else
        tmp = x * (y + 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.55e+22) {
		tmp = x * y;
	} else if (y <= 4.2e+17) {
		tmp = z + (x * 0.5);
	} else {
		tmp = x * (y + 0.5);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.55e+22:
		tmp = x * y
	elif y <= 4.2e+17:
		tmp = z + (x * 0.5)
	else:
		tmp = x * (y + 0.5)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.55e+22)
		tmp = Float64(x * y);
	elseif (y <= 4.2e+17)
		tmp = Float64(z + Float64(x * 0.5));
	else
		tmp = Float64(x * Float64(y + 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1.55e+22)
		tmp = x * y;
	elseif (y <= 4.2e+17)
		tmp = z + (x * 0.5);
	else
		tmp = x * (y + 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.55e+22], N[(x * y), $MachinePrecision], If[LessEqual[y, 4.2e+17], N[(z + N[(x * 0.5), $MachinePrecision]), $MachinePrecision], N[(x * N[(y + 0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.5500000000000001e22

   1. Initial program 100.0%

      \[\left(\frac{x}{2} + y \cdot x\right) + z \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 74.7%

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

   5. Simplified 74.7%

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

   if -1.5500000000000001e22 < y < 4.2e17

   1. Initial program 100.0%

      \[\left(\frac{x}{2} + y \cdot x\right) + z \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(x \cdot \frac{1}{2}\right), z\right) \]

      2. *-lowering-*.f64 99.6%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), z\right) \]

   5. Simplified 99.6%

      \[\leadsto \color{blue}{x \cdot 0.5} + z \]

   if 4.2e17 < y

   1. Initial program 100.0%

      \[\left(\frac{x}{2} + y \cdot x\right) + z \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

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

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

      2. +-lowering-+.f64 72.4%

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

   5. Simplified 72.4%

      \[\leadsto \color{blue}{x \cdot \left(0.5 + y\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.55 \cdot 10^{+22}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{+17}:\\ \;\;\;\;z + x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(y + 0.5\right)\\ \mathbf{if}\;x \leq -3.2 \cdot 10^{-39}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{+19}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (+ y 0.5))))
   (if (<= x -3.2e-39) t_0 (if (<= x 1.55e+19) z t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x * (y + 0.5);
	double tmp;
	if (x <= -3.2e-39) {
		tmp = t_0;
	} else if (x <= 1.55e+19) {
		tmp = z;
	} else {
		tmp = t_0;
	}
	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) :: tmp
    t_0 = x * (y + 0.5d0)
    if (x <= (-3.2d-39)) then
        tmp = t_0
    else if (x <= 1.55d+19) then
        tmp = z
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x * (y + 0.5);
	double tmp;
	if (x <= -3.2e-39) {
		tmp = t_0;
	} else if (x <= 1.55e+19) {
		tmp = z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * (y + 0.5)
	tmp = 0
	if x <= -3.2e-39:
		tmp = t_0
	elif x <= 1.55e+19:
		tmp = z
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(y + 0.5))
	tmp = 0.0
	if (x <= -3.2e-39)
		tmp = t_0;
	elseif (x <= 1.55e+19)
		tmp = z;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * (y + 0.5);
	tmp = 0.0;
	if (x <= -3.2e-39)
		tmp = t_0;
	elseif (x <= 1.55e+19)
		tmp = z;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[(y + 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.2e-39], t$95$0, If[LessEqual[x, 1.55e+19], z, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.1999999999999998e-39 or 1.55e19 < x

   1. Initial program 100.0%

      \[\left(\frac{x}{2} + y \cdot x\right) + z \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

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

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

      2. +-lowering-+.f64 78.6%

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

   5. Simplified 78.6%

      \[\leadsto \color{blue}{x \cdot \left(0.5 + y\right)} \]

   if -3.1999999999999998e-39 < x < 1.55e19

   1. Initial program 100.0%

      \[\left(\frac{x}{2} + y \cdot x\right) + z \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

   5. Simplified 68.3%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{-39}:\\ \;\;\;\;x \cdot \left(y + 0.5\right)\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{+19}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 51.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.36 \cdot 10^{-11}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-60}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.36e-11) z (if (<= z 6e-60) (* x 0.5) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.36e-11) {
		tmp = z;
	} else if (z <= 6e-60) {
		tmp = x * 0.5;
	} else {
		tmp = 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 (z <= (-1.36d-11)) then
        tmp = z
    else if (z <= 6d-60) then
        tmp = x * 0.5d0
    else
        tmp = z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.36e-11) {
		tmp = z;
	} else if (z <= 6e-60) {
		tmp = x * 0.5;
	} else {
		tmp = z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.36e-11:
		tmp = z
	elif z <= 6e-60:
		tmp = x * 0.5
	else:
		tmp = z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.36e-11)
		tmp = z;
	elseif (z <= 6e-60)
		tmp = Float64(x * 0.5);
	else
		tmp = z;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.36e-11)
		tmp = z;
	elseif (z <= 6e-60)
		tmp = x * 0.5;
	else
		tmp = z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.36e-11], z, If[LessEqual[z, 6e-60], N[(x * 0.5), $MachinePrecision], z]]

Derivation

1. Split input into 2 regimes

2. if z < -1.36e-11 or 6.00000000000000038e-60 < z

   1. Initial program 100.0%

      \[\left(\frac{x}{2} + y \cdot x\right) + z \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

   5. Simplified 61.3%

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

   if -1.36e-11 < z < 6.00000000000000038e-60

   1. Initial program 100.0%

      \[\left(\frac{x}{2} + y \cdot x\right) + z \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

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

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

      2. +-lowering-+.f64 81.9%

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

   5. Simplified 81.9%

      \[\leadsto \color{blue}{x \cdot \left(0.5 + y\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot x} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto x \cdot \color{blue}{\frac{1}{2}} \]

      2. *-lowering-*.f64 40.1%

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{2}}\right) \]

   8. Simplified 40.1%

      \[\leadsto \color{blue}{x \cdot 0.5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 40.9% accurate, 9.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := z

Derivation

1. Initial program 100.0%

   \[\left(\frac{x}{2} + y \cdot x\right) + z \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

5. Simplified 44.6%

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

Reproduce

herbie shell --seed 2024160 
(FPCore (x y z)
  :name "Data.Histogram.Bin.BinF:$cfromIndex from histogram-fill-0.8.4.1"
  :precision binary64
  (+ (+ (/ x 2.0) (* y x)) z))