Statistics.Sample:$swelfordMean from math-functions-0.1.5.2

Percentage Accurate: 100.0% → 100.0%

Time: 5.8s

Alternatives: 7

Speedup: 1.0×

• 20.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[x + \frac{y - x}{z} \]
2. Add Preprocessing

3. Final simplification 100.0%

   \[\leadsto x + \frac{y - x}{z} \]
4. Add Preprocessing

Alternative 2: 60.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{z}\\ \mathbf{if}\;z \leq -3.2 \cdot 10^{+127}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -1.6 \cdot 10^{-8}:\\ \;\;\;\;\frac{y}{z}\\ \mathbf{elif}\;z \leq -3.1 \cdot 10^{-56}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -8.8 \cdot 10^{-249}:\\ \;\;\;\;\frac{y}{z}\\ \mathbf{elif}\;z \leq -4.2 \cdot 10^{-270}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{-292}:\\ \;\;\;\;\frac{y}{z}\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{-105}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 10^{+33}:\\ \;\;\;\;\frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (- x) z)))
   (if (<= z -3.2e+127)
     x
     (if (<= z -1.6e-8)
       (/ y z)
       (if (<= z -3.1e-56)
         t_0
         (if (<= z -8.8e-249)
           (/ y z)
           (if (<= z -4.2e-270)
             t_0
             (if (<= z 1.4e-292)
               (/ y z)
               (if (<= z 3.3e-105) t_0 (if (<= z 1e+33) (/ y z) x))))))))))

C

double code(double x, double y, double z) {
	double t_0 = -x / z;
	double tmp;
	if (z <= -3.2e+127) {
		tmp = x;
	} else if (z <= -1.6e-8) {
		tmp = y / z;
	} else if (z <= -3.1e-56) {
		tmp = t_0;
	} else if (z <= -8.8e-249) {
		tmp = y / z;
	} else if (z <= -4.2e-270) {
		tmp = t_0;
	} else if (z <= 1.4e-292) {
		tmp = y / z;
	} else if (z <= 3.3e-105) {
		tmp = t_0;
	} else if (z <= 1e+33) {
		tmp = y / z;
	} else {
		tmp = 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) :: t_0
    real(8) :: tmp
    t_0 = -x / z
    if (z <= (-3.2d+127)) then
        tmp = x
    else if (z <= (-1.6d-8)) then
        tmp = y / z
    else if (z <= (-3.1d-56)) then
        tmp = t_0
    else if (z <= (-8.8d-249)) then
        tmp = y / z
    else if (z <= (-4.2d-270)) then
        tmp = t_0
    else if (z <= 1.4d-292) then
        tmp = y / z
    else if (z <= 3.3d-105) then
        tmp = t_0
    else if (z <= 1d+33) then
        tmp = y / z
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = -x / z;
	double tmp;
	if (z <= -3.2e+127) {
		tmp = x;
	} else if (z <= -1.6e-8) {
		tmp = y / z;
	} else if (z <= -3.1e-56) {
		tmp = t_0;
	} else if (z <= -8.8e-249) {
		tmp = y / z;
	} else if (z <= -4.2e-270) {
		tmp = t_0;
	} else if (z <= 1.4e-292) {
		tmp = y / z;
	} else if (z <= 3.3e-105) {
		tmp = t_0;
	} else if (z <= 1e+33) {
		tmp = y / z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = -x / z
	tmp = 0
	if z <= -3.2e+127:
		tmp = x
	elif z <= -1.6e-8:
		tmp = y / z
	elif z <= -3.1e-56:
		tmp = t_0
	elif z <= -8.8e-249:
		tmp = y / z
	elif z <= -4.2e-270:
		tmp = t_0
	elif z <= 1.4e-292:
		tmp = y / z
	elif z <= 3.3e-105:
		tmp = t_0
	elif z <= 1e+33:
		tmp = y / z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(-x) / z)
	tmp = 0.0
	if (z <= -3.2e+127)
		tmp = x;
	elseif (z <= -1.6e-8)
		tmp = Float64(y / z);
	elseif (z <= -3.1e-56)
		tmp = t_0;
	elseif (z <= -8.8e-249)
		tmp = Float64(y / z);
	elseif (z <= -4.2e-270)
		tmp = t_0;
	elseif (z <= 1.4e-292)
		tmp = Float64(y / z);
	elseif (z <= 3.3e-105)
		tmp = t_0;
	elseif (z <= 1e+33)
		tmp = Float64(y / z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = -x / z;
	tmp = 0.0;
	if (z <= -3.2e+127)
		tmp = x;
	elseif (z <= -1.6e-8)
		tmp = y / z;
	elseif (z <= -3.1e-56)
		tmp = t_0;
	elseif (z <= -8.8e-249)
		tmp = y / z;
	elseif (z <= -4.2e-270)
		tmp = t_0;
	elseif (z <= 1.4e-292)
		tmp = y / z;
	elseif (z <= 3.3e-105)
		tmp = t_0;
	elseif (z <= 1e+33)
		tmp = y / z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[((-x) / z), $MachinePrecision]}, If[LessEqual[z, -3.2e+127], x, If[LessEqual[z, -1.6e-8], N[(y / z), $MachinePrecision], If[LessEqual[z, -3.1e-56], t$95$0, If[LessEqual[z, -8.8e-249], N[(y / z), $MachinePrecision], If[LessEqual[z, -4.2e-270], t$95$0, If[LessEqual[z, 1.4e-292], N[(y / z), $MachinePrecision], If[LessEqual[z, 3.3e-105], t$95$0, If[LessEqual[z, 1e+33], N[(y / z), $MachinePrecision], x]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.19999999999999976e127 or 9.9999999999999995e32 < z

