Numeric.SpecFunctions:choose from math-functions-0.1.5.2

Percentage Accurate: 84.7% → 95.6%

Time: 4.4s

Alternatives: 8

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 84.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.8 \cdot 10^{+65}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{y}{z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(y + z\right)}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2.8e+65) (fma x (/ y z) x) (/ (* x (+ y z)) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.8e+65) {
		tmp = fma(x, (y / z), x);
	} else {
		tmp = (x * (y + z)) / z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2.8e+65)
		tmp = fma(x, Float64(y / z), x);
	else
		tmp = Float64(Float64(x * Float64(y + z)) / z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2.8e+65], N[(x * N[(y / z), $MachinePrecision] + x), $MachinePrecision], N[(N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.7999999999999999e65

   1. Initial program 83.3%

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

      1. remove-double-neg 83.3%

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

      2. distribute-lft-neg-out 83.3%

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

      3. *-commutative 83.3%

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

      4. distribute-lft-neg-in 83.3%

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

      5. associate-/l* 84.2%

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

      6. distribute-neg-in 84.2%

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

      7. unsub-neg 84.2%

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

      8. div-sub 79.8%

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

      9. distribute-frac-neg 79.8%

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

      10. associate-/r/ 81.0%

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

      11. distribute-rgt-neg-out 81.0%

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

      12. remove-double-neg 81.0%

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

      13. associate-/r/ 99.0%

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

      14. *-inverses 99.0%

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

      15. *-lft-identity 99.0%

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

      16. *-commutative 99.0%

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

      17. fma-neg 99.0%

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

      18. remove-double-neg 99.0%

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

   3. Simplified 99.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{y}{z}, x\right)} \]
   4. Add Preprocessing

   if 2.7999999999999999e65 < y

   1. Initial program 96.1%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.8 \cdot 10^{+65}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{y}{z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(y + z\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 70.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{+84} \lor \neg \left(y \leq -5.6 \cdot 10^{+47}\right) \land \left(y \leq -7.2 \cdot 10^{+14} \lor \neg \left(y \leq 2.55 \cdot 10^{-27}\right)\right):\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -2e+84)
         (and (not (<= y -5.6e+47))
              (or (<= y -7.2e+14) (not (<= y 2.55e-27)))))
   (* x (/ y z))
   x))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -2e+84) || (!(y <= -5.6e+47) && ((y <= -7.2e+14) || !(y <= 2.55e-27)))) {
		tmp = x * (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 ((y <= (-2d+84)) .or. (.not. (y <= (-5.6d+47))) .and. (y <= (-7.2d+14)) .or. (.not. (y <= 2.55d-27))) then
        tmp = x * (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 ((y <= -2e+84) || (!(y <= -5.6e+47) && ((y <= -7.2e+14) || !(y <= 2.55e-27)))) {
		tmp = x * (y / z);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -2e+84) or (not (y <= -5.6e+47) and ((y <= -7.2e+14) or not (y <= 2.55e-27))):
		tmp = x * (y / z)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -2e+84) || (!(y <= -5.6e+47) && ((y <= -7.2e+14) || !(y <= 2.55e-27))))
		tmp = Float64(x * Float64(y / z));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -2e+84) || (~((y <= -5.6e+47)) && ((y <= -7.2e+14) || ~((y <= 2.55e-27)))))
		tmp = x * (y / z);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -2e+84], And[N[Not[LessEqual[y, -5.6e+47]], $MachinePrecision], Or[LessEqual[y, -7.2e+14], N[Not[LessEqual[y, 2.55e-27]], $MachinePrecision]]]], N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if y < -2.00000000000000012e84 or -5.59999999999999976e47 < y < -7.2e14 or 2.55e-27 < y

   1. Initial program 92.0%

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

      1. associate-*l/ 87.2%

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

      2. *-commutative 87.2%

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

   3. Simplified 87.2%

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

   5. Taylor expanded in y around inf 80.5%

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

      1. associate-*r/ 78.0%

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

   7. Simplified 78.0%

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

   if -2.00000000000000012e84 < y < -5.59999999999999976e47 or -7.2e14 < y < 2.55e-27

