Numeric.SpecFunctions:choose from math-functions-0.1.5.2

Percentage Accurate: 84.2% → 95.9%

Time: 4.4s

Alternatives: 5

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: 84.2% 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: 95.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 84.3%

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

   1. associate-*l/ 80.7%

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

   2. distribute-lft-in 76.9%

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

   3. cancel-sign-sub 76.9%

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

   4. associate-*l/ 72.8%

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

   5. associate-*r/ 77.4%

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

   6. distribute-frac-neg 77.4%

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

   7. fma-neg 77.4%

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

   8. distribute-rgt-neg-out 77.4%

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

   9. neg-mul-1 77.4%

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

   10. associate-/l* 76.3%

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

   11. associate-*l/ 77.1%

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

   12. neg-mul-1 77.1%

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

   13. remove-double-neg 77.1%

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

   14. associate-/r/ 97.7%

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

   15. *-inverses 97.7%

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

   16. *-lft-identity 97.7%

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

3. Simplified 97.7%

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

5. Final simplification 97.7%

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

Alternative 2: 68.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.5 \cdot 10^{+21}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{+102}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -4.5e+21) x (if (<= z 1.45e+102) (* x (/ y z)) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -4.5e+21) {
		tmp = x;
	} else if (z <= 1.45e+102) {
		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 (z <= (-4.5d+21)) then
        tmp = x
    else if (z <= 1.45d+102) 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 (z <= -4.5e+21) {
		tmp = x;
	} else if (z <= 1.45e+102) {
		tmp = x * (y / z);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -4.5e+21:
		tmp = x
	elif z <= 1.45e+102:
		tmp = x * (y / z)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -4.5e+21)
		tmp = x;
	elseif (z <= 1.45e+102)
		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 (z <= -4.5e+21)
		tmp = x;
	elseif (z <= 1.45e+102)
		tmp = x * (y / z);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -4.5e+21], x, If[LessEqual[z, 1.45e+102], N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if z < -4.5e21 or 1.4500000000000001e102 < z

   1. Initial program 72.3%

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

      1. *-commutative 72.3%

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

      2. associate-*l/ 99.9%

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

      3. *-commutative 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in y around 0 79.9%

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

   if -4.5e21 < z < 1.4500000000000001e102

   1. Initial program 92.7%

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

      1. *-commutative 92.7%

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

      2. associate-*l/ 96.1%

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

      3. *-commutative 96.1%

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

   3. Simplified 96.1%

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

   5. Taylor expanded in y around inf 72.0%

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

      1. associate-*r/ 72.9%

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

   7. Simplified 72.9%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.5 \cdot 10^{+21}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{+102}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.9% accurate, 1.0× speedup

Math

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

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(x * Float64(Float64(y + z) / z))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.3%

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

   1. *-commutative 84.3%

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

   2. associate-*l/ 97.7%

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

   3. *-commutative 97.7%

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

3. Simplified 97.7%

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

5. Final simplification 97.7%

   \[\leadsto x \cdot \frac{y + z}{z} \]
6. Add Preprocessing

Alternative 4: 95.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.3%

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

   1. associate-*l/ 80.7%

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

   2. distribute-lft-in 76.9%

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

   3. cancel-sign-sub 76.9%

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

   4. associate-*l/ 72.8%

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

   5. associate-*r/ 77.4%

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

   6. distribute-frac-neg 77.4%

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

   7. fma-neg 77.4%

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

   8. distribute-rgt-neg-out 77.4%

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

   9. neg-mul-1 77.4%

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

   10. associate-/l* 76.3%

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

   11. associate-*l/ 77.1%

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

   12. neg-mul-1 77.1%

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

   13. remove-double-neg 77.1%

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

   14. associate-/r/ 97.7%

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

   15. *-inverses 97.7%

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

   16. *-lft-identity 97.7%

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

3. Simplified 97.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{y}{z}, x\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. fma-undefine 97.7%

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

6. Applied egg-rr 97.7%

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

7. Final simplification 97.7%

   \[\leadsto x + x \cdot \frac{y}{z} \]
8. Add Preprocessing

Alternative 5: 51.1% 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 84.3%

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

   1. *-commutative 84.3%

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

   2. associate-*l/ 97.7%

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

   3. *-commutative 97.7%

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

3. Simplified 97.7%

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

5. Taylor expanded in y around 0 47.2%

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

6. Final simplification 47.2%

   \[\leadsto x \]
7. Add Preprocessing

Developer target: 96.3% 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 2024034 
(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))