   1. Initial program 100.0%

      \[x + \frac{y - x}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 75.0%

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

   if -3.19999999999999976e127 < z < -1.6000000000000001e-8 or -3.09999999999999987e-56 < z < -8.8e-249 or -4.19999999999999992e-270 < z < 1.4000000000000001e-292 or 3.2999999999999999e-105 < z < 9.9999999999999995e32

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. clear-num 99.8%

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

      3. associate-/r/ 99.7%

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

      4. fma-def 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in y around inf 65.0%

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

   if -1.6000000000000001e-8 < z < -3.09999999999999987e-56 or -8.8e-249 < z < -4.19999999999999992e-270 or 1.4000000000000001e-292 < z < 3.2999999999999999e-105

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. clear-num 99.5%

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

      3. associate-/r/ 99.6%

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

      4. fma-def 99.5%

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

   4. Applied egg-rr 99.5%

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

   5. Taylor expanded in z around 0 98.6%

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

   6. Taylor expanded in y around 0 75.2%

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

      1. neg-mul-1 75.2%

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

      2. distribute-neg-frac 75.2%

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

   8. Simplified 75.2%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.2 \cdot 10^{+127}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -1.6 \cdot 10^{-8}:\\ \;\;\;\;\frac{y}{z}\\ \mathbf{elif}\;z \leq -3.1 \cdot 10^{-56}:\\ \;\;\;\;\frac{-x}{z}\\ \mathbf{elif}\;z \leq -8.8 \cdot 10^{-249}:\\ \;\;\;\;\frac{y}{z}\\ \mathbf{elif}\;z \leq -4.2 \cdot 10^{-270}:\\ \;\;\;\;\frac{-x}{z}\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{-292}:\\ \;\;\;\;\frac{y}{z}\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{-105}:\\ \;\;\;\;\frac{-x}{z}\\ \mathbf{elif}\;z \leq 10^{+33}:\\ \;\;\;\;\frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{y}{z}\\ t_1 := \frac{-x}{z}\\ \mathbf{if}\;z \leq -6 \cdot 10^{-8}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -9.5 \cdot 10^{-57}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -5.4 \cdot 10^{-250}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -2.5 \cdot 10^{-268}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 8 \cdot 10^{-293}:\\ \;\;\;\;\frac{y}{z}\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{-105}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (/ y z))) (t_1 (/ (- x) z)))
   (if (<= z -6e-8)
     t_0
     (if (<= z -9.5e-57)
       t_1
       (if (<= z -5.4e-250)
         t_0
         (if (<= z -2.5e-268)
           t_1
           (if (<= z 8e-293) (/ y z) (if (<= z 1.7e-105) t_1 t_0))))))))

