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

Percentage Accurate: 98.5% → 99.0%

Time: 2.7s

Alternatives: 11

Speedup: 1.8×

• 33.8% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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]

Initial Program: 98.5% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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]

Alternative 1: 99.0% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.5%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. +-commutative N/A

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

   10. associate-+r+ N/A

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

   11. distribute-lft-out N/A

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

   12. lower-fma.f64 N/A

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

   13. lower-+.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-fma.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 99.0

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

3. Applied rewrites 99.0%

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

Alternative 2: 99.0% accurate, 1.8× speedup

Math

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

FPCore

(FPCore (x y z) :precision binary64 (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(Float64(3.0 * z), z, Float64(y * x))
end

Wolfram

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

Derivation

1. Initial program 98.5%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. associate-+l+ N/A

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

   9. count-2-rev N/A

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

   10. pow2 N/A

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

   11. +-commutative N/A

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

   12. +-commutative N/A

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

   13. associate-+r+ N/A

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

   14. pow2 N/A

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

   15. distribute-lft1-in N/A

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

   16. metadata-eval N/A

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

   17. associate-*r* N/A

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

3. Applied rewrites 99.0%

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

Alternative 3: 84.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(z, z + z, x \cdot y\right)\\ \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{-201}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{-132}:\\ \;\;\;\;\mathsf{fma}\left(z, z + z, z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma z (+ z z) (* x y))))
   (if (<= (* x y) -5e-201)
     t_0
     (if (<= (* x y) 2e-132) (fma z (+ z z) (* z z)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma(z, (z + z), (x * y));
	double tmp;
	if ((x * y) <= -5e-201) {
		tmp = t_0;
	} else if ((x * y) <= 2e-132) {
		tmp = fma(z, (z + z), (z * z));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(z, Float64(z + z), Float64(x * y))
	tmp = 0.0
	if (Float64(x * y) <= -5e-201)
		tmp = t_0;
	elseif (Float64(x * y) <= 2e-132)
		tmp = fma(z, Float64(z + z), Float64(z * z));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(z + z), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * y), $MachinePrecision], -5e-201], t$95$0, If[LessEqual[N[(x * y), $MachinePrecision], 2e-132], N[(z * N[(z + z), $MachinePrecision] + N[(z * z), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -4.9999999999999999e-201 or 2e-132 < (*.f64 x y)

   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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. +-commutative N/A

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

      10. associate-+r+ N/A

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

      11. distribute-lft-out N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 98.7

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

   3. Applied rewrites 98.7%

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

   4. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(z, z + z, \color{blue}{x \cdot y}\right) \]
   5. Step-by-step derivation

      1. lower-*.f64 84.9

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

   6. Applied rewrites 84.9%

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

   if -4.9999999999999999e-201 < (*.f64 x y) < 2e-132

   1. Initial program 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. +-commutative N/A

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

      10. associate-+r+ N/A

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

      11. distribute-lft-out N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 99.9

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(z, z + z, \color{blue}{x \cdot y}\right) \]
   5. Step-by-step derivation

      1. lower-*.f64 61.4

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

   6. Applied rewrites 61.4%

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

   7. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(z, z + z, \color{blue}{{z}^{2}}\right) \]
   8. Step-by-step derivation

      1. pow2 N/A

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

      2. lift-*.f64 83.8

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

   9. Applied rewrites 83.8%

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

Alternative 4: 84.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(z, z + z, x \cdot y\right)\\ \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{-201}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{-132}:\\ \;\;\;\;3 \cdot \left(z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma z (+ z z) (* x y))))
   (if (<= (* x y) -5e-201) t_0 (if (<= (* x y) 2e-132) (* 3.0 (* z z)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma(z, (z + z), (x * y));
	double tmp;
	if ((x * y) <= -5e-201) {
		tmp = t_0;
	} else if ((x * y) <= 2e-132) {
		tmp = 3.0 * (z * z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(z, Float64(z + z), Float64(x * y))
	tmp = 0.0
	if (Float64(x * y) <= -5e-201)
		tmp = t_0;
	elseif (Float64(x * y) <= 2e-132)
		tmp = Float64(3.0 * Float64(z * z));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(z + z), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * y), $MachinePrecision], -5e-201], t$95$0, If[LessEqual[N[(x * y), $MachinePrecision], 2e-132], N[(3.0 * N[(z * z), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -4.9999999999999999e-201 or 2e-132 < (*.f64 x y)

