Migdal et al, Equation (64)

Percentage Accurate: 99.5% → 99.6%

Time: 5.5s

Alternatives: 8

Speedup: 1.9×

• 56.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ t\_1 \cdot \left(a1 \cdot a1\right) + t\_1 \cdot \left(a2 \cdot a2\right) \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (/ (cos th) (sqrt 2.0))))
   (+ (* t_1 (* a1 a1)) (* t_1 (* a2 a2)))))

C

double code(double a1, double a2, double th) {
	double t_1 = cos(th) / sqrt(2.0);
	return (t_1 * (a1 * a1)) + (t_1 * (a2 * a2));
}

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(a1, a2, th)
use fmin_fmax_functions
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: t_1
    t_1 = cos(th) / sqrt(2.0d0)
    code = (t_1 * (a1 * a1)) + (t_1 * (a2 * a2))
end function

Java

public static double code(double a1, double a2, double th) {
	double t_1 = Math.cos(th) / Math.sqrt(2.0);
	return (t_1 * (a1 * a1)) + (t_1 * (a2 * a2));
}

Python

def code(a1, a2, th):
	t_1 = math.cos(th) / math.sqrt(2.0)
	return (t_1 * (a1 * a1)) + (t_1 * (a2 * a2))

Julia

function code(a1, a2, th)
	t_1 = Float64(cos(th) / sqrt(2.0))
	return Float64(Float64(t_1 * Float64(a1 * a1)) + Float64(t_1 * Float64(a2 * a2)))
end

MATLAB

function tmp = code(a1, a2, th)
	t_1 = cos(th) / sqrt(2.0);
	tmp = (t_1 * (a1 * a1)) + (t_1 * (a2 * a2));
end

Wolfram

code[a1_, a2_, th_] := Block[{t$95$1 = N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, N[(N[(t$95$1 * N[(a1 * a1), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * N[(a2 * a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Initial Program: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ t\_1 \cdot \left(a1 \cdot a1\right) + t\_1 \cdot \left(a2 \cdot a2\right) \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (/ (cos th) (sqrt 2.0))))
   (+ (* t_1 (* a1 a1)) (* t_1 (* a2 a2)))))

C

double code(double a1, double a2, double th) {
	double t_1 = cos(th) / sqrt(2.0);
	return (t_1 * (a1 * a1)) + (t_1 * (a2 * a2));
}

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(a1, a2, th)
use fmin_fmax_functions
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: t_1
    t_1 = cos(th) / sqrt(2.0d0)
    code = (t_1 * (a1 * a1)) + (t_1 * (a2 * a2))
end function

Java

public static double code(double a1, double a2, double th) {
	double t_1 = Math.cos(th) / Math.sqrt(2.0);
	return (t_1 * (a1 * a1)) + (t_1 * (a2 * a2));
}

Python

def code(a1, a2, th):
	t_1 = math.cos(th) / math.sqrt(2.0)
	return (t_1 * (a1 * a1)) + (t_1 * (a2 * a2))

Julia

function code(a1, a2, th)
	t_1 = Float64(cos(th) / sqrt(2.0))
	return Float64(Float64(t_1 * Float64(a1 * a1)) + Float64(t_1 * Float64(a2 * a2)))
end

MATLAB

function tmp = code(a1, a2, th)
	t_1 = cos(th) / sqrt(2.0);
	tmp = (t_1 * (a1 * a1)) + (t_1 * (a2 * a2));
end

Wolfram

code[a1_, a2_, th_] := Block[{t$95$1 = N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, N[(N[(t$95$1 * N[(a1 * a1), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * N[(a2 * a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Alternative 1: 99.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{max}\left(\left|a1\right|, \left|a2\right|\right)\\ t_2 := \mathsf{min}\left(\left|a1\right|, \left|a2\right|\right)\\ \left(\left(1.4142135623730951 \cdot \mathsf{fma}\left(t\_1, t\_1, t\_2 \cdot t\_2\right)\right) \cdot \cos th\right) \cdot 0.5 \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (fmax (fabs a1) (fabs a2))) (t_2 (fmin (fabs a1) (fabs a2))))
   (* (* (* 1.4142135623730951 (fma t_1 t_1 (* t_2 t_2))) (cos th)) 0.5)))

C

double code(double a1, double a2, double th) {
	double t_1 = fmax(fabs(a1), fabs(a2));
	double t_2 = fmin(fabs(a1), fabs(a2));
	return ((1.4142135623730951 * fma(t_1, t_1, (t_2 * t_2))) * cos(th)) * 0.5;
}

Julia

function code(a1, a2, th)
	t_1 = fmax(abs(a1), abs(a2))
	t_2 = fmin(abs(a1), abs(a2))
	return Float64(Float64(Float64(1.4142135623730951 * fma(t_1, t_1, Float64(t_2 * t_2))) * cos(th)) * 0.5)
end

Wolfram

code[a1_, a2_, th_] := Block[{t$95$1 = N[Max[N[Abs[a1], $MachinePrecision], N[Abs[a2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Min[N[Abs[a1], $MachinePrecision], N[Abs[a2], $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(1.4142135623730951 * N[(t$95$1 * t$95$1 + N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[th], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]]