C

double code(double x, double y, double z) {
	double t_0 = x + (y / z);
	double t_1 = -x / z;
	double tmp;
	if (z <= -6e-8) {
		tmp = t_0;
	} else if (z <= -9.5e-57) {
		tmp = t_1;
	} else if (z <= -5.4e-250) {
		tmp = t_0;
	} else if (z <= -2.5e-268) {
		tmp = t_1;
	} else if (z <= 8e-293) {
		tmp = y / z;
	} else if (z <= 1.7e-105) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_0 = x + (y / z)
    t_1 = -x / z
    if (z <= (-6d-8)) then
        tmp = t_0
    else if (z <= (-9.5d-57)) then
        tmp = t_1
    else if (z <= (-5.4d-250)) then
        tmp = t_0
    else if (z <= (-2.5d-268)) then
        tmp = t_1
    else if (z <= 8d-293) then
        tmp = y / z
    else if (z <= 1.7d-105) then
        tmp = t_1
    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 / z);
	double t_1 = -x / z;
	double tmp;
	if (z <= -6e-8) {
		tmp = t_0;
	} else if (z <= -9.5e-57) {
		tmp = t_1;
	} else if (z <= -5.4e-250) {
		tmp = t_0;
	} else if (z <= -2.5e-268) {
		tmp = t_1;
	} else if (z <= 8e-293) {
		tmp = y / z;
	} else if (z <= 1.7e-105) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x + (y / z)
	t_1 = -x / z
	tmp = 0
	if z <= -6e-8:
		tmp = t_0
	elif z <= -9.5e-57:
		tmp = t_1
	elif z <= -5.4e-250:
		tmp = t_0
	elif z <= -2.5e-268:
		tmp = t_1
	elif z <= 8e-293:
		tmp = y / z
	elif z <= 1.7e-105:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x + Float64(y / z))
	t_1 = Float64(Float64(-x) / z)
	tmp = 0.0
	if (z <= -6e-8)
		tmp = t_0;
	elseif (z <= -9.5e-57)
		tmp = t_1;
	elseif (z <= -5.4e-250)
		tmp = t_0;
	elseif (z <= -2.5e-268)
		tmp = t_1;
	elseif (z <= 8e-293)
		tmp = Float64(y / z);
	elseif (z <= 1.7e-105)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x + (y / z);
	t_1 = -x / z;
	tmp = 0.0;
	if (z <= -6e-8)
		tmp = t_0;
	elseif (z <= -9.5e-57)
		tmp = t_1;
	elseif (z <= -5.4e-250)
		tmp = t_0;
	elseif (z <= -2.5e-268)
		tmp = t_1;
	elseif (z <= 8e-293)
		tmp = y / z;
	elseif (z <= 1.7e-105)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x + N[(y / z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[((-x) / z), $MachinePrecision]}, If[LessEqual[z, -6e-8], t$95$0, If[LessEqual[z, -9.5e-57], t$95$1, If[LessEqual[z, -5.4e-250], t$95$0, If[LessEqual[z, -2.5e-268], t$95$1, If[LessEqual[z, 8e-293], N[(y / z), $MachinePrecision], If[LessEqual[z, 1.7e-105], t$95$1, t$95$0]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.99999999999999946e-8 or -9.5000000000000005e-57 < z < -5.40000000000000004e-250 or 1.69999999999999996e-105 < z

   1. Initial program 100.0%

      \[x + \frac{y - x}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 86.4%

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

   if -5.99999999999999946e-8 < z < -9.5000000000000005e-57 or -5.40000000000000004e-250 < z < -2.5e-268 or 8.0000000000000004e-293 < z < 1.69999999999999996e-105