   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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. +-commutative N/A

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

      10. associate-+r+ N/A

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

      11. distribute-lft-out N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 98.7

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

   3. Applied rewrites 98.7%

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

   4. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(z, z + z, \color{blue}{x \cdot y}\right) \]
   5. Step-by-step derivation

      1. lower-*.f64 84.9

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

   6. Applied rewrites 84.9%

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

   if -4.9999999999999999e-201 < (*.f64 x y) < 2e-132

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{2 \cdot {z}^{2} + {z}^{2}} \]
   3. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto 2 \cdot \left(z \cdot z\right) + {z}^{2} \]

      2. pow2 N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

         \[\leadsto 3 \cdot \left(\color{blue}{z} \cdot z\right) \]

      5. lower-*.f64 N/A

         \[\leadsto 3 \cdot \color{blue}{\left(z \cdot z\right)} \]

      6. lift-*.f64 83.7

         \[\leadsto 3 \cdot \left(z \cdot \color{blue}{z}\right) \]

   4. Applied rewrites 83.7%

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

Alternative 5: 84.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(z, z, x \cdot y\right)\\ \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{-201}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{-132}:\\ \;\;\;\;3 \cdot \left(z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma z z (* x y))))
   (if (<= (* x y) -5e-201) t_0 (if (<= (* x y) 2e-132) (* 3.0 (* z z)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma(z, z, (x * y));
	double tmp;
	if ((x * y) <= -5e-201) {
		tmp = t_0;
	} else if ((x * y) <= 2e-132) {
		tmp = 3.0 * (z * z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(z, z, Float64(x * y))
	tmp = 0.0
	if (Float64(x * y) <= -5e-201)
		tmp = t_0;
	elseif (Float64(x * y) <= 2e-132)
		tmp = Float64(3.0 * Float64(z * z));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * z + N[(x * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * y), $MachinePrecision], -5e-201], t$95$0, If[LessEqual[N[(x * y), $MachinePrecision], 2e-132], N[(3.0 * N[(z * z), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -4.9999999999999999e-201 or 2e-132 < (*.f64 x y)

   1. Initial program 98.1%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 83.9

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

   4. Applied rewrites 83.9%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{2 \cdot {z}^{2}} + z \cdot z \]
   6. Step-by-step derivation

      1. pow2 N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

         \[\leadsto \left(z \cdot 2\right) \cdot z + z \cdot z \]

      4. *-commutative N/A

         \[\leadsto \left(2 \cdot z\right) \cdot z + z \cdot z \]

      5. count-2-rev N/A

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

      6. flip-+ N/A

         \[\leadsto \frac{z \cdot z - z \cdot z}{z - z} \cdot z + z \cdot z \]

      7. +-inverses N/A

         \[\leadsto \frac{0}{z - z} \cdot z + z \cdot z \]

      8. +-inverses N/A

         \[\leadsto \frac{0}{0} \cdot z + z \cdot z \]

      9. metadata-eval N/A

         \[\leadsto \frac{1 - 1}{0} \cdot z + z \cdot z \]

      10. metadata-eval N/A

          \[\leadsto \frac{1 \cdot 1 - 1}{0} \cdot z + z \cdot z \]

      11. metadata-eval N/A

          \[\leadsto \frac{1 \cdot 1 - 1 \cdot 1}{0} \cdot z + z \cdot z \]

      12. metadata-eval N/A

          \[\leadsto \frac{1 \cdot 1 - 1 \cdot 1}{1 - 1} \cdot z + z \cdot z \]

