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

Percentage Accurate: 100.0% → 100.0%

Time: 5.9s

Alternatives: 7

Speedup: 1.0×

• 20.7% 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: 74.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{-75} \lor \neg \left(x \leq 1.18 \cdot 10^{-163}\right) \land \left(x \leq 1.7 \cdot 10^{-119} \lor \neg \left(x \leq 8.5 \cdot 10^{-50}\right)\right):\\ \;\;\;\;x - \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -2.1e-75)
         (and (not (<= x 1.18e-163))
              (or (<= x 1.7e-119) (not (<= x 8.5e-50)))))
   (- x (/ x z))
   (/ y z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2.1e-75) || (!(x <= 1.18e-163) && ((x <= 1.7e-119) || !(x <= 8.5e-50)))) {
		tmp = x - (x / z);
	} else {
		tmp = y / 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 <= (-2.1d-75)) .or. (.not. (x <= 1.18d-163)) .and. (x <= 1.7d-119) .or. (.not. (x <= 8.5d-50))) then
        tmp = x - (x / z)
    else
        tmp = y / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2.1e-75) || (!(x <= 1.18e-163) && ((x <= 1.7e-119) || !(x <= 8.5e-50)))) {
		tmp = x - (x / z);
	} else {
		tmp = y / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -2.1e-75) or (not (x <= 1.18e-163) and ((x <= 1.7e-119) or not (x <= 8.5e-50))):
		tmp = x - (x / z)
	else:
		tmp = y / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -2.1e-75) || (!(x <= 1.18e-163) && ((x <= 1.7e-119) || !(x <= 8.5e-50))))
		tmp = Float64(x - Float64(x / z));
	else
		tmp = Float64(y / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -2.1e-75) || (~((x <= 1.18e-163)) && ((x <= 1.7e-119) || ~((x <= 8.5e-50)))))
		tmp = x - (x / z);
	else
		tmp = y / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -2.1e-75], And[N[Not[LessEqual[x, 1.18e-163]], $MachinePrecision], Or[LessEqual[x, 1.7e-119], N[Not[LessEqual[x, 8.5e-50]], $MachinePrecision]]]], N[(x - N[(x / z), $MachinePrecision]), $MachinePrecision], N[(y / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.1000000000000001e-75 or 1.18000000000000004e-163 < x < 1.70000000000000012e-119 or 8.50000000000000012e-50 < x

   1. Initial program 100.0%

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

      1. remove-double-neg 100.0%

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

      2. neg-mul-1 100.0%

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

      3. fma-define 100.0%

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

      4. sub-neg 100.0%

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

      5. +-commutative 100.0%

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

      6. neg-sub0 100.0%

         \[\leadsto \mathsf{fma}\left(-1, -x, \frac{\color{blue}{\left(0 - x\right)} + y}{z}\right) \]

      7. associate-+l- 100.0%

         \[\leadsto \mathsf{fma}\left(-1, -x, \frac{\color{blue}{0 - \left(x - y\right)}}{z}\right) \]

      8. sub0-neg 100.0%

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

      9. distribute-frac-neg 100.0%

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

      10. fma-neg 100.0%

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

      11. neg-mul-1 100.0%

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

      12. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 83.8%

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

   if -2.1000000000000001e-75 < x < 1.18000000000000004e-163 or 1.70000000000000012e-119 < x < 8.50000000000000012e-50

   1. Initial program 100.0%

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

      1. remove-double-neg 100.0%

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

      2. neg-mul-1 100.0%

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

      3. fma-define 100.0%

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

      4. sub-neg 100.0%

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

      5. +-commutative 100.0%

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

      6. neg-sub0 100.0%

         \[\leadsto \mathsf{fma}\left(-1, -x, \frac{\color{blue}{\left(0 - x\right)} + y}{z}\right) \]

      7. associate-+l- 100.0%

         \[\leadsto \mathsf{fma}\left(-1, -x, \frac{\color{blue}{0 - \left(x - y\right)}}{z}\right) \]

      8. sub0-neg 100.0%

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

      9. distribute-frac-neg 100.0%

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

      10. fma-neg 100.0%

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

      11. neg-mul-1 100.0%

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

      12. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 75.6%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{-75} \lor \neg \left(x \leq 1.18 \cdot 10^{-163}\right) \land \left(x \leq 1.7 \cdot 10^{-119} \lor \neg \left(x \leq 8.5 \cdot 10^{-50}\right)\right):\\ \;\;\;\;x - \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \frac{x}{z}\\ \mathbf{if}\;z \leq -0.000136:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 0.065:\\ \;\;\;\;\frac{y - x}{z}\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{+82} \lor \neg \left(z \leq 9 \cdot 10^{+119}\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- x (/ x z))))
   (if (<= z -0.000136)
     t_0
     (if (<= z 0.065)
       (/ (- y x) z)
       (if (or (<= z 5.2e+82) (not (<= z 9e+119))) t_0 (/ y z))))))

