Numeric.SpecFunctions:choose from math-functions-0.1.5.2

Percentage Accurate: 84.5% → 96.1%

Time: 6.2s

Alternatives: 3

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: 84.5% 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: 96.1% 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 87.1%

   \[\frac{x \cdot \left(y + z\right)}{z} \]
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. lower-fma.f64 N/A

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

   17. lower-/.f64 96.9

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

5. Applied rewrites 96.9%

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

Alternative 2: 71.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot y}{z}\\ \mathbf{if}\;y \leq -1.36 \cdot 10^{-124}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{-46}:\\ \;\;\;\;\frac{x}{1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* x y) z)))
   (if (<= y -1.36e-124) t_0 (if (<= y 1.5e-46) (/ x 1.0) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (x * y) / z;
	double tmp;
	if (y <= -1.36e-124) {
		tmp = t_0;
	} else if (y <= 1.5e-46) {
		tmp = x / 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x * y) / z
    if (y <= (-1.36d-124)) then
        tmp = t_0
    else if (y <= 1.5d-46) then
        tmp = x / 1.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x * y) / z;
	double tmp;
	if (y <= -1.36e-124) {
		tmp = t_0;
	} else if (y <= 1.5e-46) {
		tmp = x / 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x * y) / z
	tmp = 0
	if y <= -1.36e-124:
		tmp = t_0
	elif y <= 1.5e-46:
		tmp = x / 1.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x * y) / z)
	tmp = 0.0
	if (y <= -1.36e-124)
		tmp = t_0;
	elseif (y <= 1.5e-46)
		tmp = Float64(x / 1.0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x * y) / z;
	tmp = 0.0;
	if (y <= -1.36e-124)
		tmp = t_0;
	elseif (y <= 1.5e-46)
		tmp = x / 1.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[y, -1.36e-124], t$95$0, If[LessEqual[y, 1.5e-46], N[(x / 1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.3599999999999999e-124 or 1.49999999999999994e-46 < y

   1. Initial program 90.3%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 73.8

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

   5. Applied rewrites 73.8%

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

   if -1.3599999999999999e-124 < y < 1.49999999999999994e-46

   1. Initial program 81.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around inf

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

   7. Applied rewrites 89.6%

      \[\leadsto \frac{x}{\color{blue}{1}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 50.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \frac{x}{1} \end{array} \]

FPCore

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

C

double code(double x, double y, double z) {
	return x / 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 / 1.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.1%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. clear-num N/A

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

   5. un-div-inv N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 96.9

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

4. Applied rewrites 96.9%

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

5. Taylor expanded in z around inf

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

7. Applied rewrites 46.8%

   \[\leadsto \frac{x}{\color{blue}{1}} \]
8. Add Preprocessing

Developer Target 1: 96.4% 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 2024219 
(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))