      13. flip-+ N/A

          \[\leadsto \left(1 + 1\right) \cdot z + z \cdot z \]

      14. metadata-eval N/A

          \[\leadsto 2 \cdot z + z \cdot z \]

      15. count-2-rev N/A

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

      16. flip-+ N/A

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

      17. +-inverses N/A

          \[\leadsto \frac{0}{\color{blue}{z} - z} + z \cdot z \]

      18. +-inverses N/A

          \[\leadsto \frac{0}{0} + z \cdot z \]

      19. metadata-eval N/A

          \[\leadsto \frac{1 - 1}{0} + z \cdot z \]

      20. metadata-eval N/A

          \[\leadsto \frac{1 \cdot 1 - 1}{0} + z \cdot z \]

      21. metadata-eval N/A

          \[\leadsto \frac{1 \cdot 1 - 1 \cdot 1}{0} + z \cdot z \]

      22. metadata-eval N/A

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

      23. flip-+ N/A

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

      24. metadata-eval 28.7

          \[\leadsto 2 + z \cdot z \]

   7. Applied rewrites 28.7%

      \[\leadsto \color{blue}{2} + z \cdot z \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. pow2 N/A

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

      4. +-commutative N/A

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

      5. pow2 N/A

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

      6. lower-fma.f64 28.7

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

   9. Applied rewrites 28.7%

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

   10. Taylor expanded in x around inf

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

       1. lift-*.f64 84.5

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

   12. Applied rewrites 84.5%

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

   if -4.9999999999999999e-201 < (*.f64 x y) < 2e-132

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{2 \cdot {z}^{2} + {z}^{2}} \]
   3. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto 2 \cdot \left(z \cdot z\right) + {z}^{2} \]

      2. pow2 N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

         \[\leadsto 3 \cdot \left(\color{blue}{z} \cdot z\right) \]

      5. lower-*.f64 N/A

         \[\leadsto 3 \cdot \color{blue}{\left(z \cdot z\right)} \]

      6. lift-*.f64 83.7

         \[\leadsto 3 \cdot \left(z \cdot \color{blue}{z}\right) \]

   4. Applied rewrites 83.7%

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

Alternative 6: 84.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{-201}:\\ \;\;\;\;\mathsf{fma}\left(z, z, x \cdot y\right)\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{-132}:\\ \;\;\;\;3 \cdot \left(z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x + z \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* x y) -5e-201)
   (fma z z (* x y))
   (if (<= (* x y) 2e-132) (* 3.0 (* z z)) (+ (* y x) (* z z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x * y) <= -5e-201) {
		tmp = fma(z, z, (x * y));
	} else if ((x * y) <= 2e-132) {
		tmp = 3.0 * (z * z);
	} else {
		tmp = (y * x) + (z * z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(x * y) <= -5e-201)
		tmp = fma(z, z, Float64(x * y));
	elseif (Float64(x * y) <= 2e-132)
		tmp = Float64(3.0 * Float64(z * z));
	else
		tmp = Float64(Float64(y * x) + Float64(z * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(x * y), $MachinePrecision], -5e-201], N[(z * z + N[(x * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 2e-132], N[(3.0 * N[(z * z), $MachinePrecision]), $MachinePrecision], N[(N[(y * x), $MachinePrecision] + N[(z * z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -4.9999999999999999e-201

   1. Initial program 96.4%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 80.8

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

   4. Applied rewrites 80.8%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{2 \cdot {z}^{2}} + z \cdot z \]
   6. Step-by-step derivation

      1. pow2 N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

         \[\leadsto \left(z \cdot 2\right) \cdot z + z \cdot z \]

      4. *-commutative N/A

         \[\leadsto \left(2 \cdot z\right) \cdot z + z \cdot z \]

      5. count-2-rev N/A

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

      6. flip-+ N/A

         \[\leadsto \frac{z \cdot z - z \cdot z}{z - z} \cdot z + z \cdot z \]