   1. Initial program 79.3%

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

      1. associate-*l/ 85.0%

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

      2. *-commutative 85.0%

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

   3. Simplified 85.0%

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

   5. Taylor expanded in y around 0 84.7%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{+84} \lor \neg \left(y \leq -5.6 \cdot 10^{+47}\right) \land \left(y \leq -7.2 \cdot 10^{+14} \lor \neg \left(y \leq 2.55 \cdot 10^{-27}\right)\right):\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 71.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{+84}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;y \leq -3.3 \cdot 10^{+48} \lor \neg \left(y \leq -110000000000\right) \land y \leq 2.4 \cdot 10^{-27}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2.5e+84)
   (* x (/ y z))
   (if (or (<= y -3.3e+48) (and (not (<= y -110000000000.0)) (<= y 2.4e-27)))
     x
     (/ y (/ z x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.5e+84) {
		tmp = x * (y / z);
	} else if ((y <= -3.3e+48) || (!(y <= -110000000000.0) && (y <= 2.4e-27))) {
		tmp = x;
	} else {
		tmp = y / (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 <= (-2.5d+84)) then
        tmp = x * (y / z)
    else if ((y <= (-3.3d+48)) .or. (.not. (y <= (-110000000000.0d0))) .and. (y <= 2.4d-27)) then
        tmp = x
    else
        tmp = y / (z / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.5e+84) {
		tmp = x * (y / z);
	} else if ((y <= -3.3e+48) || (!(y <= -110000000000.0) && (y <= 2.4e-27))) {
		tmp = x;
	} else {
		tmp = y / (z / x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -2.5e+84:
		tmp = x * (y / z)
	elif (y <= -3.3e+48) or (not (y <= -110000000000.0) and (y <= 2.4e-27)):
		tmp = x
	else:
		tmp = y / (z / x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2.5e+84)
		tmp = Float64(x * Float64(y / z));
	elseif ((y <= -3.3e+48) || (!(y <= -110000000000.0) && (y <= 2.4e-27)))
		tmp = x;
	else
		tmp = Float64(y / Float64(z / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -2.5e+84)
		tmp = x * (y / z);
	elseif ((y <= -3.3e+48) || (~((y <= -110000000000.0)) && (y <= 2.4e-27)))
		tmp = x;
	else
		tmp = y / (z / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2.5e+84], N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[y, -3.3e+48], And[N[Not[LessEqual[y, -110000000000.0]], $MachinePrecision], LessEqual[y, 2.4e-27]]], x, N[(y / N[(z / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.5e84

   1. Initial program 88.9%

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

      1. associate-*l/ 86.3%

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

      2. *-commutative 86.3%

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

   3. Simplified 86.3%

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

   5. Taylor expanded in y around inf 74.4%

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

      1. associate-*r/ 79.4%

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

   7. Simplified 79.4%

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

   if -2.5e84 < y < -3.30000000000000023e48 or -1.1e11 < y < 2.40000000000000002e-27

