Linear.Quaternion:$c/ from linear-1.19.1.3, A

Percentage Accurate: 98.4% → 99.4%

Time: 7.4s

Alternatives: 9

Speedup: 0.8×

• 34.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(z, z, \mathsf{fma}\left(x, y, 2 \cdot \left(z \cdot z\right)\right)\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fma z z (fma x y (* 2.0 (* z z)))))

C

double code(double x, double y, double z) {
	return fma(z, z, fma(x, y, (2.0 * (z * z))));
}

Julia

function code(x, y, z)
	return fma(z, z, fma(x, y, Float64(2.0 * Float64(z * z))))
end

Wolfram

code[x_, y_, z_] := N[(z * z + N[(x * y + N[(2.0 * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.7%

   \[\left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z \]
2. Step-by-step derivation

   1. +-commutative 96.7%

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

   2. fma-def 96.8%

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

   3. associate-+l+ 96.8%

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

   4. fma-def 98.8%

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

   5. count-2 98.8%

      \[\leadsto \mathsf{fma}\left(z, z, \mathsf{fma}\left(x, y, \color{blue}{2 \cdot \left(z \cdot z\right)}\right)\right) \]

3. Simplified 98.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(z, z, \mathsf{fma}\left(x, y, 2 \cdot \left(z \cdot z\right)\right)\right)} \]

4. Final simplification 98.8%

   \[\leadsto \mathsf{fma}\left(z, z, \mathsf{fma}\left(x, y, 2 \cdot \left(z \cdot z\right)\right)\right) \]

Alternative 2: 99.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+307}:\\ \;\;\;\;z \cdot z + \left(z \cdot z + \left(z \cdot z + x \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot {z}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 2e+307)
   (+ (* z z) (+ (* z z) (+ (* z z) (* x y))))
   (* 3.0 (pow z 2.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e+307) {
		tmp = (z * z) + ((z * z) + ((z * z) + (x * y)));
	} else {
		tmp = 3.0 * pow(z, 2.0);
	}
	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 * z) <= 2d+307) then
        tmp = (z * z) + ((z * z) + ((z * z) + (x * y)))
    else
        tmp = 3.0d0 * (z ** 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e+307) {
		tmp = (z * z) + ((z * z) + ((z * z) + (x * y)));
	} else {
		tmp = 3.0 * Math.pow(z, 2.0);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z * z) <= 2e+307:
		tmp = (z * z) + ((z * z) + ((z * z) + (x * y)))
	else:
		tmp = 3.0 * math.pow(z, 2.0)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 2e+307)
		tmp = Float64(Float64(z * z) + Float64(Float64(z * z) + Float64(Float64(z * z) + Float64(x * y))));
	else
		tmp = Float64(3.0 * (z ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 2e+307)
		tmp = (z * z) + ((z * z) + ((z * z) + (x * y)));
	else
		tmp = 3.0 * (z ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 2e+307], N[(N[(z * z), $MachinePrecision] + N[(N[(z * z), $MachinePrecision] + N[(N[(z * z), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(3.0 * N[Power[z, 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.99999999999999997e307

   1. Initial program 99.8%

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

   if 1.99999999999999997e307 < (*.f64 z z)

   1. Initial program 88.7%

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

   2. Taylor expanded in x around 0 95.8%

      \[\leadsto \color{blue}{2 \cdot {z}^{2} + {z}^{2}} \]

   4. Simplified 95.8%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+307}:\\ \;\;\;\;z \cdot z + \left(z \cdot z + \left(z \cdot z + x \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot {z}^{2}\\ \end{array} \]

Alternative 3: 99.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, y, \left(z \cdot z\right) \cdot 3\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fma x y (* (* z z) 3.0)))

C

double code(double x, double y, double z) {
	return fma(x, y, ((z * z) * 3.0));
}

Julia

function code(x, y, z)
	return fma(x, y, Float64(Float64(z * z) * 3.0))
end

Wolfram

code[x_, y_, z_] := N[(x * y + N[(N[(z * z), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.7%

   \[\left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z \]
2. Step-by-step derivation

   1. associate-+l+ 96.7%

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

   2. associate-+l+ 96.7%

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

   3. fma-def 98.7%

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

   4. cancel-sign-sub 98.7%

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

   5. neg-mul-1 98.7%

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

   6. associate-*l* 98.7%

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

   7. count-2 98.7%

      \[\leadsto \mathsf{fma}\left(x, y, \color{blue}{2 \cdot \left(z \cdot z\right)} - -1 \cdot \left(z \cdot z\right)\right) \]

   8. distribute-rgt-out-- 98.7%

      \[\leadsto \mathsf{fma}\left(x, y, \color{blue}{\left(z \cdot z\right) \cdot \left(2 - -1\right)}\right) \]