Derivation

1. Initial program 99.5%

   \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
2. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \color{blue}{\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right)} + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]

   3. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\cos th}{\sqrt{2}}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]

   4. associate-*l/ N/A

      \[\leadsto \color{blue}{\frac{\cos th \cdot \left(a1 \cdot a1\right)}{\sqrt{2}}} + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{\cos th \cdot \left(a1 \cdot a1\right)}{\sqrt{2}} + \color{blue}{\frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)} \]

   6. lift-/.f64 N/A

      \[\leadsto \frac{\cos th \cdot \left(a1 \cdot a1\right)}{\sqrt{2}} + \color{blue}{\frac{\cos th}{\sqrt{2}}} \cdot \left(a2 \cdot a2\right) \]

   7. associate-*l/ N/A

      \[\leadsto \frac{\cos th \cdot \left(a1 \cdot a1\right)}{\sqrt{2}} + \color{blue}{\frac{\cos th \cdot \left(a2 \cdot a2\right)}{\sqrt{2}}} \]

   8. common-denominator N/A

      \[\leadsto \color{blue}{\frac{\left(\cos th \cdot \left(a1 \cdot a1\right)\right) \cdot \sqrt{2} + \left(\cos th \cdot \left(a2 \cdot a2\right)\right) \cdot \sqrt{2}}{\sqrt{2} \cdot \sqrt{2}}} \]

   9. lift-sqrt.f64 N/A

      \[\leadsto \frac{\left(\cos th \cdot \left(a1 \cdot a1\right)\right) \cdot \sqrt{2} + \left(\cos th \cdot \left(a2 \cdot a2\right)\right) \cdot \sqrt{2}}{\color{blue}{\sqrt{2}} \cdot \sqrt{2}} \]

   10. lift-sqrt.f64 N/A

       \[\leadsto \frac{\left(\cos th \cdot \left(a1 \cdot a1\right)\right) \cdot \sqrt{2} + \left(\cos th \cdot \left(a2 \cdot a2\right)\right) \cdot \sqrt{2}}{\sqrt{2} \cdot \color{blue}{\sqrt{2}}} \]

   11. rem-square-sqrt N/A

       \[\leadsto \frac{\left(\cos th \cdot \left(a1 \cdot a1\right)\right) \cdot \sqrt{2} + \left(\cos th \cdot \left(a2 \cdot a2\right)\right) \cdot \sqrt{2}}{\color{blue}{2}} \]

   12. mult-flip N/A

       \[\leadsto \color{blue}{\left(\left(\cos th \cdot \left(a1 \cdot a1\right)\right) \cdot \sqrt{2} + \left(\cos th \cdot \left(a2 \cdot a2\right)\right) \cdot \sqrt{2}\right) \cdot \frac{1}{2}} \]

   13. +-commutative N/A

       \[\leadsto \color{blue}{\left(\left(\cos th \cdot \left(a2 \cdot a2\right)\right) \cdot \sqrt{2} + \left(\cos th \cdot \left(a1 \cdot a1\right)\right) \cdot \sqrt{2}\right)} \cdot \frac{1}{2} \]

   14. metadata-eval N/A

       \[\leadsto \left(\left(\cos th \cdot \left(a2 \cdot a2\right)\right) \cdot \sqrt{2} + \left(\cos th \cdot \left(a1 \cdot a1\right)\right) \cdot \sqrt{2}\right) \cdot \color{blue}{\frac{1}{2}} \]

3. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot \cos th\right)\right) \cdot 0.5} \]
4. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot \cos th\right)\right)} \cdot \frac{1}{2} \]

   2. lift-*.f64 N/A

      \[\leadsto \left(\sqrt{2} \cdot \color{blue}{\left(\mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot \cos th\right)}\right) \cdot \frac{1}{2} \]

   3. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\left(\sqrt{2} \cdot \mathsf{fma}\left(a2, a2, a1 \cdot a1\right)\right) \cdot \cos th\right)} \cdot \frac{1}{2} \]

   4. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(\sqrt{2} \cdot \mathsf{fma}\left(a2, a2, a1 \cdot a1\right)\right) \cdot \cos th\right)} \cdot \frac{1}{2} \]

   5. lower-*.f64 99.5%

      \[\leadsto \left(\color{blue}{\left(\sqrt{2} \cdot \mathsf{fma}\left(a2, a2, a1 \cdot a1\right)\right)} \cdot \cos th\right) \cdot 0.5 \]

5. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\left(\left(\sqrt{2} \cdot \mathsf{fma}\left(a2, a2, a1 \cdot a1\right)\right) \cdot \cos th\right)} \cdot 0.5 \]

6. Evaluated real constant 99.5%

   \[\leadsto \left(\left(\color{blue}{1.4142135623730951} \cdot \mathsf{fma}\left(a2, a2, a1 \cdot a1\right)\right) \cdot \cos th\right) \cdot 0.5 \]
7. Add Preprocessing

Alternative 2: 99.5% accurate, 1.9× speedup

Math

\[\frac{\mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot \cos th}{1.4142135623730951} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (/ (* (fma a2 a2 (* a1 a1)) (cos th)) 1.4142135623730951))