   1. Initial program 79.3%

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

      1. associate-*l/ 85.0%

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

      2. *-commutative 85.0%

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

   3. Simplified 85.0%

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

   5. Taylor expanded in y around 0 84.7%

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

   if -3.30000000000000023e48 < y < -1.1e11 or 2.40000000000000002e-27 < y

   1. Initial program 94.1%

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

      1. associate-*l/ 87.8%

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

      2. *-commutative 87.8%

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

   3. Simplified 87.8%

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

   5. Taylor expanded in y around inf 84.6%

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

      1. associate-*r/ 77.0%

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

   7. Simplified 77.0%

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

      1. associate-*r/ 84.6%

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

      2. *-commutative 84.6%

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

      3. associate-/l* 83.1%

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

   9. Applied egg-rr 83.1%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{+84}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;y \leq -3.3 \cdot 10^{+48} \lor \neg \left(y \leq -110000000000\right) \land y \leq 2.4 \cdot 10^{-27}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{+84}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;y \leq -2.25 \cdot 10^{+48} \lor \neg \left(y \leq -145000000000\right) \land y \leq 1.7 \cdot 10^{-27}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2e+84)
   (* x (/ y z))
   (if (or (<= y -2.25e+48) (and (not (<= y -145000000000.0)) (<= y 1.7e-27)))
     x
     (/ (* y x) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2e+84) {
		tmp = x * (y / z);
	} else if ((y <= -2.25e+48) || (!(y <= -145000000000.0) && (y <= 1.7e-27))) {
		tmp = x;
	} 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 (y <= (-2d+84)) then
        tmp = x * (y / z)
    else if ((y <= (-2.25d+48)) .or. (.not. (y <= (-145000000000.0d0))) .and. (y <= 1.7d-27)) then
        tmp = x
    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 (y <= -2e+84) {
		tmp = x * (y / z);
	} else if ((y <= -2.25e+48) || (!(y <= -145000000000.0) && (y <= 1.7e-27))) {
		tmp = x;
	} else {
		tmp = (y * x) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -2e+84:
		tmp = x * (y / z)
	elif (y <= -2.25e+48) or (not (y <= -145000000000.0) and (y <= 1.7e-27)):
		tmp = x
	else:
		tmp = (y * x) / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2e+84)
		tmp = Float64(x * Float64(y / z));
	elseif ((y <= -2.25e+48) || (!(y <= -145000000000.0) && (y <= 1.7e-27)))
		tmp = x;
	else
		tmp = Float64(Float64(y * x) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -2e+84)
		tmp = x * (y / z);
	elseif ((y <= -2.25e+48) || (~((y <= -145000000000.0)) && (y <= 1.7e-27)))
		tmp = x;
	else
		tmp = (y * x) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2e+84], N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[y, -2.25e+48], And[N[Not[LessEqual[y, -145000000000.0]], $MachinePrecision], LessEqual[y, 1.7e-27]]], x, N[(N[(y * x), $MachinePrecision] / z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.00000000000000012e84

   1. Initial program 88.9%

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

      1. associate-*l/ 86.3%

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

      2. *-commutative 86.3%

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

   3. Simplified 86.3%

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

   5. Taylor expanded in y around inf 74.4%

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

      1. associate-*r/ 79.4%

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

   7. Simplified 79.4%

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

   if -2.00000000000000012e84 < y < -2.24999999999999998e48 or -1.45e11 < y < 1.69999999999999985e-27

   1. Initial program 79.3%

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

      1. associate-*l/ 85.0%

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

      2. *-commutative 85.0%

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

   3. Simplified 85.0%

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

   5. Taylor expanded in y around 0 84.7%

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

   if -2.24999999999999998e48 < y < -1.45e11 or 1.69999999999999985e-27 < y

   1. Initial program 94.1%

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

      1. associate-*l/ 87.8%

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

      2. *-commutative 87.8%

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

   3. Simplified 87.8%

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

   5. Taylor expanded in y around inf 84.6%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{+84}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;y \leq -2.25 \cdot 10^{+48} \lor \neg \left(y \leq -145000000000\right) \land y \leq 1.7 \cdot 10^{-27}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 70.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{y}{z}\\ \mathbf{if}\;y \leq -2 \cdot 10^{+84}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -2.9 \cdot 10^{+48}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -150000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 2.45 \cdot 10^{-27}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (/ y z))))
   (if (<= y -2e+84)
     t_0
     (if (<= y -2.9e+48)
       x
       (if (<= y -150000000000.0) t_0 (if (<= y 2.45e-27) x (/ x (/ z y))))))))