   9. metadata-eval 98.7%

      \[\leadsto \mathsf{fma}\left(x, y, \left(z \cdot z\right) \cdot \color{blue}{3}\right) \]

3. Simplified 98.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, \left(z \cdot z\right) \cdot 3\right)} \]

4. Final simplification 98.7%

   \[\leadsto \mathsf{fma}\left(x, y, \left(z \cdot z\right) \cdot 3\right) \]

Alternative 4: 99.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+307}:\\ \;\;\;\;z \cdot z + \left(z \cdot z + \left(z \cdot z + x \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot z + \left(z \cdot z + \left(z \cdot z + -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 2e+307)
   (+ (* z z) (+ (* z z) (+ (* z z) (* x y))))
   (+ (* z z) (+ (* z z) (+ (* z z) -1.0)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e+307) {
		tmp = (z * z) + ((z * z) + ((z * z) + (x * y)));
	} else {
		tmp = (z * z) + ((z * z) + ((z * z) + -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 * z) <= 2d+307) then
        tmp = (z * z) + ((z * z) + ((z * z) + (x * y)))
    else
        tmp = (z * z) + ((z * z) + ((z * z) + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e+307) {
		tmp = (z * z) + ((z * z) + ((z * z) + (x * y)));
	} else {
		tmp = (z * z) + ((z * z) + ((z * z) + -1.0));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z * z) <= 2e+307:
		tmp = (z * z) + ((z * z) + ((z * z) + (x * y)))
	else:
		tmp = (z * z) + ((z * z) + ((z * z) + -1.0))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 2e+307)
		tmp = Float64(Float64(z * z) + Float64(Float64(z * z) + Float64(Float64(z * z) + Float64(x * y))));
	else
		tmp = Float64(Float64(z * z) + Float64(Float64(z * z) + Float64(Float64(z * z) + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 2e+307)
		tmp = (z * z) + ((z * z) + ((z * z) + (x * y)));
	else
		tmp = (z * z) + ((z * z) + ((z * z) + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 2e+307], N[(N[(z * z), $MachinePrecision] + N[(N[(z * z), $MachinePrecision] + N[(N[(z * z), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(z * z), $MachinePrecision] + N[(N[(z * z), $MachinePrecision] + N[(N[(z * z), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.99999999999999997e307

   1. Initial program 99.8%

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

   if 1.99999999999999997e307 < (*.f64 z z)

   1. Initial program 88.7%

      \[\left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z \]
   2. Step-by-step derivation

      1. expm1-log1p-u 88.7%

         \[\leadsto \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot y + z \cdot z\right)\right)} + z \cdot z\right) + z \cdot z \]

      2. fma-def 95.8%

         \[\leadsto \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(x, y, z \cdot z\right)}\right)\right) + z \cdot z\right) + z \cdot z \]

      3. pow2 95.8%

         \[\leadsto \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(x, y, \color{blue}{{z}^{2}}\right)\right)\right) + z \cdot z\right) + z \cdot z \]

   3. Applied egg-rr 95.8%

      \[\leadsto \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(x, y, {z}^{2}\right)\right)\right)} + z \cdot z\right) + z \cdot z \]

   4. Taylor expanded in z around inf 39.4%

      \[\leadsto \left(\mathsf{expm1}\left(\color{blue}{-2 \cdot \log \left(\frac{1}{z}\right)}\right) + z \cdot z\right) + z \cdot z \]
   5. Step-by-step derivation

      1. log-rec 39.4%

         \[\leadsto \left(\mathsf{expm1}\left(-2 \cdot \color{blue}{\left(-\log z\right)}\right) + z \cdot z\right) + z \cdot z \]

   6. Simplified 39.4%

      \[\leadsto \left(\mathsf{expm1}\left(\color{blue}{-2 \cdot \left(-\log z\right)}\right) + z \cdot z\right) + z \cdot z \]

   7. Taylor expanded in z around inf 95.8%

      \[\leadsto \left(\color{blue}{\left({\left(\frac{1}{z}\right)}^{-2} - 1\right)} + z \cdot z\right) + z \cdot z \]
   8. Step-by-step derivation

      1. inv-pow 95.8%

         \[\leadsto \left(\left({\color{blue}{\left({z}^{-1}\right)}}^{-2} - 1\right) + z \cdot z\right) + z \cdot z \]

      2. pow-pow 95.8%

         \[\leadsto \left(\left(\color{blue}{{z}^{\left(-1 \cdot -2\right)}} - 1\right) + z \cdot z\right) + z \cdot z \]