C

double code(double a1, double a2, double th) {
	return (fma(a2, a2, (a1 * a1)) * cos(th)) / 1.4142135623730951;
}

Julia

function code(a1, a2, th)
	return Float64(Float64(fma(a2, a2, Float64(a1 * a1)) * cos(th)) / 1.4142135623730951)
end

Wolfram

code[a1_, a2_, th_] := N[(N[(N[(a2 * a2 + N[(a1 * a1), $MachinePrecision]), $MachinePrecision] * N[Cos[th], $MachinePrecision]), $MachinePrecision] / 1.4142135623730951), $MachinePrecision]

Derivation

1. Initial program 99.5%

   \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
2. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \color{blue}{\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right)} + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]

   3. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\cos th}{\sqrt{2}}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]

   4. associate-*l/ N/A

      \[\leadsto \color{blue}{\frac{\cos th \cdot \left(a1 \cdot a1\right)}{\sqrt{2}}} + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{\cos th \cdot \left(a1 \cdot a1\right)}{\sqrt{2}} + \color{blue}{\frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)} \]

   6. lift-/.f64 N/A

      \[\leadsto \frac{\cos th \cdot \left(a1 \cdot a1\right)}{\sqrt{2}} + \color{blue}{\frac{\cos th}{\sqrt{2}}} \cdot \left(a2 \cdot a2\right) \]

   7. associate-*l/ N/A

      \[\leadsto \frac{\cos th \cdot \left(a1 \cdot a1\right)}{\sqrt{2}} + \color{blue}{\frac{\cos th \cdot \left(a2 \cdot a2\right)}{\sqrt{2}}} \]

   8. div-add-rev N/A

      \[\leadsto \color{blue}{\frac{\cos th \cdot \left(a1 \cdot a1\right) + \cos th \cdot \left(a2 \cdot a2\right)}{\sqrt{2}}} \]

   9. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\cos th \cdot \left(a1 \cdot a1\right) + \cos th \cdot \left(a2 \cdot a2\right)}{\sqrt{2}}} \]

   10. distribute-lft-out N/A

       \[\leadsto \frac{\color{blue}{\cos th \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)}}{\sqrt{2}} \]

   11. *-commutative N/A

       \[\leadsto \frac{\color{blue}{\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \cos th}}{\sqrt{2}} \]

   12. lower-*.f64 N/A

       \[\leadsto \frac{\color{blue}{\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \cos th}}{\sqrt{2}} \]

   13. +-commutative N/A

       \[\leadsto \frac{\color{blue}{\left(a2 \cdot a2 + a1 \cdot a1\right)} \cdot \cos th}{\sqrt{2}} \]

   14. lift-*.f64 N/A

       \[\leadsto \frac{\left(\color{blue}{a2 \cdot a2} + a1 \cdot a1\right) \cdot \cos th}{\sqrt{2}} \]

   15. lower-fma.f64 99.5%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(a2, a2, a1 \cdot a1\right)} \cdot \cos th}{\sqrt{2}} \]

3. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot \cos th}{\sqrt{2}}} \]

4. Evaluated real constant 99.5%

   \[\leadsto \frac{\mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot \cos th}{\color{blue}{1.4142135623730951}} \]
5. Add Preprocessing

Alternative 3: 99.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{max}\left(\left|a1\right|, \left|a2\right|\right)\\ t_2 := \mathsf{min}\left(\left|a1\right|, \left|a2\right|\right)\\ 0.7071067811865476 \cdot \left(\cos th \cdot \mathsf{fma}\left(t\_1, t\_1, t\_2 \cdot t\_2\right)\right) \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (fmax (fabs a1) (fabs a2))) (t_2 (fmin (fabs a1) (fabs a2))))
   (* 0.7071067811865476 (* (cos th) (fma t_1 t_1 (* t_2 t_2))))))

C

double code(double a1, double a2, double th) {
	double t_1 = fmax(fabs(a1), fabs(a2));
	double t_2 = fmin(fabs(a1), fabs(a2));
	return 0.7071067811865476 * (cos(th) * fma(t_1, t_1, (t_2 * t_2)));
}

Julia

function code(a1, a2, th)
	t_1 = fmax(abs(a1), abs(a2))
	t_2 = fmin(abs(a1), abs(a2))
	return Float64(0.7071067811865476 * Float64(cos(th) * fma(t_1, t_1, Float64(t_2 * t_2))))
end

Wolfram

code[a1_, a2_, th_] := Block[{t$95$1 = N[Max[N[Abs[a1], $MachinePrecision], N[Abs[a2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Min[N[Abs[a1], $MachinePrecision], N[Abs[a2], $MachinePrecision]], $MachinePrecision]}, N[(0.7071067811865476 * N[(N[Cos[th], $MachinePrecision] * N[(t$95$1 * t$95$1 + N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 99.5%

   \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
2. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \color{blue}{\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right)} + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]

   3. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\cos th}{\sqrt{2}}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]

   4. associate-*l/ N/A

      \[\leadsto \color{blue}{\frac{\cos th \cdot \left(a1 \cdot a1\right)}{\sqrt{2}}} + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{\cos th \cdot \left(a1 \cdot a1\right)}{\sqrt{2}} + \color{blue}{\frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)} \]