C

double code(double x, double y, double z) {
	double t_0 = x * (y / z);
	double tmp;
	if (y <= -2e+84) {
		tmp = t_0;
	} else if (y <= -2.9e+48) {
		tmp = x;
	} else if (y <= -150000000000.0) {
		tmp = t_0;
	} else if (y <= 2.45e-27) {
		tmp = x;
	} else {
		tmp = x / (z / 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) :: t_0
    real(8) :: tmp
    t_0 = x * (y / z)
    if (y <= (-2d+84)) then
        tmp = t_0
    else if (y <= (-2.9d+48)) then
        tmp = x
    else if (y <= (-150000000000.0d0)) then
        tmp = t_0
    else if (y <= 2.45d-27) then
        tmp = x
    else
        tmp = x / (z / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x * (y / z);
	double tmp;
	if (y <= -2e+84) {
		tmp = t_0;
	} else if (y <= -2.9e+48) {
		tmp = x;
	} else if (y <= -150000000000.0) {
		tmp = t_0;
	} else if (y <= 2.45e-27) {
		tmp = x;
	} else {
		tmp = x / (z / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * (y / z)
	tmp = 0
	if y <= -2e+84:
		tmp = t_0
	elif y <= -2.9e+48:
		tmp = x
	elif y <= -150000000000.0:
		tmp = t_0
	elif y <= 2.45e-27:
		tmp = x
	else:
		tmp = x / (z / y)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(y / z))
	tmp = 0.0
	if (y <= -2e+84)
		tmp = t_0;
	elseif (y <= -2.9e+48)
		tmp = x;
	elseif (y <= -150000000000.0)
		tmp = t_0;
	elseif (y <= 2.45e-27)
		tmp = x;
	else
		tmp = Float64(x / Float64(z / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * (y / z);
	tmp = 0.0;
	if (y <= -2e+84)
		tmp = t_0;
	elseif (y <= -2.9e+48)
		tmp = x;
	elseif (y <= -150000000000.0)
		tmp = t_0;
	elseif (y <= 2.45e-27)
		tmp = x;
	else
		tmp = x / (z / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2e+84], t$95$0, If[LessEqual[y, -2.9e+48], x, If[LessEqual[y, -150000000000.0], t$95$0, If[LessEqual[y, 2.45e-27], x, N[(x / N[(z / y), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.00000000000000012e84 or -2.8999999999999999e48 < y < -1.5e11

   1. Initial program 89.2%

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

      1. associate-*l/ 88.4%

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

      2. *-commutative 88.4%

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

   3. Simplified 88.4%

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

   5. Taylor expanded in y around inf 77.0%

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

      1. associate-*r/ 79.6%

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

   7. Simplified 79.6%

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

   if -2.00000000000000012e84 < y < -2.8999999999999999e48 or -1.5e11 < y < 2.44999999999999988e-27