      7. +-inverses N/A

         \[\leadsto \frac{0}{z - z} \cdot z + z \cdot z \]

      8. +-inverses N/A

         \[\leadsto \frac{0}{0} \cdot z + z \cdot z \]

      9. metadata-eval N/A

         \[\leadsto \frac{1 - 1}{0} \cdot z + z \cdot z \]

      10. metadata-eval N/A

          \[\leadsto \frac{1 \cdot 1 - 1}{0} \cdot z + z \cdot z \]

      11. metadata-eval N/A

          \[\leadsto \frac{1 \cdot 1 - 1 \cdot 1}{0} \cdot z + z \cdot z \]

      12. metadata-eval N/A

          \[\leadsto \frac{1 \cdot 1 - 1 \cdot 1}{1 - 1} \cdot z + z \cdot z \]

      13. flip-+ N/A

          \[\leadsto \left(1 + 1\right) \cdot z + z \cdot z \]

      14. metadata-eval N/A

          \[\leadsto 2 \cdot z + z \cdot z \]

      15. count-2-rev N/A

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

      16. flip-+ N/A

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

      17. +-inverses N/A

          \[\leadsto \frac{0}{\color{blue}{z} - z} + z \cdot z \]

      18. +-inverses N/A

          \[\leadsto \frac{0}{0} + z \cdot z \]

      19. metadata-eval N/A

          \[\leadsto \frac{1 - 1}{0} + z \cdot z \]

      20. metadata-eval N/A

          \[\leadsto \frac{1 \cdot 1 - 1}{0} + z \cdot z \]

      21. metadata-eval N/A

          \[\leadsto \frac{1 \cdot 1 - 1 \cdot 1}{0} + z \cdot z \]

      22. metadata-eval N/A

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

      23. flip-+ N/A

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

      24. metadata-eval 27.0

          \[\leadsto 2 + z \cdot z \]

   7. Applied rewrites 27.0%

      \[\leadsto \color{blue}{2} + z \cdot z \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. pow2 N/A

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

      4. +-commutative N/A

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

      5. pow2 N/A

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

      6. lower-fma.f64 27.0

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

   9. Applied rewrites 27.0%

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

   10. Taylor expanded in x around inf

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

       1. lift-*.f64 81.8

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

   12. Applied rewrites 81.8%

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

   if -4.9999999999999999e-201 < (*.f64 x y) < 2e-132

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{2 \cdot {z}^{2} + {z}^{2}} \]
   3. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto 2 \cdot \left(z \cdot z\right) + {z}^{2} \]

      2. pow2 N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

         \[\leadsto 3 \cdot \left(\color{blue}{z} \cdot z\right) \]

      5. lower-*.f64 N/A

         \[\leadsto 3 \cdot \color{blue}{\left(z \cdot z\right)} \]

      6. lift-*.f64 83.7

         \[\leadsto 3 \cdot \left(z \cdot \color{blue}{z}\right) \]

   4. Applied rewrites 83.7%

      \[\leadsto \color{blue}{3 \cdot \left(z \cdot z\right)} \]

   if 2e-132 < (*.f64 x y)

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 87.4

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

   4. Applied rewrites 87.4%

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

Alternative 7: 69.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1.02 \cdot 10^{-8}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(z \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 1.02e-8) (* y x) (* 3.0 (* z z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.02e-8) {
		tmp = y * x;
	} else {
		tmp = 3.0 * (z * z);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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 <= 1.02d-8) then
        tmp = y * x
    else
        tmp = 3.0d0 * (z * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.02e-8) {
		tmp = y * x;
	} else {
		tmp = 3.0 * (z * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 1.02e-8:
		tmp = y * x
	else:
		tmp = 3.0 * (z * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 1.02e-8)
		tmp = Float64(y * x);
	else
		tmp = Float64(3.0 * Float64(z * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 1.02e-8)
		tmp = y * x;
	else
		tmp = 3.0 * (z * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 1.02e-8], N[(y * x), $MachinePrecision], N[(3.0 * N[(z * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.02000000000000003e-8