   6. lift-/.f64 N/A

      \[\leadsto \frac{\cos th \cdot \left(a1 \cdot a1\right)}{\sqrt{2}} + \color{blue}{\frac{\cos th}{\sqrt{2}}} \cdot \left(a2 \cdot a2\right) \]

   7. associate-*l/ N/A

      \[\leadsto \frac{\cos th \cdot \left(a1 \cdot a1\right)}{\sqrt{2}} + \color{blue}{\frac{\cos th \cdot \left(a2 \cdot a2\right)}{\sqrt{2}}} \]

   8. common-denominator N/A

      \[\leadsto \color{blue}{\frac{\left(\cos th \cdot \left(a1 \cdot a1\right)\right) \cdot \sqrt{2} + \left(\cos th \cdot \left(a2 \cdot a2\right)\right) \cdot \sqrt{2}}{\sqrt{2} \cdot \sqrt{2}}} \]

   9. lift-sqrt.f64 N/A

      \[\leadsto \frac{\left(\cos th \cdot \left(a1 \cdot a1\right)\right) \cdot \sqrt{2} + \left(\cos th \cdot \left(a2 \cdot a2\right)\right) \cdot \sqrt{2}}{\color{blue}{\sqrt{2}} \cdot \sqrt{2}} \]

   10. lift-sqrt.f64 N/A

       \[\leadsto \frac{\left(\cos th \cdot \left(a1 \cdot a1\right)\right) \cdot \sqrt{2} + \left(\cos th \cdot \left(a2 \cdot a2\right)\right) \cdot \sqrt{2}}{\sqrt{2} \cdot \color{blue}{\sqrt{2}}} \]

   11. rem-square-sqrt N/A

       \[\leadsto \frac{\left(\cos th \cdot \left(a1 \cdot a1\right)\right) \cdot \sqrt{2} + \left(\cos th \cdot \left(a2 \cdot a2\right)\right) \cdot \sqrt{2}}{\color{blue}{2}} \]

   12. mult-flip N/A

       \[\leadsto \color{blue}{\left(\left(\cos th \cdot \left(a1 \cdot a1\right)\right) \cdot \sqrt{2} + \left(\cos th \cdot \left(a2 \cdot a2\right)\right) \cdot \sqrt{2}\right) \cdot \frac{1}{2}} \]

   13. +-commutative N/A

       \[\leadsto \color{blue}{\left(\left(\cos th \cdot \left(a2 \cdot a2\right)\right) \cdot \sqrt{2} + \left(\cos th \cdot \left(a1 \cdot a1\right)\right) \cdot \sqrt{2}\right)} \cdot \frac{1}{2} \]

   14. metadata-eval N/A

       \[\leadsto \left(\left(\cos th \cdot \left(a2 \cdot a2\right)\right) \cdot \sqrt{2} + \left(\cos th \cdot \left(a1 \cdot a1\right)\right) \cdot \sqrt{2}\right) \cdot \color{blue}{\frac{1}{2}} \]

3. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot \cos th\right)\right) \cdot 0.5} \]
4. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot \cos th\right)\right)} \cdot \frac{1}{2} \]

   2. lift-*.f64 N/A

      \[\leadsto \left(\sqrt{2} \cdot \color{blue}{\left(\mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot \cos th\right)}\right) \cdot \frac{1}{2} \]

   3. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\left(\sqrt{2} \cdot \mathsf{fma}\left(a2, a2, a1 \cdot a1\right)\right) \cdot \cos th\right)} \cdot \frac{1}{2} \]

   4. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(\sqrt{2} \cdot \mathsf{fma}\left(a2, a2, a1 \cdot a1\right)\right) \cdot \cos th\right)} \cdot \frac{1}{2} \]

   5. lower-*.f64 99.5%

      \[\leadsto \left(\color{blue}{\left(\sqrt{2} \cdot \mathsf{fma}\left(a2, a2, a1 \cdot a1\right)\right)} \cdot \cos th\right) \cdot 0.5 \]

5. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\left(\left(\sqrt{2} \cdot \mathsf{fma}\left(a2, a2, a1 \cdot a1\right)\right) \cdot \cos th\right)} \cdot 0.5 \]

6. Evaluated real constant 99.5%

   \[\leadsto \left(\left(\color{blue}{1.4142135623730951} \cdot \mathsf{fma}\left(a2, a2, a1 \cdot a1\right)\right) \cdot \cos th\right) \cdot 0.5 \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(\frac{6369051672525773}{4503599627370496} \cdot \mathsf{fma}\left(a2, a2, a1 \cdot a1\right)\right) \cdot \cos th\right) \cdot \frac{1}{2}} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\left(\frac{6369051672525773}{4503599627370496} \cdot \mathsf{fma}\left(a2, a2, a1 \cdot a1\right)\right) \cdot \cos th\right)} \]

   3. lift-cos.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\left(\frac{6369051672525773}{4503599627370496} \cdot \mathsf{fma}\left(a2, a2, a1 \cdot a1\right)\right) \cdot \color{blue}{\cos th}\right) \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\left(\frac{6369051672525773}{4503599627370496} \cdot \mathsf{fma}\left(a2, a2, a1 \cdot a1\right)\right) \cdot \cos th\right)} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\left(\frac{6369051672525773}{4503599627370496} \cdot \mathsf{fma}\left(a2, a2, a1 \cdot a1\right)\right)} \cdot \cos th\right) \]

