Numeric.SpecFunctions:choose from math-functions-0.1.5.2

Percentage Accurate: 84.5% → 95.6%

Time: 5.4s

Alternatives: 5

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: 95.6% 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 84.1%

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

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

   3. +-commutative N/A

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

   4. *-lft-identity N/A

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

   5. metadata-eval N/A

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

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

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

   7. div-sub N/A

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

   8. *-inverses N/A

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

   9. associate-*r/ N/A

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

   10. unsub-neg N/A

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

   11. mul-1-neg N/A

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

   12. remove-double-neg N/A

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

   13. +-commutative 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. lower-fma.f64 N/A

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

   16. lower-/.f64 98.5

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

5. Applied rewrites 98.5%

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

Alternative 2: 71.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7.5 \cdot 10^{-93}:\\ \;\;\;\;\frac{x}{1}\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{-38}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -7.5e-93) (/ x 1.0) (if (<= z 2.5e-38) (/ (* y x) z) (/ x 1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -7.5e-93) {
		tmp = x / 1.0;
	} else if (z <= 2.5e-38) {
		tmp = (y * x) / z;
	} else {
		tmp = x / 1.0;
	}
	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) :: tmp
    if (z <= (-7.5d-93)) then
        tmp = x / 1.0d0
    else if (z <= 2.5d-38) then
        tmp = (y * x) / z
    else
        tmp = x / 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -7.5e-93) {
		tmp = x / 1.0;
	} else if (z <= 2.5e-38) {
		tmp = (y * x) / z;
	} else {
		tmp = x / 1.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -7.5e-93:
		tmp = x / 1.0
	elif z <= 2.5e-38:
		tmp = (y * x) / z
	else:
		tmp = x / 1.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -7.5e-93)
		tmp = Float64(x / 1.0);
	elseif (z <= 2.5e-38)
		tmp = Float64(Float64(y * x) / z);
	else
		tmp = Float64(x / 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -7.5e-93)
		tmp = x / 1.0;
	elseif (z <= 2.5e-38)
		tmp = (y * x) / z;
	else
		tmp = x / 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -7.5e-93], N[(x / 1.0), $MachinePrecision], If[LessEqual[z, 2.5e-38], N[(N[(y * x), $MachinePrecision] / z), $MachinePrecision], N[(x / 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -7.50000000000000034e-93 or 2.50000000000000017e-38 < z

   1. Initial program 79.8%

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

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

      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 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 73.6%

      \[\leadsto \frac{x}{\color{blue}{1}} \]

   if -7.50000000000000034e-93 < z < 2.50000000000000017e-38

   1. Initial program 91.5%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 84.7

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

   5. Applied rewrites 84.7%

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

Alternative 3: 69.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7.4 \cdot 10^{-93}:\\ \;\;\;\;\frac{x}{1}\\ \mathbf{elif}\;z \leq 2 \cdot 10^{-38}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -7.4e-93) (/ x 1.0) (if (<= z 2e-38) (* (/ y z) x) (/ x 1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -7.4e-93) {
		tmp = x / 1.0;
	} else if (z <= 2e-38) {
		tmp = (y / z) * x;
	} else {
		tmp = x / 1.0;
	}
	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) :: tmp
    if (z <= (-7.4d-93)) then
        tmp = x / 1.0d0
    else if (z <= 2d-38) then
        tmp = (y / z) * x
    else
        tmp = x / 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -7.4e-93) {
		tmp = x / 1.0;
	} else if (z <= 2e-38) {
		tmp = (y / z) * x;
	} else {
		tmp = x / 1.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -7.4e-93:
		tmp = x / 1.0
	elif z <= 2e-38:
		tmp = (y / z) * x
	else:
		tmp = x / 1.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -7.4e-93)
		tmp = Float64(x / 1.0);
	elseif (z <= 2e-38)
		tmp = Float64(Float64(y / z) * x);
	else
		tmp = Float64(x / 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -7.4e-93)
		tmp = x / 1.0;
	elseif (z <= 2e-38)
		tmp = (y / z) * x;
	else
		tmp = x / 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -7.4e-93], N[(x / 1.0), $MachinePrecision], If[LessEqual[z, 2e-38], N[(N[(y / z), $MachinePrecision] * x), $MachinePrecision], N[(x / 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -7.40000000000000005e-93 or 1.9999999999999999e-38 < z