   1. Initial program 99.0%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 64.1

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

   4. Applied rewrites 64.1%

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

   if 1.02000000000000003e-8 < z

   1. Initial program 97.2%

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

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{2 \cdot {z}^{2} + {z}^{2}} \]
   3. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto 2 \cdot \left(z \cdot z\right) + {z}^{2} \]

      2. pow2 N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

         \[\leadsto 3 \cdot \left(\color{blue}{z} \cdot z\right) \]

      5. lower-*.f64 N/A

         \[\leadsto 3 \cdot \color{blue}{\left(z \cdot z\right)} \]

      6. lift-*.f64 83.2

         \[\leadsto 3 \cdot \left(z \cdot \color{blue}{z}\right) \]

   4. Applied rewrites 83.2%

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

Alternative 8: 63.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 4.5 \cdot 10^{+152}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(z + z\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= z 4.5e+152) (* y x) (* (+ z z) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 4.5e+152) {
		tmp = y * x;
	} else {
		tmp = (z + z) * z;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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 <= 4.5d+152) then
        tmp = y * x
    else
        tmp = (z + z) * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= 4.5e+152) {
		tmp = y * x;
	} else {
		tmp = (z + z) * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 4.5e+152:
		tmp = y * x
	else:
		tmp = (z + z) * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 4.5e+152)
		tmp = Float64(y * x);
	else
		tmp = Float64(Float64(z + z) * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 4.5e+152)
		tmp = y * x;
	else
		tmp = (z + z) * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 4.5e+152], N[(y * x), $MachinePrecision], N[(N[(z + z), $MachinePrecision] * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 4.5000000000000001e152

   1. Initial program 99.1%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 59.2

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

   4. Applied rewrites 59.2%

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

   if 4.5000000000000001e152 < z

   1. Initial program 94.6%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 93.8

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

   4. Applied rewrites 93.8%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. fp-cancel-sign-sub-inv N/A

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

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

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

      5. *-lft-identity N/A

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

      6. metadata-eval N/A

         \[\leadsto y \cdot x - \left(\mathsf{neg}\left(\color{blue}{\frac{2}{2}} \cdot \left(z \cdot z\right)\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto y \cdot x - \left(\mathsf{neg}\left(\color{blue}{\frac{2 \cdot \left(z \cdot z\right)}{2}}\right)\right) \]

   6. Applied rewrites 10.9%

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

   7. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

         \[\leadsto y \cdot x + 1 \]

      3. lower-fma.f64 10.9

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

   9. Applied rewrites 10.9%

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

   10. Taylor expanded in z around inf

       \[\leadsto \color{blue}{2 \cdot {z}^{2}} \]
   11. Step-by-step derivation

       1. count-2-rev N/A

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

       2. pow2 N/A

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

       3. pow2 N/A

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

       4. distribute-lft-in N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. lift-+.f64 97.4

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

   12. Applied rewrites 97.4%

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

Alternative 9: 63.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 4.5 \cdot 10^{+152}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, z, 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= z 4.5e+152) (* y x) (fma z z 2.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 4.5e+152) {
		tmp = y * x;
	} else {
		tmp = fma(z, z, 2.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 4.5e+152)
		tmp = Float64(y * x);
	else
		tmp = fma(z, z, 2.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 4.5e+152], N[(y * x), $MachinePrecision], N[(z * z + 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 4.5000000000000001e152

   1. Initial program 99.1%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 59.2

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

   4. Applied rewrites 59.2%

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

   if 4.5000000000000001e152 < z

   1. Initial program 94.6%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 93.8

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

   4. Applied rewrites 93.8%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{2 \cdot {z}^{2}} + z \cdot z \]
   6. Step-by-step derivation

      1. pow2 N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

         \[\leadsto \left(z \cdot 2\right) \cdot z + z \cdot z \]

      4. *-commutative N/A

         \[\leadsto \left(2 \cdot z\right) \cdot z + z \cdot z \]