   6. associate-*l* N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\frac{6369051672525773}{4503599627370496} \cdot \left(\mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot \cos th\right)\right)} \]

   7. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \frac{6369051672525773}{4503599627370496}\right) \cdot \left(\mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot \cos th\right)} \]

   8. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \frac{6369051672525773}{4503599627370496}\right) \cdot \left(\mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot \cos th\right)} \]

   9. metadata-eval N/A

      \[\leadsto \color{blue}{\frac{6369051672525773}{9007199254740992}} \cdot \left(\mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot \cos th\right) \]

   10. *-commutative N/A

       \[\leadsto \frac{6369051672525773}{9007199254740992} \cdot \color{blue}{\left(\cos th \cdot \mathsf{fma}\left(a2, a2, a1 \cdot a1\right)\right)} \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{6369051672525773}{9007199254740992} \cdot \color{blue}{\left(\cos th \cdot \mathsf{fma}\left(a2, a2, a1 \cdot a1\right)\right)} \]

   12. lift-cos.f64 99.6%

       \[\leadsto 0.7071067811865476 \cdot \left(\color{blue}{\cos th} \cdot \mathsf{fma}\left(a2, a2, a1 \cdot a1\right)\right) \]

8. Applied rewrites 99.6%

   \[\leadsto \color{blue}{0.7071067811865476 \cdot \left(\cos th \cdot \mathsf{fma}\left(a2, a2, a1 \cdot a1\right)\right)} \]
9. Add Preprocessing

Alternative 4: 99.5% accurate, 1.9× speedup

Math

\[\mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot \left(0.7071067811865475 \cdot \cos th\right) \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (* (fma a2 a2 (* a1 a1)) (* 0.7071067811865475 (cos th))))

C

double code(double a1, double a2, double th) {
	return fma(a2, a2, (a1 * a1)) * (0.7071067811865475 * cos(th));
}

Julia

function code(a1, a2, th)
	return Float64(fma(a2, a2, Float64(a1 * a1)) * Float64(0.7071067811865475 * cos(th)))
end

Wolfram

code[a1_, a2_, th_] := N[(N[(a2 * a2 + N[(a1 * a1), $MachinePrecision]), $MachinePrecision] * N[(0.7071067811865475 * N[Cos[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

   \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]

2. Taylor expanded in th around 0

   \[\leadsto \frac{\color{blue}{1}}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
3. Step-by-step derivation

4. Applied rewrites 82.3%

   \[\leadsto \frac{\color{blue}{1}}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]

5. Taylor expanded in th around 0

   \[\leadsto \frac{1}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\color{blue}{1}}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
6. Step-by-step derivation

7. Applied rewrites 66.5%

   \[\leadsto \frac{1}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\color{blue}{1}}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
8. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{1}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{\sqrt{2}} \cdot \left(a1 \cdot a1\right)} + \frac{1}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{1}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \color{blue}{\frac{1}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)} \]

   4. distribute-lft-out N/A

      \[\leadsto \color{blue}{\frac{1}{\sqrt{2}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)} \]

   5. +-commutative N/A

      \[\leadsto \frac{1}{\sqrt{2}} \cdot \color{blue}{\left(a2 \cdot a2 + a1 \cdot a1\right)} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{1}{\sqrt{2}} \cdot \left(\color{blue}{a2 \cdot a2} + a1 \cdot a1\right) \]

   7. lift-fma.f64 N/A

      \[\leadsto \frac{1}{\sqrt{2}} \cdot \color{blue}{\mathsf{fma}\left(a2, a2, a1 \cdot a1\right)} \]

   8. *-commutative N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot \frac{1}{\sqrt{2}}} \]

   9. lower-*.f64 66.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot \frac{1}{\sqrt{2}}} \]

9. Applied rewrites 66.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot \frac{1}{\sqrt{2}}} \]

10. Evaluated real constant 66.5%

    \[\leadsto \mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot \frac{1}{\color{blue}{1.4142135623730951}} \]

11. Taylor expanded in th around inf

    \[\leadsto \mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot \color{blue}{\left(\frac{4503599627370496}{6369051672525773} \cdot \cos th\right)} \]
12. Step-by-step derivation

    1. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot \left(\frac{4503599627370496}{6369051672525773} \cdot \color{blue}{\cos th}\right) \]

    2. lower-cos.f64 99.5%

       \[\leadsto \mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot \left(0.7071067811865475 \cdot \cos th\right) \]

13. Applied rewrites 99.5%

    \[\leadsto \mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot \color{blue}{\left(0.7071067811865475 \cdot \cos th\right)} \]
14. Add Preprocessing