C

double code(double x, double y, double z) {
	double t_0 = x - (x / z);
	double tmp;
	if (z <= -0.000136) {
		tmp = t_0;
	} else if (z <= 0.065) {
		tmp = (y - x) / z;
	} else if ((z <= 5.2e+82) || !(z <= 9e+119)) {
		tmp = t_0;
	} else {
		tmp = y / 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) :: t_0
    real(8) :: tmp
    t_0 = x - (x / z)
    if (z <= (-0.000136d0)) then
        tmp = t_0
    else if (z <= 0.065d0) then
        tmp = (y - x) / z
    else if ((z <= 5.2d+82) .or. (.not. (z <= 9d+119))) then
        tmp = t_0
    else
        tmp = y / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x - (x / z);
	double tmp;
	if (z <= -0.000136) {
		tmp = t_0;
	} else if (z <= 0.065) {
		tmp = (y - x) / z;
	} else if ((z <= 5.2e+82) || !(z <= 9e+119)) {
		tmp = t_0;
	} else {
		tmp = y / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x - (x / z)
	tmp = 0
	if z <= -0.000136:
		tmp = t_0
	elif z <= 0.065:
		tmp = (y - x) / z
	elif (z <= 5.2e+82) or not (z <= 9e+119):
		tmp = t_0
	else:
		tmp = y / z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x - Float64(x / z))
	tmp = 0.0
	if (z <= -0.000136)
		tmp = t_0;
	elseif (z <= 0.065)
		tmp = Float64(Float64(y - x) / z);
	elseif ((z <= 5.2e+82) || !(z <= 9e+119))
		tmp = t_0;
	else
		tmp = Float64(y / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x - (x / z);
	tmp = 0.0;
	if (z <= -0.000136)
		tmp = t_0;
	elseif (z <= 0.065)
		tmp = (y - x) / z;
	elseif ((z <= 5.2e+82) || ~((z <= 9e+119)))
		tmp = t_0;
	else
		tmp = y / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x - N[(x / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -0.000136], t$95$0, If[LessEqual[z, 0.065], N[(N[(y - x), $MachinePrecision] / z), $MachinePrecision], If[Or[LessEqual[z, 5.2e+82], N[Not[LessEqual[z, 9e+119]], $MachinePrecision]], t$95$0, N[(y / z), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.36e-4 or 0.065000000000000002 < z < 5.1999999999999997e82 or 9.00000000000000039e119 < z

   1. Initial program 100.0%

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

      1. remove-double-neg 100.0%

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

      2. neg-mul-1 100.0%

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

      3. fma-define 100.0%

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

      4. sub-neg 100.0%

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

      5. +-commutative 100.0%

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

      6. neg-sub0 100.0%

         \[\leadsto \mathsf{fma}\left(-1, -x, \frac{\color{blue}{\left(0 - x\right)} + y}{z}\right) \]

      7. associate-+l- 100.0%

         \[\leadsto \mathsf{fma}\left(-1, -x, \frac{\color{blue}{0 - \left(x - y\right)}}{z}\right) \]

      8. sub0-neg 100.0%

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

      9. distribute-frac-neg 100.0%

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

      10. fma-neg 100.0%

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

      11. neg-mul-1 100.0%

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

      12. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 71.7%

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

   if -1.36e-4 < z < 0.065000000000000002

   1. Initial program 100.0%

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

      1. remove-double-neg 100.0%

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

      2. neg-mul-1 100.0%

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

      3. fma-define 100.0%

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

      4. sub-neg 100.0%

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

      5. +-commutative 100.0%

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

      6. neg-sub0 100.0%

         \[\leadsto \mathsf{fma}\left(-1, -x, \frac{\color{blue}{\left(0 - x\right)} + y}{z}\right) \]

      7. associate-+l- 100.0%

         \[\leadsto \mathsf{fma}\left(-1, -x, \frac{\color{blue}{0 - \left(x - y\right)}}{z}\right) \]