   1. Initial program 79.3%

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

      1. associate-*l/ 85.0%

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

      2. *-commutative 85.0%

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

   3. Simplified 85.0%

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

   5. Taylor expanded in y around 0 84.7%

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

   if 2.44999999999999988e-27 < y

   1. Initial program 94.6%

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

      1. associate-*l/ 86.1%

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

      2. *-commutative 86.1%

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

   3. Simplified 86.1%

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

   5. Taylor expanded in y around inf 83.8%

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

      1. associate-*r/ 76.5%

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

   7. Simplified 76.5%

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

      1. associate-*r/ 83.8%

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

      2. associate-/l* 76.6%

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

   9. Applied egg-rr 76.6%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{+84}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;y \leq -2.9 \cdot 10^{+48}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -150000000000:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;y \leq 2.45 \cdot 10^{-27}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 95.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 6 \cdot 10^{+58}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(y + z\right)}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 6e+58) (* x (+ (/ y z) 1.0)) (/ (* x (+ y z)) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 6e+58) {
		tmp = x * ((y / z) + 1.0);
	} else {
		tmp = (x * (y + z)) / 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 <= 6d+58) then
        tmp = x * ((y / z) + 1.0d0)
    else
        tmp = (x * (y + z)) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 6e+58) {
		tmp = x * ((y / z) + 1.0);
	} else {
		tmp = (x * (y + z)) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 6e+58:
		tmp = x * ((y / z) + 1.0)
	else:
		tmp = (x * (y + z)) / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 6e+58)
		tmp = Float64(x * Float64(Float64(y / z) + 1.0));
	else
		tmp = Float64(Float64(x * Float64(y + z)) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 6e+58)
		tmp = x * ((y / z) + 1.0);
	else
		tmp = (x * (y + z)) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 6e+58], N[(x * N[(N[(y / z), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.0000000000000005e58

   1. Initial program 83.3%

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

      1. remove-double-neg 83.3%

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

      2. distribute-lft-neg-out 83.3%

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

      3. *-commutative 83.3%

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

      4. distribute-lft-neg-in 83.3%

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

      5. associate-/l* 84.2%

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

      6. distribute-neg-in 84.2%

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

      7. unsub-neg 84.2%

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

      8. div-sub 79.8%

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

      9. distribute-frac-neg 79.8%

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

      10. associate-/r/ 81.0%

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

      11. distribute-rgt-neg-out 81.0%

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

      12. remove-double-neg 81.0%

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

      13. associate-/r/ 99.0%

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

      14. *-inverses 99.0%

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

      15. *-lft-identity 99.0%

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

      16. *-commutative 99.0%

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

      17. fma-neg 99.0%

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

      18. remove-double-neg 99.0%

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

   3. Simplified 99.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{y}{z}, x\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 99.0%

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

   if 6.0000000000000005e58 < y

   1. Initial program 96.1%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6 \cdot 10^{+58}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(y + z\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 96.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(\frac{y}{z} + 1\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* x (+ (/ y z) 1.0)))

C

double code(double x, double y, double z) {
	return x * ((y / z) + 1.0);
}

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 / z) + 1.0d0)
end function

Java

public static double code(double x, double y, double z) {
	return x * ((y / z) + 1.0);
}

Python

def code(x, y, z):
	return x * ((y / z) + 1.0)

Julia

function code(x, y, z)
	return Float64(x * Float64(Float64(y / z) + 1.0))
end

MATLAB

function tmp = code(x, y, z)
	tmp = x * ((y / z) + 1.0);
end

Wolfram

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

Derivation

1. Initial program 85.7%

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

   1. remove-double-neg 85.7%

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

   2. distribute-lft-neg-out 85.7%

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

   3. *-commutative 85.7%

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

   4. distribute-lft-neg-in 85.7%

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

   5. associate-/l* 85.3%

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

   6. distribute-neg-in 85.3%

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

   7. unsub-neg 85.3%

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

   8. div-sub 80.2%

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

   9. distribute-frac-neg 80.2%

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

   10. associate-/r/ 79.7%

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

   11. distribute-rgt-neg-out 79.7%

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

   12. remove-double-neg 79.7%

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

   13. associate-/r/ 96.6%

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

   14. *-inverses 96.6%

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

   15. *-lft-identity 96.6%

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

   16. *-commutative 96.6%

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

   17. fma-neg 96.6%

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

   18. remove-double-neg 96.6%

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

3. Simplified 96.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{y}{z}, x\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 96.6%

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

6. Final simplification 96.6%

   \[\leadsto x \cdot \left(\frac{y}{z} + 1\right) \]
7. Add Preprocessing

Alternative 8: 50.3% 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 85.7%

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

   1. associate-*l/ 86.1%

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

   2. *-commutative 86.1%

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

3. Simplified 86.1%

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

5. Taylor expanded in y around 0 50.5%

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

6. Final simplification 50.5%

   \[\leadsto x \]
7. Add Preprocessing

Developer target: 96.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024021 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:choose from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (/ x (/ z (+ y z)))

  (/ (* x (+ y z)) z))