Alternative 5: 78.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{max}\left(\left|a1\right|, \left|a2\right|\right)\\ t_2 := \frac{\cos th}{\sqrt{2}}\\ t_3 := \mathsf{min}\left(\left|a1\right|, \left|a2\right|\right)\\ t_4 := t\_3 \cdot t\_3\\ \mathbf{if}\;t\_2 \cdot t\_4 + t\_2 \cdot \left(t\_1 \cdot t\_1\right) \leq -2 \cdot 10^{-132}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, t\_1, t\_4\right) \cdot \frac{\mathsf{fma}\left(th \cdot th, -0.5, 1\right)}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1 \cdot t\_1}{\sqrt{2}}, t\_1, \frac{1}{\sqrt{2}} \cdot t\_4\right)\\ \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (fmax (fabs a1) (fabs a2)))
        (t_2 (/ (cos th) (sqrt 2.0)))
        (t_3 (fmin (fabs a1) (fabs a2)))
        (t_4 (* t_3 t_3)))
   (if (<= (+ (* t_2 t_4) (* t_2 (* t_1 t_1))) -2e-132)
     (* (fma t_1 t_1 t_4) (/ (fma (* th th) -0.5 1.0) (sqrt 2.0)))
     (fma (/ (* 1.0 t_1) (sqrt 2.0)) t_1 (* (/ 1.0 (sqrt 2.0)) t_4)))))

C

double code(double a1, double a2, double th) {
	double t_1 = fmax(fabs(a1), fabs(a2));
	double t_2 = cos(th) / sqrt(2.0);
	double t_3 = fmin(fabs(a1), fabs(a2));
	double t_4 = t_3 * t_3;
	double tmp;
	if (((t_2 * t_4) + (t_2 * (t_1 * t_1))) <= -2e-132) {
		tmp = fma(t_1, t_1, t_4) * (fma((th * th), -0.5, 1.0) / sqrt(2.0));
	} else {
		tmp = fma(((1.0 * t_1) / sqrt(2.0)), t_1, ((1.0 / sqrt(2.0)) * t_4));
	}
	return tmp;
}

Julia

function code(a1, a2, th)
	t_1 = fmax(abs(a1), abs(a2))
	t_2 = Float64(cos(th) / sqrt(2.0))
	t_3 = fmin(abs(a1), abs(a2))
	t_4 = Float64(t_3 * t_3)
	tmp = 0.0
	if (Float64(Float64(t_2 * t_4) + Float64(t_2 * Float64(t_1 * t_1))) <= -2e-132)
		tmp = Float64(fma(t_1, t_1, t_4) * Float64(fma(Float64(th * th), -0.5, 1.0) / sqrt(2.0)));
	else
		tmp = fma(Float64(Float64(1.0 * t_1) / sqrt(2.0)), t_1, Float64(Float64(1.0 / sqrt(2.0)) * t_4));
	end
	return tmp
end

Wolfram

code[a1_, a2_, th_] := Block[{t$95$1 = N[Max[N[Abs[a1], $MachinePrecision], N[Abs[a2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Min[N[Abs[a1], $MachinePrecision], N[Abs[a2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 * t$95$3), $MachinePrecision]}, If[LessEqual[N[(N[(t$95$2 * t$95$4), $MachinePrecision] + N[(t$95$2 * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-132], N[(N[(t$95$1 * t$95$1 + t$95$4), $MachinePrecision] * N[(N[(N[(th * th), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 * t$95$1), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$1 + N[(N[(1.0 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$4), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a1 a1)) (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a2 a2))) < -2e-132

   1. Initial program 99.5%

      \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]

   2. Taylor expanded in th around 0

      \[\leadsto \frac{\color{blue}{1}}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
   3. Step-by-step derivation

   4. Applied rewrites 82.3%

      \[\leadsto \frac{\color{blue}{1}}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]

   5. Taylor expanded in th around 0

      \[\leadsto \frac{1}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\color{blue}{1}}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 66.5%

      \[\leadsto \frac{1}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\color{blue}{1}}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{1}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\sqrt{2}} \cdot \left(a1 \cdot a1\right)} + \frac{1}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{1}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \color{blue}{\frac{1}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)} \]

      4. distribute-lft-out N/A

         \[\leadsto \color{blue}{\frac{1}{\sqrt{2}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)} \]

      5. +-commutative N/A

         \[\leadsto \frac{1}{\sqrt{2}} \cdot \color{blue}{\left(a2 \cdot a2 + a1 \cdot a1\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{1}{\sqrt{2}} \cdot \left(\color{blue}{a2 \cdot a2} + a1 \cdot a1\right) \]

      7. lift-fma.f64 N/A

         \[\leadsto \frac{1}{\sqrt{2}} \cdot \color{blue}{\mathsf{fma}\left(a2, a2, a1 \cdot a1\right)} \]

      8. *-commutative N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot \frac{1}{\sqrt{2}}} \]

      9. lower-*.f64 66.5%

         \[\leadsto \color{blue}{\mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot \frac{1}{\sqrt{2}}} \]

   9. Applied rewrites 66.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot \frac{1}{\sqrt{2}}} \]

   10. Taylor expanded in th around 0

       \[\leadsto \mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot \color{blue}{\left(\frac{-1}{2} \cdot \frac{{th}^{2}}{\sqrt{2}} + \frac{1}{\sqrt{2}}\right)} \]
   11. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\frac{{th}^{2}}{\sqrt{2}}}, \frac{1}{\sqrt{2}}\right) \]

