Numeric.SpecFunctions:choose from math-functions-0.1.5.2

Percentage Accurate: 84.3% → 96.2%

Time: 5.3s

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: 84.3% 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: 96.2% 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]

Derivation

1. Initial program 81.3%

   \[\frac{x \cdot \left(y + z\right)}{z} \]
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 96.8

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-+.f64 96.8

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

4. Applied rewrites 96.8%

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

5. Final simplification 96.8%

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

Alternative 2: 51.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(y + z\right) \cdot x}{z} \leq -5 \cdot 10^{-256}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (/ (* (+ y z) x) z) -5e-256) (/ (* y x) z) (* 1.0 x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((((y + z) * x) / z) <= -5e-256) {
		tmp = (y * x) / z;
	} else {
		tmp = 1.0 * 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 + z) * x) / z) <= (-5d-256)) then
        tmp = (y * x) / z
    else
        tmp = 1.0d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((((y + z) * x) / z) <= -5e-256) {
		tmp = (y * x) / z;
	} else {
		tmp = 1.0 * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (((y + z) * x) / z) <= -5e-256:
		tmp = (y * x) / z
	else:
		tmp = 1.0 * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(Float64(y + z) * x) / z) <= -5e-256)
		tmp = Float64(Float64(y * x) / z);
	else
		tmp = Float64(1.0 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((((y + z) * x) / z) <= -5e-256)
		tmp = (y * x) / z;
	else
		tmp = 1.0 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[(N[(y + z), $MachinePrecision] * x), $MachinePrecision] / z), $MachinePrecision], -5e-256], N[(N[(y * x), $MachinePrecision] / z), $MachinePrecision], N[(1.0 * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (+.f64 y z)) z) < -5e-256

   1. Initial program 80.5%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 43.4

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

   5. Applied rewrites 43.4%

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

   if -5e-256 < (/.f64 (*.f64 x (+.f64 y z)) z)

   1. Initial program 82.0%

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

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 95.9

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 95.9

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

   4. Applied rewrites 95.9%

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

   5. Taylor expanded in z around inf

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

   7. Applied rewrites 55.5%

      \[\leadsto \color{blue}{1} \cdot x \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(y + z\right) \cdot x}{z} \leq -5 \cdot 10^{-256}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 50.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(y + z\right) \cdot x}{z} \leq -5 \cdot 10^{-256}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (/ (* (+ y z) x) z) -5e-256) (* (/ y z) x) (* 1.0 x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((((y + z) * x) / z) <= -5e-256) {
		tmp = (y / z) * x;
	} else {
		tmp = 1.0 * 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 + z) * x) / z) <= (-5d-256)) then
        tmp = (y / z) * x
    else
        tmp = 1.0d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((((y + z) * x) / z) <= -5e-256) {
		tmp = (y / z) * x;
	} else {
		tmp = 1.0 * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (((y + z) * x) / z) <= -5e-256:
		tmp = (y / z) * x
	else:
		tmp = 1.0 * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(Float64(y + z) * x) / z) <= -5e-256)
		tmp = Float64(Float64(y / z) * x);
	else
		tmp = Float64(1.0 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((((y + z) * x) / z) <= -5e-256)
		tmp = (y / z) * x;
	else
		tmp = 1.0 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[(N[(y + z), $MachinePrecision] * x), $MachinePrecision] / z), $MachinePrecision], -5e-256], N[(N[(y / z), $MachinePrecision] * x), $MachinePrecision], N[(1.0 * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (+.f64 y z)) z) < -5e-256

   1. Initial program 80.5%

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

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 97.4

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 97.4

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

   4. Applied rewrites 97.4%

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

   5. Taylor expanded in z around 0

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

      1. lower-/.f64 42.1

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

   7. Applied rewrites 42.1%

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

   if -5e-256 < (/.f64 (*.f64 x (+.f64 y z)) z)

   1. Initial program 82.0%

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

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 95.9

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 95.9

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

   4. Applied rewrites 95.9%

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

   5. Taylor expanded in z around inf

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

   7. Applied rewrites 55.5%

      \[\leadsto \color{blue}{1} \cdot x \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(y + z\right) \cdot x}{z} \leq -5 \cdot 10^{-256}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.9% 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 81.3%

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

3. Taylor expanded in z around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 42.7

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

5. Applied rewrites 42.7%

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

6. Taylor expanded in z around 0

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

   1. distribute-lft-in N/A

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

   2. associate-/l* N/A

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

   3. +-commutative N/A

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

   4. remove-double-neg N/A

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

   5. mul-1-neg N/A

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

   6. unsub-neg N/A

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

   7. div-sub N/A

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

   8. *-inverses N/A

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

   9. associate-*r/ N/A

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

   10. distribute-lft-out-- N/A

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

   11. *-rgt-identity N/A

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

   12. mul-1-neg N/A

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

   13. distribute-rgt-neg-in N/A

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

   14. associate-/l* N/A

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

   15. mul-1-neg N/A

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

   16. sub-neg N/A

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

   17. *-lft-identity N/A

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

   18. distribute-lft-neg-in N/A

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

   19. metadata-eval N/A

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

   20. distribute-lft-out N/A

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

   21. *-lft-identity N/A

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

   22. +-commutative N/A

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

   23. *-commutative N/A

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

   24. associate-*l/ N/A

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

   25. lower-fma.f64 N/A

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

8. Applied rewrites 96.6%

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

Alternative 5: 94.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 81.3%

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

3. Taylor expanded in z around 0

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

   1. distribute-lft-in N/A

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

   2. associate-/l* N/A

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

   3. +-commutative N/A

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

   4. *-lft-identity N/A

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

   5. metadata-eval N/A

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

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

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

   7. div-sub N/A

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

   8. *-inverses N/A

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

   9. associate-*r/ N/A

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

   10. unsub-neg N/A

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

   11. mul-1-neg N/A

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

   12. remove-double-neg N/A

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

   13. +-commutative N/A

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

   14. metadata-eval N/A

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

   15. sub-neg N/A

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

   16. sub-neg N/A

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

   17. metadata-eval N/A

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

   18. distribute-lft-out N/A

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

   19. associate-/l* N/A

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

   20. associate-*l/ N/A

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

   21. *-rgt-identity N/A

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

   22. lower-fma.f64 N/A

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

   23. lower-/.f64 94.3

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

5. Applied rewrites 94.3%

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

Alternative 6: 50.7% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.3%

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

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 96.6

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

   7. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-+.f64 96.6

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

4. Applied rewrites 96.6%

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

5. Taylor expanded in z around inf

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

7. Applied rewrites 57.1%

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

Developer Target 1: 96.2% 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 2024235 
(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))