      3. metadata-eval 95.8%

         \[\leadsto \left(\left({z}^{\color{blue}{2}} - 1\right) + z \cdot z\right) + z \cdot z \]

      4. pow2 95.8%

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

   9. Applied egg-rr 95.8%

      \[\leadsto \left(\left(\color{blue}{z \cdot z} - 1\right) + z \cdot z\right) + z \cdot z \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+307}:\\ \;\;\;\;z \cdot z + \left(z \cdot z + \left(z \cdot z + x \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot z + \left(z \cdot z + \left(z \cdot z + -1\right)\right)\\ \end{array} \]

Alternative 5: 86.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 4.5 \cdot 10^{+33}:\\ \;\;\;\;z \cdot z + \left(z \cdot z + x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot z + \left(z \cdot z + \left(z \cdot z + -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 4.5e+33)
   (+ (* z z) (+ (* z z) (* x y)))
   (+ (* z z) (+ (* z z) (+ (* z z) -1.0)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 4.5e+33) {
		tmp = (z * z) + ((z * z) + (x * y));
	} else {
		tmp = (z * z) + ((z * z) + ((z * z) + -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 * z) <= 4.5d+33) then
        tmp = (z * z) + ((z * z) + (x * y))
    else
        tmp = (z * z) + ((z * z) + ((z * z) + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 4.5e+33) {
		tmp = (z * z) + ((z * z) + (x * y));
	} else {
		tmp = (z * z) + ((z * z) + ((z * z) + -1.0));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z * z) <= 4.5e+33:
		tmp = (z * z) + ((z * z) + (x * y))
	else:
		tmp = (z * z) + ((z * z) + ((z * z) + -1.0))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 4.5e+33)
		tmp = Float64(Float64(z * z) + Float64(Float64(z * z) + Float64(x * y)));
	else
		tmp = Float64(Float64(z * z) + Float64(Float64(z * z) + Float64(Float64(z * z) + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 4.5e+33)
		tmp = (z * z) + ((z * z) + (x * y));
	else
		tmp = (z * z) + ((z * z) + ((z * z) + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 4.5e+33], N[(N[(z * z), $MachinePrecision] + N[(N[(z * z), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(z * z), $MachinePrecision] + N[(N[(z * z), $MachinePrecision] + N[(N[(z * z), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 4.5e33

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 89.1%

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

   if 4.5e33 < (*.f64 z z)

   1. Initial program 93.6%

      \[\left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z \]
   2. Step-by-step derivation

      1. expm1-log1p-u 87.2%

         \[\leadsto \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot y + z \cdot z\right)\right)} + z \cdot z\right) + z \cdot z \]

      2. fma-def 91.1%

         \[\leadsto \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(x, y, z \cdot z\right)}\right)\right) + z \cdot z\right) + z \cdot z \]

      3. pow2 91.1%

         \[\leadsto \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(x, y, \color{blue}{{z}^{2}}\right)\right)\right) + z \cdot z\right) + z \cdot z \]

   3. Applied egg-rr 91.1%

      \[\leadsto \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(x, y, {z}^{2}\right)\right)\right)} + z \cdot z\right) + z \cdot z \]

   4. Taylor expanded in z around inf 33.6%

      \[\leadsto \left(\mathsf{expm1}\left(\color{blue}{-2 \cdot \log \left(\frac{1}{z}\right)}\right) + z \cdot z\right) + z \cdot z \]
   5. Step-by-step derivation

      1. log-rec 33.6%

         \[\leadsto \left(\mathsf{expm1}\left(-2 \cdot \color{blue}{\left(-\log z\right)}\right) + z \cdot z\right) + z \cdot z \]

   6. Simplified 33.6%

      \[\leadsto \left(\mathsf{expm1}\left(\color{blue}{-2 \cdot \left(-\log z\right)}\right) + z \cdot z\right) + z \cdot z \]

   7. Taylor expanded in z around inf 88.1%

      \[\leadsto \left(\color{blue}{\left({\left(\frac{1}{z}\right)}^{-2} - 1\right)} + z \cdot z\right) + z \cdot z \]
   8. Step-by-step derivation

      1. inv-pow 88.1%

         \[\leadsto \left(\left({\color{blue}{\left({z}^{-1}\right)}}^{-2} - 1\right) + z \cdot z\right) + z \cdot z \]

      2. pow-pow 88.1%

         \[\leadsto \left(\left(\color{blue}{{z}^{\left(-1 \cdot -2\right)}} - 1\right) + z \cdot z\right) + z \cdot z \]

      3. metadata-eval 88.1%

         \[\leadsto \left(\left({z}^{\color{blue}{2}} - 1\right) + z \cdot z\right) + z \cdot z \]