       2. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, \frac{{th}^{2}}{\color{blue}{\sqrt{2}}}, \frac{1}{\sqrt{2}}\right) \]

       3. lower-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, \frac{{th}^{2}}{\sqrt{\color{blue}{2}}}, \frac{1}{\sqrt{2}}\right) \]

       4. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, \frac{{th}^{2}}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right) \]

       5. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, \frac{{th}^{2}}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right) \]

       6. lower-sqrt.f64 63.3%

          \[\leadsto \mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot \mathsf{fma}\left(-0.5, \frac{{th}^{2}}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right) \]

   12. Applied rewrites 63.3%

       \[\leadsto \mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot \color{blue}{\mathsf{fma}\left(-0.5, \frac{{th}^{2}}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right)} \]
   13. Step-by-step derivation

       1. lift-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot \left(\frac{-1}{2} \cdot \frac{{th}^{2}}{\sqrt{2}} + \color{blue}{\frac{1}{\sqrt{2}}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot \left(\frac{-1}{2} \cdot \frac{{th}^{2}}{\sqrt{2}} + \frac{1}{\sqrt{2}}\right) \]

       3. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot \left(\frac{\frac{-1}{2} \cdot {th}^{2}}{\sqrt{2}} + \frac{\color{blue}{1}}{\sqrt{2}}\right) \]

       4. lift-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot \left(\frac{\frac{-1}{2} \cdot {th}^{2}}{\sqrt{2}} + \frac{1}{\color{blue}{\sqrt{2}}}\right) \]

       5. div-add-rev N/A

          \[\leadsto \mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot \frac{\frac{-1}{2} \cdot {th}^{2} + 1}{\color{blue}{\sqrt{2}}} \]

       6. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot \frac{\frac{-1}{2} \cdot {th}^{2} + 1}{\color{blue}{\sqrt{2}}} \]

       7. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot \frac{{th}^{2} \cdot \frac{-1}{2} + 1}{\sqrt{2}} \]

       8. lower-fma.f64 63.3%

          \[\leadsto \mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot \frac{\mathsf{fma}\left({th}^{2}, -0.5, 1\right)}{\sqrt{\color{blue}{2}}} \]

       9. lift-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot \frac{\mathsf{fma}\left({th}^{2}, \frac{-1}{2}, 1\right)}{\sqrt{2}} \]

       10. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot \frac{\mathsf{fma}\left(th \cdot th, \frac{-1}{2}, 1\right)}{\sqrt{2}} \]

       11. lower-*.f64 63.3%

           \[\leadsto \mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot \frac{\mathsf{fma}\left(th \cdot th, -0.5, 1\right)}{\sqrt{2}} \]

   14. Applied rewrites 63.3%

       \[\leadsto \color{blue}{\mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot \frac{\mathsf{fma}\left(th \cdot th, -0.5, 1\right)}{\sqrt{2}}} \]

   if -2e-132 < (+.f64 (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a1 a1)) (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a2 a2)))

   1. Initial program 99.5%

      \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]

   2. Taylor expanded in th around 0

      \[\leadsto \frac{\color{blue}{1}}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
   3. Step-by-step derivation

   4. Applied rewrites 82.3%

      \[\leadsto \frac{\color{blue}{1}}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]

   5. Taylor expanded in th around 0

      \[\leadsto \frac{1}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\color{blue}{1}}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 66.5%

      \[\leadsto \frac{1}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\color{blue}{1}}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{1}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) + \frac{1}{\sqrt{2}} \cdot \left(a1 \cdot a1\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)} + \frac{1}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{1}{\sqrt{2}} \cdot \color{blue}{\left(a2 \cdot a2\right)} + \frac{1}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) \]

      5. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{\sqrt{2}} \cdot a2\right) \cdot a2} + \frac{1}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{\sqrt{2}} \cdot a2, a2, \frac{1}{\sqrt{2}} \cdot \left(a1 \cdot a1\right)\right)} \]

      7. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{\sqrt{2}}} \cdot a2, a2, \frac{1}{\sqrt{2}} \cdot \left(a1 \cdot a1\right)\right) \]

      8. associate-*l/ N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1 \cdot a2}{\sqrt{2}}}, a2, \frac{1}{\sqrt{2}} \cdot \left(a1 \cdot a1\right)\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1 \cdot a2}{\sqrt{2}}}, a2, \frac{1}{\sqrt{2}} \cdot \left(a1 \cdot a1\right)\right) \]

      10. lower-*.f64 66.5%

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{1 \cdot a2}}{\sqrt{2}}, a2, \frac{1}{\sqrt{2}} \cdot \left(a1 \cdot a1\right)\right) \]