      5. count-2-rev N/A

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

      6. flip-+ N/A

         \[\leadsto \frac{z \cdot z - z \cdot z}{z - z} \cdot z + z \cdot z \]

      7. +-inverses N/A

         \[\leadsto \frac{0}{z - z} \cdot z + z \cdot z \]

      8. +-inverses N/A

         \[\leadsto \frac{0}{0} \cdot z + z \cdot z \]

      9. metadata-eval N/A

         \[\leadsto \frac{1 - 1}{0} \cdot z + z \cdot z \]

      10. metadata-eval N/A

          \[\leadsto \frac{1 \cdot 1 - 1}{0} \cdot z + z \cdot z \]

      11. metadata-eval N/A

          \[\leadsto \frac{1 \cdot 1 - 1 \cdot 1}{0} \cdot z + z \cdot z \]

      12. metadata-eval N/A

          \[\leadsto \frac{1 \cdot 1 - 1 \cdot 1}{1 - 1} \cdot z + z \cdot z \]

      13. flip-+ N/A

          \[\leadsto \left(1 + 1\right) \cdot z + z \cdot z \]

      14. metadata-eval N/A

          \[\leadsto 2 \cdot z + z \cdot z \]

      15. count-2-rev N/A

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

      16. flip-+ N/A

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

      17. +-inverses N/A

          \[\leadsto \frac{0}{\color{blue}{z} - z} + z \cdot z \]

      18. +-inverses N/A

          \[\leadsto \frac{0}{0} + z \cdot z \]

      19. metadata-eval N/A

          \[\leadsto \frac{1 - 1}{0} + z \cdot z \]

      20. metadata-eval N/A

          \[\leadsto \frac{1 \cdot 1 - 1}{0} + z \cdot z \]

      21. metadata-eval N/A

          \[\leadsto \frac{1 \cdot 1 - 1 \cdot 1}{0} + z \cdot z \]

      22. metadata-eval N/A

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

      23. flip-+ N/A

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

      24. metadata-eval 97.4

          \[\leadsto 2 + z \cdot z \]

   7. Applied rewrites 97.4%

      \[\leadsto \color{blue}{2} + z \cdot z \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. pow2 N/A

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

      4. +-commutative N/A

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

      5. pow2 N/A

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

      6. lower-fma.f64 97.4

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

   9. Applied rewrites 97.4%

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

Alternative 10: 53.2% accurate, 5.3× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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 = y * x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.5%

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

2. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 53.2

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

4. Applied rewrites 53.2%

   \[\leadsto \color{blue}{y \cdot x} \]
5. Add Preprocessing

Alternative 11: 3.8% accurate, 21.0× speedup

Math

\[\begin{array}{l} \\ 3 \end{array} \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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 = 3.0d0
end function

Java

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

Python

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

Julia

function code(x, y, z)
	return 3.0
end

MATLAB

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

Wolfram

code[x_, y_, z_] := 3.0

Derivation

1. Initial program 98.5%

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

2. Taylor expanded in x around 0

   \[\leadsto \color{blue}{2 \cdot {z}^{2} + {z}^{2}} \]
3. Step-by-step derivation

   1. pow2 N/A

      \[\leadsto 2 \cdot \left(z \cdot z\right) + {z}^{2} \]

   2. pow2 N/A

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

   3. distribute-lft1-in N/A

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

   4. metadata-eval N/A

      \[\leadsto 3 \cdot \left(\color{blue}{z} \cdot z\right) \]

   5. lower-*.f64 N/A

      \[\leadsto 3 \cdot \color{blue}{\left(z \cdot z\right)} \]

   6. lift-*.f64 53.1

      \[\leadsto 3 \cdot \left(z \cdot \color{blue}{z}\right) \]

4. Applied rewrites 53.1%

   \[\leadsto \color{blue}{3 \cdot \left(z \cdot z\right)} \]