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. clear-num 99.5%

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

      3. associate-/r/ 99.6%

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

      4. fma-def 99.5%

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

   4. Applied egg-rr 99.5%

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

   5. Taylor expanded in z around 0 98.6%

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

   6. Taylor expanded in y around 0 75.2%

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

      1. neg-mul-1 75.2%

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

      2. distribute-neg-frac 75.2%

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

   8. Simplified 75.2%

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

   if -2.5e-268 < z < 8.0000000000000004e-293

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. clear-num 100.0%

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

      3. associate-/r/ 99.9%

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

      4. fma-def 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in y around inf 79.2%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{-8}:\\ \;\;\;\;x + \frac{y}{z}\\ \mathbf{elif}\;z \leq -9.5 \cdot 10^{-57}:\\ \;\;\;\;\frac{-x}{z}\\ \mathbf{elif}\;z \leq -5.4 \cdot 10^{-250}:\\ \;\;\;\;x + \frac{y}{z}\\ \mathbf{elif}\;z \leq -2.5 \cdot 10^{-268}:\\ \;\;\;\;\frac{-x}{z}\\ \mathbf{elif}\;z \leq 8 \cdot 10^{-293}:\\ \;\;\;\;\frac{y}{z}\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{-105}:\\ \;\;\;\;\frac{-x}{z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{-17} \lor \neg \left(y \leq -5.2 \cdot 10^{-45} \lor \neg \left(y \leq -2.4 \cdot 10^{-98}\right) \land y \leq 1.55 \cdot 10^{-17}\right):\\ \;\;\;\;x + \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -8.5e-17)
         (not
          (or (<= y -5.2e-45) (and (not (<= y -2.4e-98)) (<= y 1.55e-17)))))
   (+ x (/ y z))
   (- x (/ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -8.5e-17) || !((y <= -5.2e-45) || (!(y <= -2.4e-98) && (y <= 1.55e-17)))) {
		tmp = x + (y / z);
	} else {
		tmp = x - (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 ((y <= (-8.5d-17)) .or. (.not. (y <= (-5.2d-45)) .or. (.not. (y <= (-2.4d-98))) .and. (y <= 1.55d-17))) then
        tmp = x + (y / z)
    else
        tmp = x - (x / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -8.5e-17) || !((y <= -5.2e-45) || (!(y <= -2.4e-98) && (y <= 1.55e-17)))) {
		tmp = x + (y / z);
	} else {
		tmp = x - (x / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -8.5e-17) or not ((y <= -5.2e-45) or (not (y <= -2.4e-98) and (y <= 1.55e-17))):
		tmp = x + (y / z)
	else:
		tmp = x - (x / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -8.5e-17) || !((y <= -5.2e-45) || (!(y <= -2.4e-98) && (y <= 1.55e-17))))
		tmp = Float64(x + Float64(y / z));
	else
		tmp = Float64(x - Float64(x / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -8.5e-17) || ~(((y <= -5.2e-45) || (~((y <= -2.4e-98)) && (y <= 1.55e-17)))))
		tmp = x + (y / z);
	else
		tmp = x - (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -8.5e-17], N[Not[Or[LessEqual[y, -5.2e-45], And[N[Not[LessEqual[y, -2.4e-98]], $MachinePrecision], LessEqual[y, 1.55e-17]]]], $MachinePrecision]], N[(x + N[(y / z), $MachinePrecision]), $MachinePrecision], N[(x - N[(x / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -8.5e-17 or -5.19999999999999973e-45 < y < -2.40000000000000005e-98 or 1.5499999999999999e-17 < y

   1. Initial program 100.0%

      \[x + \frac{y - x}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 90.9%

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

   if -8.5e-17 < y < -5.19999999999999973e-45 or -2.40000000000000005e-98 < y < 1.5499999999999999e-17