   1. Initial program 79.8%

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

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

      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 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 73.6%

      \[\leadsto \frac{x}{\color{blue}{1}} \]

   if -7.40000000000000005e-93 < z < 1.9999999999999999e-38

   1. Initial program 91.5%

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

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

      3. +-commutative N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

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

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

      7. div-sub N/A

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

      8. *-inverses N/A

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

      9. associate-*r/ N/A

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

      10. unsub-neg N/A

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

      11. mul-1-neg N/A

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

      12. remove-double-neg N/A

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

      13. +-commutative 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. lower-fma.f64 N/A

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

      16. lower-/.f64 95.9

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

   5. Applied rewrites 95.9%

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

   6. Taylor expanded in y around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 79.3

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

   8. Applied rewrites 79.3%

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

   10. Applied rewrites 83.6%

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

Alternative 4: 71.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7.5 \cdot 10^{-93}:\\ \;\;\;\;\frac{x}{1}\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{-38}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -7.5e-93) (/ x 1.0) (if (<= z 2.5e-38) (* (/ x z) y) (/ x 1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -7.5e-93) {
		tmp = x / 1.0;
	} else if (z <= 2.5e-38) {
		tmp = (x / z) * y;
	} else {
		tmp = x / 1.0;
	}
	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) :: tmp
    if (z <= (-7.5d-93)) then
        tmp = x / 1.0d0
    else if (z <= 2.5d-38) then
        tmp = (x / z) * y
    else
        tmp = x / 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -7.5e-93) {
		tmp = x / 1.0;
	} else if (z <= 2.5e-38) {
		tmp = (x / z) * y;
	} else {
		tmp = x / 1.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -7.5e-93:
		tmp = x / 1.0
	elif z <= 2.5e-38:
		tmp = (x / z) * y
	else:
		tmp = x / 1.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -7.5e-93)
		tmp = Float64(x / 1.0);
	elseif (z <= 2.5e-38)
		tmp = Float64(Float64(x / z) * y);
	else
		tmp = Float64(x / 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -7.5e-93)
		tmp = x / 1.0;
	elseif (z <= 2.5e-38)
		tmp = (x / z) * y;
	else
		tmp = x / 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -7.5e-93], N[(x / 1.0), $MachinePrecision], If[LessEqual[z, 2.5e-38], N[(N[(x / z), $MachinePrecision] * y), $MachinePrecision], N[(x / 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -7.50000000000000034e-93 or 2.50000000000000017e-38 < z

   1. Initial program 79.8%

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

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

      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 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 73.6%

      \[\leadsto \frac{x}{\color{blue}{1}} \]

   if -7.50000000000000034e-93 < z < 2.50000000000000017e-38

   1. Initial program 91.5%

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

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

      3. +-commutative N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

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

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

      7. div-sub N/A

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

      8. *-inverses N/A

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

      9. associate-*r/ N/A

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

      10. unsub-neg N/A

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

      11. mul-1-neg N/A

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

      12. remove-double-neg N/A

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

      13. +-commutative 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. lower-fma.f64 N/A

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

      16. lower-/.f64 95.9

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

   5. Applied rewrites 95.9%

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

   6. Taylor expanded in y around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 79.3

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

   8. Applied rewrites 79.3%

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

Alternative 5: 50.1% 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 84.1%

   \[\frac{x \cdot \left(y + z\right)}{z} \]
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 98.4

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

   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 98.4

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

4. Applied rewrites 98.4%

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

5. Taylor expanded in y around 0

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

7. Applied rewrites 51.9%

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

Developer Target 1: 95.8% 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 2024296 
(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))