5. Taylor expanded in z around 0

   \[\leadsto 3 \cdot \color{blue}{{z}^{2}} \]
6. Step-by-step derivation

   1. pow2 N/A

      \[\leadsto 3 \cdot \left(z \cdot z\right) \]

   2. *-lft-identity N/A

      \[\leadsto 3 \cdot \left(1 \cdot \left(z \cdot \color{blue}{z}\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto 3 \cdot \left(\frac{2}{2} \cdot \left(z \cdot z\right)\right) \]

   4. associate-*l/ N/A

      \[\leadsto 3 \cdot \frac{2 \cdot \left(z \cdot z\right)}{2} \]

   5. associate-*r* N/A

      \[\leadsto 3 \cdot \frac{\left(2 \cdot z\right) \cdot z}{2} \]

   6. *-commutative N/A

      \[\leadsto 3 \cdot \frac{\left(z \cdot 2\right) \cdot z}{2} \]

   7. *-commutative N/A

      \[\leadsto 3 \cdot \frac{\left(2 \cdot z\right) \cdot z}{2} \]

   8. count-2-rev N/A

      \[\leadsto 3 \cdot \frac{\left(z + z\right) \cdot z}{2} \]

   9. flip-+ N/A

      \[\leadsto 3 \cdot \frac{\frac{z \cdot z - z \cdot z}{z - z} \cdot z}{2} \]

   10. +-inverses N/A

       \[\leadsto 3 \cdot \frac{\frac{0}{z - z} \cdot z}{2} \]

   11. +-inverses N/A

       \[\leadsto 3 \cdot \frac{\frac{0}{0} \cdot z}{2} \]

   12. metadata-eval N/A

       \[\leadsto 3 \cdot \frac{\frac{1 - 1}{0} \cdot z}{2} \]

   13. metadata-eval N/A

       \[\leadsto 3 \cdot \frac{\frac{1 \cdot 1 - 1}{0} \cdot z}{2} \]

   14. metadata-eval N/A

       \[\leadsto 3 \cdot \frac{\frac{1 \cdot 1 - 1 \cdot 1}{0} \cdot z}{2} \]

   15. metadata-eval N/A

       \[\leadsto 3 \cdot \frac{\frac{1 \cdot 1 - 1 \cdot 1}{1 - 1} \cdot z}{2} \]

   16. flip-+ N/A

       \[\leadsto 3 \cdot \frac{\left(1 + 1\right) \cdot z}{2} \]

   17. metadata-eval N/A

       \[\leadsto 3 \cdot \frac{2 \cdot z}{2} \]

   18. count-2-rev N/A

       \[\leadsto 3 \cdot \frac{z + z}{2} \]

   19. flip-+ N/A

       \[\leadsto 3 \cdot \frac{\frac{z \cdot z - z \cdot z}{z - z}}{2} \]

   20. +-inverses N/A

       \[\leadsto 3 \cdot \frac{\frac{0}{z - z}}{2} \]

   21. +-inverses N/A

       \[\leadsto 3 \cdot \frac{\frac{0}{0}}{2} \]

   22. metadata-eval N/A

       \[\leadsto 3 \cdot \frac{\frac{1 - 1}{0}}{2} \]

   23. metadata-eval N/A

       \[\leadsto 3 \cdot \frac{\frac{1 \cdot 1 - 1}{0}}{2} \]

   24. metadata-eval N/A

       \[\leadsto 3 \cdot \frac{\frac{1 \cdot 1 - 1 \cdot 1}{0}}{2} \]

   25. metadata-eval N/A

       \[\leadsto 3 \cdot \frac{\frac{1 \cdot 1 - 1 \cdot 1}{1 - 1}}{2} \]

   26. flip-+ N/A

       \[\leadsto 3 \cdot \frac{1 + 1}{2} \]

   27. metadata-eval N/A

       \[\leadsto 3 \cdot \frac{2}{2} \]

   28. metadata-eval N/A

       \[\leadsto 3 \cdot 1 \]

7. Applied rewrites 3.8%

   \[\leadsto 3 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025110 
(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)))