   1. Initial program 100.0%

      \[x + \frac{y - x}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 88.5%

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

      1. sub-neg 88.5%

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

      2. distribute-lft-in 88.5%

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

      3. *-rgt-identity 88.5%

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

      4. distribute-rgt-neg-out 88.5%

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

      5. associate-*r/ 88.7%

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

      6. *-rgt-identity 88.7%

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

      7. sub-neg 88.7%

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

   5. Simplified 88.7%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{-17} \lor \neg \left(y \leq -5.2 \cdot 10^{-45} \lor \neg \left(y \leq -2.4 \cdot 10^{-98}\right) \land y \leq 1.55 \cdot 10^{-17}\right):\\ \;\;\;\;x + \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -210000 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;x + \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -210000.0) (not (<= z 1.0))) (+ x (/ y z)) (/ (- y x) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -210000.0) || !(z <= 1.0)) {
		tmp = x + (y / z);
	} else {
		tmp = (y - 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 ((z <= (-210000.0d0)) .or. (.not. (z <= 1.0d0))) then
        tmp = x + (y / z)
    else
        tmp = (y - x) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -210000.0) || !(z <= 1.0)) {
		tmp = x + (y / z);
	} else {
		tmp = (y - x) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -210000.0) or not (z <= 1.0):
		tmp = x + (y / z)
	else:
		tmp = (y - x) / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -210000.0) || !(z <= 1.0))
		tmp = Float64(x + Float64(y / z));
	else
		tmp = Float64(Float64(y - x) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -210000.0) || ~((z <= 1.0)))
		tmp = x + (y / z);
	else
		tmp = (y - x) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -210000.0], N[Not[LessEqual[z, 1.0]], $MachinePrecision]], N[(x + N[(y / z), $MachinePrecision]), $MachinePrecision], N[(N[(y - x), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.1e5 or 1 < z

   1. Initial program 100.0%

      \[x + \frac{y - x}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 98.9%

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

   if -2.1e5 < z < 1

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. clear-num 99.6%

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

      3. associate-/r/ 99.7%

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

      4. fma-def 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in z around 0 98.3%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -210000 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;x + \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 60.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.2 \cdot 10^{+127}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 7.2 \cdot 10^{+25}:\\ \;\;\;\;\frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -3.2e+127) x (if (<= z 7.2e+25) (/ y z) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -3.2e+127) {
		tmp = x;
	} else if (z <= 7.2e+25) {
		tmp = y / z;
	} else {
		tmp = 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 (z <= (-3.2d+127)) then
        tmp = x
    else if (z <= 7.2d+25) then
        tmp = y / z
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -3.2e+127) {
		tmp = x;
	} else if (z <= 7.2e+25) {
		tmp = y / z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -3.2e+127:
		tmp = x
	elif z <= 7.2e+25:
		tmp = y / z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -3.2e+127)
		tmp = x;
	elseif (z <= 7.2e+25)
		tmp = Float64(y / z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -3.2e+127)
		tmp = x;
	elseif (z <= 7.2e+25)
		tmp = y / z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -3.2e+127], x, If[LessEqual[z, 7.2e+25], N[(y / z), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if z < -3.19999999999999976e127 or 7.20000000000000031e25 < z

   1. Initial program 100.0%

      \[x + \frac{y - x}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 75.0%

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

   if -3.19999999999999976e127 < z < 7.20000000000000031e25

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. clear-num 99.7%

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

      3. associate-/r/ 99.7%

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

      4. fma-def 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in y around inf 50.4%

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

4. Final simplification 58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.2 \cdot 10^{+127}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 7.2 \cdot 10^{+25}:\\ \;\;\;\;\frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 37.0% accurate, 7.0× speedup

Math

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

FPCore

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

C

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

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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 100.0%

   \[x + \frac{y - x}{z} \]
2. Add Preprocessing

3. Taylor expanded in z around inf 31.3%

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

4. Final simplification 31.3%

   \[\leadsto x \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024010 
(FPCore (x y z)
  :name "Statistics.Sample:$swelfordMean from math-functions-0.1.5.2"
  :precision binary64
  (+ x (/ (- y x) z)))