      8. sub0-neg 100.0%

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

      9. distribute-frac-neg 100.0%

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

      10. fma-neg 100.0%

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

      11. neg-mul-1 100.0%

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

      12. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in z around 0 98.3%

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

   if 5.1999999999999997e82 < z < 9.00000000000000039e119

   1. Initial program 100.0%

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

      1. remove-double-neg 100.0%

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

      2. neg-mul-1 100.0%

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

      3. fma-define 100.0%

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

      4. sub-neg 100.0%

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

      5. +-commutative 100.0%

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

      6. neg-sub0 100.0%

         \[\leadsto \mathsf{fma}\left(-1, -x, \frac{\color{blue}{\left(0 - x\right)} + y}{z}\right) \]

      7. associate-+l- 100.0%

         \[\leadsto \mathsf{fma}\left(-1, -x, \frac{\color{blue}{0 - \left(x - y\right)}}{z}\right) \]

      8. sub0-neg 100.0%

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

      9. distribute-frac-neg 100.0%

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

      10. fma-neg 100.0%

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

      11. neg-mul-1 100.0%

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

      12. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 100.0%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.000136:\\ \;\;\;\;x - \frac{x}{z}\\ \mathbf{elif}\;z \leq 0.065:\\ \;\;\;\;\frac{y - x}{z}\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{+82} \lor \neg \left(z \leq 9 \cdot 10^{+119}\right):\\ \;\;\;\;x - \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 60.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -0.000136:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 190 \lor \neg \left(z \leq 2.3 \cdot 10^{+91}\right) \land z \leq 1.35 \cdot 10^{+121}:\\ \;\;\;\;\frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -0.000136)
   x
   (if (or (<= z 190.0) (and (not (<= z 2.3e+91)) (<= z 1.35e+121)))
     (/ y z)
     x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -0.000136) {
		tmp = x;
	} else if ((z <= 190.0) || (!(z <= 2.3e+91) && (z <= 1.35e+121))) {
		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 <= (-0.000136d0)) then
        tmp = x
    else if ((z <= 190.0d0) .or. (.not. (z <= 2.3d+91)) .and. (z <= 1.35d+121)) 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 <= -0.000136) {
		tmp = x;
	} else if ((z <= 190.0) || (!(z <= 2.3e+91) && (z <= 1.35e+121))) {
		tmp = y / z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -0.000136:
		tmp = x
	elif (z <= 190.0) or (not (z <= 2.3e+91) and (z <= 1.35e+121)):
		tmp = y / z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -0.000136)
		tmp = x;
	elseif ((z <= 190.0) || (!(z <= 2.3e+91) && (z <= 1.35e+121)))
		tmp = Float64(y / z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -0.000136)
		tmp = x;
	elseif ((z <= 190.0) || (~((z <= 2.3e+91)) && (z <= 1.35e+121)))
		tmp = y / z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -0.000136], x, If[Or[LessEqual[z, 190.0], And[N[Not[LessEqual[z, 2.3e+91]], $MachinePrecision], LessEqual[z, 1.35e+121]]], N[(y / z), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if z < -1.36e-4 or 190 < z < 2.29999999999999991e91 or 1.3500000000000001e121 < z

   1. Initial program 100.0%

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

      1. remove-double-neg 100.0%

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

      2. neg-mul-1 100.0%

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

      3. fma-define 100.0%

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

      4. sub-neg 100.0%

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

      5. +-commutative 100.0%

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

      6. neg-sub0 100.0%

         \[\leadsto \mathsf{fma}\left(-1, -x, \frac{\color{blue}{\left(0 - x\right)} + y}{z}\right) \]

      7. associate-+l- 100.0%

         \[\leadsto \mathsf{fma}\left(-1, -x, \frac{\color{blue}{0 - \left(x - y\right)}}{z}\right) \]

      8. sub0-neg 100.0%

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

      9. distribute-frac-neg 100.0%

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

      10. fma-neg 100.0%

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

      11. neg-mul-1 100.0%

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

      12. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in z around inf 69.8%

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

   if -1.36e-4 < z < 190 or 2.29999999999999991e91 < z < 1.3500000000000001e121

   1. Initial program 100.0%

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

      1. remove-double-neg 100.0%

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

      2. neg-mul-1 100.0%

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

      3. fma-define 100.0%

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

      4. sub-neg 100.0%

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

      5. +-commutative 100.0%

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

      6. neg-sub0 100.0%

         \[\leadsto \mathsf{fma}\left(-1, -x, \frac{\color{blue}{\left(0 - x\right)} + y}{z}\right) \]

