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

Percentage Accurate: 98.1% → 99.3%

Time: 1.4s

Alternatives: 5

Speedup: 1.7×

• 66.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

(FPCore (x y z)
  :precision binary64
  :pre TRUE
  (+ (+ (+ (* 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)
use fmin_fmax_functions
    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]

ReFlow

f(x, y, z):
	x in [-inf, +inf],
	y in [-inf, +inf],
	z in [-inf, +inf]
code: THEORY
BEGIN
f(x, y, z: real): real =
	(((x * y) + (z * z)) + (z * z)) + (z * z)
END code

Initial Program: 98.1% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z)
  :precision binary64
  :pre TRUE
  (+ (+ (+ (* 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)
use fmin_fmax_functions
    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]

ReFlow

f(x, y, z):
	x in [-inf, +inf],
	y in [-inf, +inf],
	z in [-inf, +inf]
code: THEORY
BEGIN
f(x, y, z: real): real =
	(((x * y) + (z * z)) + (z * z)) + (z * z)
END code

Alternative 1: 99.3% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

ReFlow

f(x, y, z):
	x in [-inf, +inf],
	y in [-inf, +inf],
	z in [-inf, +inf]
code: THEORY
BEGIN
f(x, y, z: real): real =
	(x * y) + ((3) * (z * z))
END code

Derivation

1. Initial program 98.1%

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

3. Applied rewrites 99.3%

   \[\leadsto \mathsf{fma}\left(x, y, \left(3 \cdot z\right) \cdot z\right) \]
4. Step-by-step derivation

5. Applied rewrites 99.3%

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

Alternative 2: 98.2% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

ReFlow

f(x, y, z):
	x in [-inf, +inf],
	y in [-inf, +inf],
	z in [-inf, +inf]
code: THEORY
BEGIN
f(x, y, z: real): real =
	((3) * (z * z)) + (y * x)
END code

Derivation

1. Initial program 98.1%

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

3. Applied rewrites 98.2%

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

Alternative 3: 83.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+109}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot z\right) \cdot 3\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  :pre TRUE
  (if (<= (* z z) 5e+109) (* x y) (* (* z z) 3.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 5e+109) {
		tmp = x * y;
	} else {
		tmp = (z * z) * 3.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z * z) <= 5d+109) then
        tmp = x * y
    else
        tmp = (z * z) * 3.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 5e+109) {
		tmp = x * y;
	} else {
		tmp = (z * z) * 3.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z * z) <= 5e+109:
		tmp = x * y
	else:
		tmp = (z * z) * 3.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 5e+109)
		tmp = Float64(x * y);
	else
		tmp = Float64(Float64(z * z) * 3.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 5e+109)
		tmp = x * y;
	else
		tmp = (z * z) * 3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 5e+109], N[(x * y), $MachinePrecision], N[(N[(z * z), $MachinePrecision] * 3.0), $MachinePrecision]]

ReFlow

f(x, y, z):
	x in [-inf, +inf],
	y in [-inf, +inf],
	z in [-inf, +inf]
code: THEORY
BEGIN
f(x, y, z: real): real =
	LET tmp = IF ((z * z) <= (50000000000000001178468375708512791662476639752844093156495626963414083423308086629915468079622475513115705344)) THEN (x * y) ELSE ((z * z) * (3)) ENDIF IN
	tmp
END code

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 5.0000000000000001e109

   1. Initial program 98.1%

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

   3. Applied rewrites 98.2%

      \[\leadsto \mathsf{fma}\left(3, z \cdot z, y \cdot x\right) \]
   4. Step-by-step derivation

   5. Applied rewrites 97.9%

      \[\leadsto \mathsf{fma}\left(3, {\left(\sqrt{\left|z\right|}\right)}^{4}, y \cdot x\right) \]

   6. Taylor expanded in x around 0

      \[\leadsto 3 \cdot {\left(\sqrt{\left|z\right|}\right)}^{4} \]
   7. Step-by-step derivation

   8. Applied rewrites 53.7%

      \[\leadsto 3 \cdot {\left(\sqrt{\left|z\right|}\right)}^{4} \]

   9. Taylor expanded in x around inf

      \[\leadsto x \cdot y \]
   10. Step-by-step derivation

   11. Applied rewrites 52.5%

       \[\leadsto x \cdot y \]

   if 5.0000000000000001e109 < (*.f64 z z)

   1. Initial program 98.1%

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

   3. Applied rewrites 98.2%

      \[\leadsto \mathsf{fma}\left(3, z \cdot z, y \cdot x\right) \]
   4. Step-by-step derivation

   5. Applied rewrites 97.9%

      \[\leadsto \mathsf{fma}\left(3, {\left(\sqrt{\left|z\right|}\right)}^{4}, y \cdot x\right) \]

   6. Taylor expanded in x around 0

      \[\leadsto 3 \cdot {\left(\sqrt{\left|z\right|}\right)}^{4} \]
   7. Step-by-step derivation

