Numeric.SpecFunctions:choose from math-functions-0.1.5.2

Percentage Accurate: 84.5% → 95.6%

Time: 6.1s

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: 84.5% 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, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 6.3 \cdot 10^{+91}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(z + y\right) \cdot x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 6.3e+91) (fma (/ y z) x x) (/ (* (+ z y) x) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 6.3e+91) {
		tmp = fma((y / z), x, x);
	} else {
		tmp = ((z + y) * x) / z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 6.3e+91)
		tmp = fma(Float64(y / z), x, x);
	else
		tmp = Float64(Float64(Float64(z + y) * x) / z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 6.3e+91], N[(N[(y / z), $MachinePrecision] * x + x), $MachinePrecision], N[(N[(N[(z + y), $MachinePrecision] * x), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.3e91

   1. Initial program 85.2%

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

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

      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 97.6

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

   5. Applied rewrites 97.6%

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

   if 6.3e91 < y

   1. Initial program 96.4%

      \[\frac{x \cdot \left(y + z\right)}{z} \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.3 \cdot 10^{+91}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(z + y\right) \cdot x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 72.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.5 \cdot 10^{+57}:\\ \;\;\;\;\frac{x}{1}\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{+26}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -4.5e+57) (/ x 1.0) (if (<= z 3.2e+26) (/ (* x y) z) (/ x 1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -4.5e+57) {
		tmp = x / 1.0;
	} else if (z <= 3.2e+26) {
		tmp = (x * y) / 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 <= (-4.5d+57)) then
        tmp = x / 1.0d0
    else if (z <= 3.2d+26) then
        tmp = (x * y) / 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 <= -4.5e+57) {
		tmp = x / 1.0;
	} else if (z <= 3.2e+26) {
		tmp = (x * y) / z;
	} else {
		tmp = x / 1.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -4.5e+57:
		tmp = x / 1.0
	elif z <= 3.2e+26:
		tmp = (x * y) / z
	else:
		tmp = x / 1.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -4.5e+57)
		tmp = Float64(x / 1.0);
	elseif (z <= 3.2e+26)
		tmp = Float64(Float64(x * y) / z);
	else
		tmp = Float64(x / 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -4.5e+57)
		tmp = x / 1.0;
	elseif (z <= 3.2e+26)
		tmp = (x * y) / z;
	else
		tmp = x / 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -4.5e+57], N[(x / 1.0), $MachinePrecision], If[LessEqual[z, 3.2e+26], N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision], N[(x / 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -4.49999999999999996e57 or 3.20000000000000029e26 < z

   1. Initial program 77.6%

      \[\frac{x \cdot \left(y + z\right)}{z} \]
   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 100.0

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

      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 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 84.2%

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

   if -4.49999999999999996e57 < z < 3.20000000000000029e26

   1. Initial program 94.2%

      \[\frac{x \cdot \left(y + z\right)}{z} \]
   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 74.3

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

   5. Applied rewrites 74.3%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.5 \cdot 10^{+57}:\\ \;\;\;\;\frac{x}{1}\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{+26}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 72.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7.5 \cdot 10^{+28}:\\ \;\;\;\;\frac{x}{1}\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{+26}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -7.5e+28) (/ x 1.0) (if (<= z 3.2e+26) (* (/ x z) y) (/ x 1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -7.5e+28) {
		tmp = x / 1.0;
	} else if (z <= 3.2e+26) {
		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+28)) then
        tmp = x / 1.0d0
    else if (z <= 3.2d+26) 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+28) {
		tmp = x / 1.0;
	} else if (z <= 3.2e+26) {
		tmp = (x / z) * y;
	} else {
		tmp = x / 1.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -7.5e+28:
		tmp = x / 1.0
	elif z <= 3.2e+26:
		tmp = (x / z) * y
	else:
		tmp = x / 1.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -7.5e+28)
		tmp = Float64(x / 1.0);
	elseif (z <= 3.2e+26)
		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+28)
		tmp = x / 1.0;
	elseif (z <= 3.2e+26)
		tmp = (x / z) * y;
	else
		tmp = x / 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -7.5e+28], N[(x / 1.0), $MachinePrecision], If[LessEqual[z, 3.2e+26], N[(N[(x / z), $MachinePrecision] * y), $MachinePrecision], N[(x / 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -7.4999999999999998e28 or 3.20000000000000029e26 < z

   1. Initial program 78.2%

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

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

      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 82.9%

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

   if -7.4999999999999998e28 < z < 3.20000000000000029e26

   1. Initial program 94.1%

      \[\frac{x \cdot \left(y + z\right)}{z} \]
   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 92.3

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

      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 92.3

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

   4. Applied rewrites 92.3%

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

   5. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{x \cdot y}{z}} \]
   6. 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 73.5

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

   7. Applied rewrites 73.5%

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

Alternative 4: 95.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.4 \cdot 10^{-57}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 3.4e-57) (fma (/ x z) y x) (fma (/ y z) x x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 3.4e-57) {
		tmp = fma((x / z), y, x);
	} else {
		tmp = fma((y / z), x, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 3.4e-57)
		tmp = fma(Float64(x / z), y, x);
	else
		tmp = fma(Float64(y / z), x, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 3.4e-57], N[(N[(x / z), $MachinePrecision] * y + x), $MachinePrecision], N[(N[(y / z), $MachinePrecision] * x + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.40000000000000016e-57

   1. Initial program 88.6%

      \[\frac{x \cdot \left(y + z\right)}{z} \]
   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 93.8

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

      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 93.8

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

   4. Applied rewrites 93.8%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/r/ N/A

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

      5. lift-+.f64 N/A

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

      6. distribute-lft-in N/A

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

      7. lft-mult-inverse N/A

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

      8. associate-/r/ N/A

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

      9. clear-num N/A

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

      10. lift-/.f64 N/A

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

      11. +-commutative N/A

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

      12. distribute-rgt-in N/A

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

      13. lift-/.f64 N/A

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

      14. associate-*l/ N/A

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

      15. *-commutative N/A

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

      16. associate-*l/ N/A

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

      17. lift-/.f64 N/A

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

      18. *-lft-identity N/A

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

      19. lower-fma.f64 93.1

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

   6. Applied rewrites 93.1%

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

   if 3.40000000000000016e-57 < x

   1. Initial program 84.7%

      \[\frac{x \cdot \left(y + z\right)}{z} \]
   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 99.9

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

   5. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{z}, x, x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 96.0% 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 87.5%

   \[\frac{x \cdot \left(y + z\right)}{z} \]
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 95.2

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

5. Applied rewrites 95.2%

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

Alternative 6: 50.8% 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 87.5%

   \[\frac{x \cdot \left(y + z\right)}{z} \]
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. 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 95.5

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

   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 95.5

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

4. Applied rewrites 95.5%

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

5. Taylor expanded in y around 0

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

7. Applied rewrites 48.0%

   \[\leadsto \frac{x}{\color{blue}{1}} \]
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 2024331 
(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))