      7. associate-+l- 100.0%

         \[\leadsto \mathsf{fma}\left(-1, -x, \frac{\color{blue}{0 - \left(x - y\right)}}{z}\right) \]

      8. sub0-neg 100.0%

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

      9. distribute-frac-neg 100.0%

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

      10. fma-neg 100.0%

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

      11. neg-mul-1 100.0%

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

      12. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 50.0%

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

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.000136:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 190 \lor \neg \left(z \leq 2.3 \cdot 10^{+91}\right) \land z \leq 1.35 \cdot 10^{+121}:\\ \;\;\;\;\frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 60.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -0.000136:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -3.4 \cdot 10^{-191}:\\ \;\;\;\;\frac{y}{z}\\ \mathbf{elif}\;z \leq 0.06:\\ \;\;\;\;\frac{-x}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -0.000136)
   x
   (if (<= z -3.4e-191) (/ y z) (if (<= z 0.06) (/ (- x) z) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -0.000136) {
		tmp = x;
	} else if (z <= -3.4e-191) {
		tmp = y / z;
	} else if (z <= 0.06) {
		tmp = -x / 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 <= (-0.000136d0)) then
        tmp = x
    else if (z <= (-3.4d-191)) then
        tmp = y / z
    else if (z <= 0.06d0) then
        tmp = -x / 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 <= -0.000136) {
		tmp = x;
	} else if (z <= -3.4e-191) {
		tmp = y / z;
	} else if (z <= 0.06) {
		tmp = -x / z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -0.000136:
		tmp = x
	elif z <= -3.4e-191:
		tmp = y / z
	elif z <= 0.06:
		tmp = -x / z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -0.000136)
		tmp = x;
	elseif (z <= -3.4e-191)
		tmp = Float64(y / z);
	elseif (z <= 0.06)
		tmp = Float64(Float64(-x) / z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -0.000136)
		tmp = x;
	elseif (z <= -3.4e-191)
		tmp = y / z;
	elseif (z <= 0.06)
		tmp = -x / z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -0.000136], x, If[LessEqual[z, -3.4e-191], N[(y / z), $MachinePrecision], If[LessEqual[z, 0.06], N[((-x) / z), $MachinePrecision], x]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.36e-4 or 0.059999999999999998 < z

   1. Initial program 100.0%

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

      1. remove-double-neg 100.0%

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

      2. neg-mul-1 100.0%

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

      3. fma-define 100.0%

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

      4. sub-neg 100.0%

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

      5. +-commutative 100.0%

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

      6. neg-sub0 100.0%

         \[\leadsto \mathsf{fma}\left(-1, -x, \frac{\color{blue}{\left(0 - x\right)} + y}{z}\right) \]

      7. associate-+l- 100.0%

         \[\leadsto \mathsf{fma}\left(-1, -x, \frac{\color{blue}{0 - \left(x - y\right)}}{z}\right) \]

      8. sub0-neg 100.0%

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

      9. distribute-frac-neg 100.0%

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

      10. fma-neg 100.0%

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

      11. neg-mul-1 100.0%

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

      12. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in z around inf 65.9%

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

   if -1.36e-4 < z < -3.39999999999999994e-191

   1. Initial program 100.0%

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

      1. remove-double-neg 100.0%

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

      2. neg-mul-1 100.0%

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

      3. fma-define 100.0%

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

      4. sub-neg 100.0%

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

      5. +-commutative 100.0%

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

      6. neg-sub0 100.0%

         \[\leadsto \mathsf{fma}\left(-1, -x, \frac{\color{blue}{\left(0 - x\right)} + y}{z}\right) \]

      7. associate-+l- 100.0%

         \[\leadsto \mathsf{fma}\left(-1, -x, \frac{\color{blue}{0 - \left(x - y\right)}}{z}\right) \]

      8. sub0-neg 100.0%

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

      9. distribute-frac-neg 100.0%

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

      10. fma-neg 100.0%

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

      11. neg-mul-1 100.0%

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

      12. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 59.7%

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

   if -3.39999999999999994e-191 < z < 0.059999999999999998

   1. Initial program 100.0%

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

      1. remove-double-neg 100.0%

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

      2. neg-mul-1 100.0%

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

      3. fma-define 100.0%

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

      4. sub-neg 100.0%

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

      5. +-commutative 100.0%

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

      6. neg-sub0 100.0%

         \[\leadsto \mathsf{fma}\left(-1, -x, \frac{\color{blue}{\left(0 - x\right)} + y}{z}\right) \]