   8. Applied rewrites 53.7%

      \[\leadsto 3 \cdot {\left(\sqrt{\left|z\right|}\right)}^{4} \]
   9. Step-by-step derivation

   10. Applied rewrites 54.0%

       \[\leadsto \left(z \cdot z\right) \cdot 3 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 83.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+109}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(3 \cdot z\right)\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  :pre TRUE
  (if (<= (* z z) 5e+109) (* x y) (* z (* 3.0 z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 5e+109) {
		tmp = x * y;
	} else {
		tmp = z * (3.0 * z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z * z) <= 5d+109) then
        tmp = x * y
    else
        tmp = z * (3.0d0 * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 5e+109) {
		tmp = x * y;
	} else {
		tmp = z * (3.0 * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z * z) <= 5e+109:
		tmp = x * y
	else:
		tmp = z * (3.0 * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 5e+109)
		tmp = Float64(x * y);
	else
		tmp = Float64(z * Float64(3.0 * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 5e+109)
		tmp = x * y;
	else
		tmp = z * (3.0 * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 5e+109], N[(x * y), $MachinePrecision], N[(z * N[(3.0 * z), $MachinePrecision]), $MachinePrecision]]

ReFlow

f(x, y, z):
	x in [-inf, +inf],
	y in [-inf, +inf],
	z in [-inf, +inf]
code: THEORY
BEGIN
f(x, y, z: real): real =
	LET tmp = IF ((z * z) <= (50000000000000001178468375708512791662476639752844093156495626963414083423308086629915468079622475513115705344)) THEN (x * y) ELSE (z * ((3) * z)) ENDIF IN
	tmp
END code

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 5.0000000000000001e109

   1. Initial program 98.1%

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

   3. Applied rewrites 98.2%

      \[\leadsto \mathsf{fma}\left(3, z \cdot z, y \cdot x\right) \]
   4. Step-by-step derivation

   5. Applied rewrites 97.9%

      \[\leadsto \mathsf{fma}\left(3, {\left(\sqrt{\left|z\right|}\right)}^{4}, y \cdot x\right) \]

   6. Taylor expanded in x around 0

      \[\leadsto 3 \cdot {\left(\sqrt{\left|z\right|}\right)}^{4} \]
   7. Step-by-step derivation

   8. Applied rewrites 53.7%

      \[\leadsto 3 \cdot {\left(\sqrt{\left|z\right|}\right)}^{4} \]

   9. Taylor expanded in x around inf

      \[\leadsto x \cdot y \]
   10. Step-by-step derivation

   11. Applied rewrites 52.5%

       \[\leadsto x \cdot y \]

   if 5.0000000000000001e109 < (*.f64 z z)

   1. Initial program 98.1%

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

   3. Applied rewrites 98.2%

      \[\leadsto \mathsf{fma}\left(3, z \cdot z, y \cdot x\right) \]
   4. Step-by-step derivation

   5. Applied rewrites 97.9%

      \[\leadsto \mathsf{fma}\left(3, {\left(\sqrt{\left|z\right|}\right)}^{4}, y \cdot x\right) \]

   6. Taylor expanded in x around 0

      \[\leadsto 3 \cdot {\left(\sqrt{\left|z\right|}\right)}^{4} \]
   7. Step-by-step derivation

   8. Applied rewrites 53.7%

      \[\leadsto 3 \cdot {\left(\sqrt{\left|z\right|}\right)}^{4} \]
   9. Step-by-step derivation

   10. Applied rewrites 54.0%

       \[\leadsto z \cdot \left(3 \cdot z\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 52.5% accurate, 5.2× speedup

Math

\[x \cdot y \]

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
use fmin_fmax_functions
    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]

ReFlow

f(x, y, z):
	x in [-inf, +inf],
	y in [-inf, +inf],
	z in [-inf, +inf]
code: THEORY
BEGIN
f(x, y, z: real): real =
	x * y
END code

Derivation

1. Initial program 98.1%

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

3. Applied rewrites 98.2%

   \[\leadsto \mathsf{fma}\left(3, z \cdot z, y \cdot x\right) \]
4. Step-by-step derivation

5. Applied rewrites 97.9%

   \[\leadsto \mathsf{fma}\left(3, {\left(\sqrt{\left|z\right|}\right)}^{4}, y \cdot x\right) \]

6. Taylor expanded in x around 0

   \[\leadsto 3 \cdot {\left(\sqrt{\left|z\right|}\right)}^{4} \]
7. Step-by-step derivation

8. Applied rewrites 53.7%

   \[\leadsto 3 \cdot {\left(\sqrt{\left|z\right|}\right)}^{4} \]

9. Taylor expanded in x around inf

   \[\leadsto x \cdot y \]
10. Step-by-step derivation

11. Applied rewrites 52.5%

    \[\leadsto x \cdot y \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2026092 
(FPCore (x y z)
  :name "Linear.Quaternion:$c/ from linear-1.19.1.3, A"
  :precision binary64
  (+ (+ (+ (* x y) (* z z)) (* z z)) (* z z)))