Numeric.SpecFunctions:choose from math-functions-0.1.5.2

Percentage Accurate: 84.1% → 95.8%

Time: 5.4s

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: 84.1% 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: 95.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 84.4%

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

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

   2. associate-*l/ N/A

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

   3. +-commutative N/A

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

   4. div-add N/A

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

   5. *-inverses N/A

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

   6. +-commutative N/A

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

   7. distribute-lft1-in N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-/.f64 97.3

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

5. Applied rewrites 97.3%

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

Alternative 2: 53.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(y + z\right)}{z}\\ \mathbf{if}\;t\_0 \leq 0 \lor \neg \left(t\_0 \leq 5 \cdot 10^{+184}\right):\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* x (+ y z)) z)))
   (if (or (<= t_0 0.0) (not (<= t_0 5e+184))) (* (/ y z) x) (/ (* z x) z))))

C

double code(double x, double y, double z) {
	double t_0 = (x * (y + z)) / z;
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= 5e+184)) {
		tmp = (y / z) * x;
	} else {
		tmp = (z * 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) :: t_0
    real(8) :: tmp
    t_0 = (x * (y + z)) / z
    if ((t_0 <= 0.0d0) .or. (.not. (t_0 <= 5d+184))) then
        tmp = (y / z) * x
    else
        tmp = (z * x) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x * (y + z)) / z;
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= 5e+184)) {
		tmp = (y / z) * x;
	} else {
		tmp = (z * x) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x * (y + z)) / z
	tmp = 0
	if (t_0 <= 0.0) or not (t_0 <= 5e+184):
		tmp = (y / z) * x
	else:
		tmp = (z * x) / z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x * Float64(y + z)) / z)
	tmp = 0.0
	if ((t_0 <= 0.0) || !(t_0 <= 5e+184))
		tmp = Float64(Float64(y / z) * x);
	else
		tmp = Float64(Float64(z * x) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x * (y + z)) / z;
	tmp = 0.0;
	if ((t_0 <= 0.0) || ~((t_0 <= 5e+184)))
		tmp = (y / z) * x;
	else
		tmp = (z * x) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, If[Or[LessEqual[t$95$0, 0.0], N[Not[LessEqual[t$95$0, 5e+184]], $MachinePrecision]], N[(N[(y / z), $MachinePrecision] * x), $MachinePrecision], N[(N[(z * x), $MachinePrecision] / z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (+.f64 y z)) z) < -0.0 or 4.9999999999999999e184 < (/.f64 (*.f64 x (+.f64 y z)) z)

   1. Initial program 76.8%

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

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 85.5

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 85.5

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

   4. Applied rewrites 85.5%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/l* N/A

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

      9. lift-*.f64 N/A

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

      10. frac-add N/A

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

      11. lift-*.f64 N/A

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

      12. lift-fma.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-/.f64 54.1

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

   6. Applied rewrites 54.7%

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

   7. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 34.0

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

   9. Applied rewrites 34.0%

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

   10. Taylor expanded in y around inf

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

       1. *-commutative N/A

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

       2. associate-*l/ N/A

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

       3. lower-*.f64 N/A

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

       4. lower-/.f64 51.0

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

   12. Applied rewrites 51.0%

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

   if -0.0 < (/.f64 (*.f64 x (+.f64 y z)) z) < 4.9999999999999999e184

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 72.1

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

   5. Applied rewrites 72.1%

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

4. Final simplification 58.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y + z\right)}{z} \leq 0 \lor \neg \left(\frac{x \cdot \left(y + z\right)}{z} \leq 5 \cdot 10^{+184}\right):\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 46.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.4%

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

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

   4. associate-/l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 80.8

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-+.f64 80.8

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

4. Applied rewrites 80.8%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. distribute-rgt-in N/A

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

   4. lift-/.f64 N/A

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

   5. associate-*r/ N/A

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

   6. lift-*.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. associate-/l* N/A

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

   9. lift-*.f64 N/A

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

   10. frac-add N/A

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

   11. lift-*.f64 N/A

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

   12. lift-fma.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. lift-/.f64 51.6

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

6. Applied rewrites 52.0%

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

7. Taylor expanded in y around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 27.1

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

9. Applied rewrites 27.1%

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

10. Taylor expanded in y around inf

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

    1. *-commutative N/A

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

    2. associate-*l/ N/A

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

    3. lower-*.f64 N/A

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

    4. lower-/.f64 44.1

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

12. Applied rewrites 44.1%

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

13. Final simplification 44.1%

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

Developer Target 1: 96.1% 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 2024326 
(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))