      7. associate-+l- 100.0%

         \[\leadsto \mathsf{fma}\left(-1, -x, \frac{\color{blue}{0 - \left(x - y\right)}}{z}\right) \]

      8. sub0-neg 100.0%

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

      9. distribute-frac-neg 100.0%

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

      10. fma-neg 100.0%

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

      11. neg-mul-1 100.0%

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

      12. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 65.4%

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

   6. Taylor expanded in z around 0 63.5%

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

      1. mul-1-neg 63.5%

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

   8. Simplified 63.5%

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

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.000136:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -3.4 \cdot 10^{-191}:\\ \;\;\;\;\frac{y}{z}\\ \mathbf{elif}\;z \leq 0.06:\\ \;\;\;\;\frac{-x}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 0.06\right):\\ \;\;\;\;x + \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.0) (not (<= z 0.06))) (+ x (/ y z)) (/ (- y x) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.0) || !(z <= 0.06)) {
		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 <= (-1.0d0)) .or. (.not. (z <= 0.06d0))) 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 <= -1.0) || !(z <= 0.06)) {
		tmp = x + (y / z);
	} else {
		tmp = (y - x) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.0) or not (z <= 0.06):
		tmp = x + (y / z)
	else:
		tmp = (y - x) / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.0) || !(z <= 0.06))
		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 <= -1.0) || ~((z <= 0.06)))
		tmp = x + (y / z);
	else
		tmp = (y - x) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.0], N[Not[LessEqual[z, 0.06]], $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 < -1 or 0.059999999999999998 < z

   1. Initial program 100.0%

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

      1. remove-double-neg 100.0%

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

      2. neg-mul-1 100.0%

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

      3. fma-define 100.0%

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

      4. sub-neg 100.0%

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

      5. +-commutative 100.0%

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

      6. neg-sub0 100.0%

         \[\leadsto \mathsf{fma}\left(-1, -x, \frac{\color{blue}{\left(0 - x\right)} + y}{z}\right) \]

      7. associate-+l- 100.0%

         \[\leadsto \mathsf{fma}\left(-1, -x, \frac{\color{blue}{0 - \left(x - y\right)}}{z}\right) \]

      8. sub0-neg 100.0%

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

      9. distribute-frac-neg 100.0%

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

      10. fma-neg 100.0%

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

      11. neg-mul-1 100.0%

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

      12. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 98.8%

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

      1. neg-mul-1 98.8%

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

      2. distribute-neg-frac 98.8%

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

   7. Simplified 98.8%

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

   if -1 < z < 0.059999999999999998

   1. Initial program 100.0%

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

      1. remove-double-neg 100.0%

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

      2. neg-mul-1 100.0%

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

      3. fma-define 100.0%

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

      4. sub-neg 100.0%

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

      5. +-commutative 100.0%

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

      6. neg-sub0 100.0%

         \[\leadsto \mathsf{fma}\left(-1, -x, \frac{\color{blue}{\left(0 - x\right)} + y}{z}\right) \]

      7. associate-+l- 100.0%

         \[\leadsto \mathsf{fma}\left(-1, -x, \frac{\color{blue}{0 - \left(x - y\right)}}{z}\right) \]

      8. sub0-neg 100.0%

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

      9. distribute-frac-neg 100.0%

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

      10. fma-neg 100.0%

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

      11. neg-mul-1 100.0%

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

      12. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in z around 0 97.8%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 0.06\right):\\ \;\;\;\;x + \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 36.9% 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. Step-by-step derivation

   1. remove-double-neg 100.0%

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

   2. neg-mul-1 100.0%

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

   3. fma-define 100.0%

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

   4. sub-neg 100.0%

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

   5. +-commutative 100.0%

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

   6. neg-sub0 100.0%

      \[\leadsto \mathsf{fma}\left(-1, -x, \frac{\color{blue}{\left(0 - x\right)} + y}{z}\right) \]

   7. associate-+l- 100.0%

      \[\leadsto \mathsf{fma}\left(-1, -x, \frac{\color{blue}{0 - \left(x - y\right)}}{z}\right) \]

   8. sub0-neg 100.0%

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

   9. distribute-frac-neg 100.0%

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

   10. fma-neg 100.0%

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

   11. neg-mul-1 100.0%

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

   12. remove-double-neg 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in z around inf 35.6%

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

6. Final simplification 35.6%

   \[\leadsto x \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024034 
(FPCore (x y z)
  :name "Statistics.Sample:$swelfordMean from math-functions-0.1.5.2"
  :precision binary64
  (+ x (/ (- y x) z)))