Numeric.SpecFunctions:choose from math-functions-0.1.5.2

Percentage Accurate: 85.2% → 96.6%

Time: 8.9s

Alternatives: 6

Speedup: 1.1×

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: 85.2% 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: 96.6% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

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 / (z + y))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.2%

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

   1. associate-/l* N/A

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

   2. clear-num N/A

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

   3. un-div-inv N/A

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

   4. /-lowering-/.f64 N/A

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

   5. /-lowering-/.f64 N/A

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

   6. +-lowering-+.f64 98.1

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

4. Applied egg-rr 98.1%

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

5. Final simplification 98.1%

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

Alternative 2: 52.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(z + y\right)}{z} \leq -2 \cdot 10^{-190}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (/ (* x (+ z y)) z) -2e-190) (/ (* x y) z) x))

C

double code(double x, double y, double z) {
	double tmp;
	if (((x * (z + y)) / z) <= -2e-190) {
		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 (((x * (z + y)) / z) <= (-2d-190)) 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 (((x * (z + y)) / z) <= -2e-190) {
		tmp = (x * y) / z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if ((x * (z + y)) / z) <= -2e-190:
		tmp = (x * y) / z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(x * Float64(z + y)) / z) <= -2e-190)
		tmp = Float64(Float64(x * y) / z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (((x * (z + y)) / z) <= -2e-190)
		tmp = (x * y) / z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[(x * N[(z + y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], -2e-190], N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (+.f64 y z)) z) < -2e-190

   1. Initial program 82.1%

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

   3. Taylor expanded in y around inf

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

   5. Simplified 43.1%

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

   if -2e-190 < (/.f64 (*.f64 x (+.f64 y z)) z)

   1. Initial program 78.7%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 59.7%

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

4. Final simplification 52.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(z + y\right)}{z} \leq -2 \cdot 10^{-190}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 51.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(z + y\right)}{z} \leq -2 \cdot 10^{-190}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (/ (* x (+ z y)) z) -2e-190) (* x (/ y z)) x))

C

double code(double x, double y, double z) {
	double tmp;
	if (((x * (z + y)) / z) <= -2e-190) {
		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 (((x * (z + y)) / z) <= (-2d-190)) 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 (((x * (z + y)) / z) <= -2e-190) {
		tmp = x * (y / z);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if ((x * (z + y)) / z) <= -2e-190:
		tmp = x * (y / z)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(x * Float64(z + y)) / z) <= -2e-190)
		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 (((x * (z + y)) / z) <= -2e-190)
		tmp = x * (y / z);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[(x * N[(z + y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], -2e-190], N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (+.f64 y z)) z) < -2e-190

   1. Initial program 82.1%

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

   3. Taylor expanded in y around inf

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

   5. Simplified 43.1%

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

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

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

      4. /-lowering-/.f64 42.2

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

   7. Applied egg-rr 42.2%

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

   if -2e-190 < (/.f64 (*.f64 x (+.f64 y z)) z)

   1. Initial program 78.7%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 59.7%

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

4. Final simplification 52.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(z + y\right)}{z} \leq -2 \cdot 10^{-190}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 52.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(z + y\right)}{z} \leq -2 \cdot 10^{-190}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (/ (* x (+ z y)) z) -2e-190) (* y (/ x z)) x))

C

double code(double x, double y, double z) {
	double tmp;
	if (((x * (z + y)) / z) <= -2e-190) {
		tmp = y * (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 (((x * (z + y)) / z) <= (-2d-190)) then
        tmp = y * (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 (((x * (z + y)) / z) <= -2e-190) {
		tmp = y * (x / z);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if ((x * (z + y)) / z) <= -2e-190:
		tmp = y * (x / z)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(x * Float64(z + y)) / z) <= -2e-190)
		tmp = Float64(y * Float64(x / z));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (((x * (z + y)) / z) <= -2e-190)
		tmp = y * (x / z);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[(x * N[(z + y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], -2e-190], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (+.f64 y z)) z) < -2e-190

   1. Initial program 82.1%

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

   3. Taylor expanded in y around inf

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

   5. Simplified 43.1%

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

      1. associate-*l/ N/A

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

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

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

      3. /-lowering-/.f64 42.4

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

   7. Applied egg-rr 42.4%

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

   if -2e-190 < (/.f64 (*.f64 x (+.f64 y z)) z)

   1. Initial program 78.7%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 59.7%

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

4. Final simplification 52.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(z + y\right)}{z} \leq -2 \cdot 10^{-190}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.3% accurate, 1.1× 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 80.2%

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

3. Taylor expanded in x around 0

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

   1. associate-/l* N/A

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

   2. +-commutative N/A

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

   3. *-lft-identity N/A

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

   4. metadata-eval N/A

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

   5. cancel-sign-sub-inv N/A

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

   6. div-sub N/A

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

   7. *-inverses N/A

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

   8. associate-*r/ N/A

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

   9. distribute-rgt-out-- N/A

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

   10. *-lft-identity N/A

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

   11. mul-1-neg N/A

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

   12. cancel-sign-sub N/A

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

   13. distribute-rgt1-in N/A

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

   14. distribute-lft1-in N/A

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

   15. *-commutative N/A

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

   16. accelerator-lowering-fma.f64 N/A

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

   17. /-lowering-/.f64 98.0

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

5. Simplified 98.0%

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

Alternative 6: 50.0% accurate, 20.0× 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 80.2%

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

3. Taylor expanded in y around 0

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

5. Simplified 58.4%

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

Developer Target 1: 96.6% accurate, 0.8× 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 2024196 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:choose from math-functions-0.1.5.2"
  :precision binary64

  :alt
  (! :herbie-platform default (/ x (/ z (+ y z))))

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