      4. pow2 88.1%

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

   9. Applied egg-rr 88.1%

      \[\leadsto \left(\left(\color{blue}{z \cdot z} - 1\right) + z \cdot z\right) + z \cdot z \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 4.5 \cdot 10^{+33}:\\ \;\;\;\;z \cdot z + \left(z \cdot z + x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot z + \left(z \cdot z + \left(z \cdot z + -1\right)\right)\\ \end{array} \]

Alternative 6: 77.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ z \cdot z + \left(z \cdot z + x \cdot y\right) \end{array} \]

FPCore

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

C

double code(double x, double y, double z) {
	return (z * z) + ((z * z) + (x * y));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (z * z) + ((z * z) + (x * y))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.7%

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

2. Taylor expanded in x around inf 77.1%

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

3. Final simplification 77.1%

   \[\leadsto z \cdot z + \left(z \cdot z + x \cdot y\right) \]

Alternative 7: 77.3% accurate, 2.1× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return (z * z) + (x * y);
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (z * z) + (x * y)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.7%

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

2. Taylor expanded in x around inf 77.1%

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

3. Taylor expanded in x around inf 76.5%

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

4. Final simplification 76.5%

   \[\leadsto z \cdot z + x \cdot y \]

Alternative 8: 52.2% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.7%

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

2. Taylor expanded in x around 0 96.7%

   \[\leadsto \color{blue}{2 \cdot {z}^{2} + \left(x \cdot y + {z}^{2}\right)} \]

4. Simplified 96.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left({z}^{2}, 3, x \cdot y\right)} \]

5. Taylor expanded in z around 0 51.3%

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

6. Final simplification 51.3%

   \[\leadsto x \cdot y \]

Alternative 9: 2.1% accurate, 15.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

function code(x, y, z)
	return -1.0
end

MATLAB

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

Wolfram

code[x_, y_, z_] := -1.0

Derivation

1. Initial program 96.7%

   \[\left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z \]
2. Step-by-step derivation

   1. expm1-log1p-u 80.0%

      \[\leadsto \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot y + z \cdot z\right)\right)} + z \cdot z\right) + z \cdot z \]

   2. fma-def 82.0%

      \[\leadsto \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(x, y, z \cdot z\right)}\right)\right) + z \cdot z\right) + z \cdot z \]

   3. pow2 82.0%

      \[\leadsto \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(x, y, \color{blue}{{z}^{2}}\right)\right)\right) + z \cdot z\right) + z \cdot z \]

3. Applied egg-rr 82.0%

   \[\leadsto \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(x, y, {z}^{2}\right)\right)\right)} + z \cdot z\right) + z \cdot z \]

4. Taylor expanded in z around inf 17.9%

   \[\leadsto \left(\mathsf{expm1}\left(\color{blue}{-2 \cdot \log \left(\frac{1}{z}\right)}\right) + z \cdot z\right) + z \cdot z \]
5. Step-by-step derivation

   1. log-rec 17.9%

      \[\leadsto \left(\mathsf{expm1}\left(-2 \cdot \color{blue}{\left(-\log z\right)}\right) + z \cdot z\right) + z \cdot z \]

6. Simplified 17.9%

   \[\leadsto \left(\mathsf{expm1}\left(\color{blue}{-2 \cdot \left(-\log z\right)}\right) + z \cdot z\right) + z \cdot z \]

7. Taylor expanded in z around inf 46.2%

   \[\leadsto \left(\color{blue}{\left({\left(\frac{1}{z}\right)}^{-2} - 1\right)} + z \cdot z\right) + z \cdot z \]

8. Taylor expanded in z around 0 2.1%

   \[\leadsto \color{blue}{-1} \]

9. Final simplification 2.1%

   \[\leadsto -1 \]

Developer target: 98.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \left(3 \cdot z\right) \cdot z + y \cdot x \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ (* (* 3.0 z) z) (* y x)))

C

double code(double x, double y, double z) {
	return ((3.0 * z) * z) + (y * 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 = ((3.0d0 * z) * z) + (y * x)
end function

Java

public static double code(double x, double y, double z) {
	return ((3.0 * z) * z) + (y * x);
}

Python

def code(x, y, z):
	return ((3.0 * z) * z) + (y * x)

Julia

function code(x, y, z)
	return Float64(Float64(Float64(3.0 * z) * z) + Float64(y * x))
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((3.0 * z) * z) + (y * x);
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(3.0 * z), $MachinePrecision] * z), $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2023308 
(FPCore (x y z)
  :name "Linear.Quaternion:$c/ from linear-1.19.1.3, A"
  :precision binary64

  :herbie-target
  (+ (* (* 3.0 z) z) (* y x))

  (+ (+ (+ (* x y) (* z z)) (* z z)) (* z z)))