   9. Applied rewrites 66.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1 \cdot a2}{\sqrt{2}}, a2, \frac{1}{\sqrt{2}} \cdot \left(a1 \cdot a1\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 78.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{fma}\left(a2, a2, a1 \cdot a1\right)\\ t_2 := \frac{\cos th}{\sqrt{2}}\\ \mathbf{if}\;t\_2 \cdot \left(a1 \cdot a1\right) + t\_2 \cdot \left(a2 \cdot a2\right) \leq -2 \cdot 10^{-132}:\\ \;\;\;\;t\_1 \cdot \frac{\mathsf{fma}\left(th \cdot th, -0.5, 1\right)}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot t\_1}{1.4142135623730951}\\ \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (fma a2 a2 (* a1 a1))) (t_2 (/ (cos th) (sqrt 2.0))))
   (if (<= (+ (* t_2 (* a1 a1)) (* t_2 (* a2 a2))) -2e-132)
     (* t_1 (/ (fma (* th th) -0.5 1.0) (sqrt 2.0)))
     (/ (* 1.0 t_1) 1.4142135623730951))))

C

double code(double a1, double a2, double th) {
	double t_1 = fma(a2, a2, (a1 * a1));
	double t_2 = cos(th) / sqrt(2.0);
	double tmp;
	if (((t_2 * (a1 * a1)) + (t_2 * (a2 * a2))) <= -2e-132) {
		tmp = t_1 * (fma((th * th), -0.5, 1.0) / sqrt(2.0));
	} else {
		tmp = (1.0 * t_1) / 1.4142135623730951;
	}
	return tmp;
}

Julia

function code(a1, a2, th)
	t_1 = fma(a2, a2, Float64(a1 * a1))
	t_2 = Float64(cos(th) / sqrt(2.0))
	tmp = 0.0
	if (Float64(Float64(t_2 * Float64(a1 * a1)) + Float64(t_2 * Float64(a2 * a2))) <= -2e-132)
		tmp = Float64(t_1 * Float64(fma(Float64(th * th), -0.5, 1.0) / sqrt(2.0)));
	else
		tmp = Float64(Float64(1.0 * t_1) / 1.4142135623730951);
	end
	return tmp
end

Wolfram

code[a1_, a2_, th_] := Block[{t$95$1 = N[(a2 * a2 + N[(a1 * a1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(t$95$2 * N[(a1 * a1), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 * N[(a2 * a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-132], N[(t$95$1 * N[(N[(N[(th * th), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 * t$95$1), $MachinePrecision] / 1.4142135623730951), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a1 a1)) (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a2 a2))) < -2e-132

   1. Initial program 99.5%

      \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]

   2. Taylor expanded in th around 0

      \[\leadsto \frac{\color{blue}{1}}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
   3. Step-by-step derivation

   4. Applied rewrites 82.3%

      \[\leadsto \frac{\color{blue}{1}}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]

   5. Taylor expanded in th around 0

      \[\leadsto \frac{1}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\color{blue}{1}}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 66.5%

      \[\leadsto \frac{1}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\color{blue}{1}}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{1}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\sqrt{2}} \cdot \left(a1 \cdot a1\right)} + \frac{1}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{1}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \color{blue}{\frac{1}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)} \]

      4. distribute-lft-out N/A

         \[\leadsto \color{blue}{\frac{1}{\sqrt{2}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)} \]

      5. +-commutative N/A

         \[\leadsto \frac{1}{\sqrt{2}} \cdot \color{blue}{\left(a2 \cdot a2 + a1 \cdot a1\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{1}{\sqrt{2}} \cdot \left(\color{blue}{a2 \cdot a2} + a1 \cdot a1\right) \]

      7. lift-fma.f64 N/A

         \[\leadsto \frac{1}{\sqrt{2}} \cdot \color{blue}{\mathsf{fma}\left(a2, a2, a1 \cdot a1\right)} \]

      8. *-commutative N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot \frac{1}{\sqrt{2}}} \]

      9. lower-*.f64 66.5%

         \[\leadsto \color{blue}{\mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot \frac{1}{\sqrt{2}}} \]

   9. Applied rewrites 66.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot \frac{1}{\sqrt{2}}} \]

   10. Taylor expanded in th around 0

       \[\leadsto \mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot \color{blue}{\left(\frac{-1}{2} \cdot \frac{{th}^{2}}{\sqrt{2}} + \frac{1}{\sqrt{2}}\right)} \]
   11. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\frac{{th}^{2}}{\sqrt{2}}}, \frac{1}{\sqrt{2}}\right) \]

       2. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, \frac{{th}^{2}}{\color{blue}{\sqrt{2}}}, \frac{1}{\sqrt{2}}\right) \]

       3. lower-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, \frac{{th}^{2}}{\sqrt{\color{blue}{2}}}, \frac{1}{\sqrt{2}}\right) \]

       4. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, \frac{{th}^{2}}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right) \]

       5. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, \frac{{th}^{2}}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right) \]

       6. lower-sqrt.f64 63.3%

          \[\leadsto \mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot \mathsf{fma}\left(-0.5, \frac{{th}^{2}}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right) \]

   12. Applied rewrites 63.3%

       \[\leadsto \mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot \color{blue}{\mathsf{fma}\left(-0.5, \frac{{th}^{2}}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right)} \]
   13. Step-by-step derivation

       1. lift-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot \left(\frac{-1}{2} \cdot \frac{{th}^{2}}{\sqrt{2}} + \color{blue}{\frac{1}{\sqrt{2}}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot \left(\frac{-1}{2} \cdot \frac{{th}^{2}}{\sqrt{2}} + \frac{1}{\sqrt{2}}\right) \]

       3. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot \left(\frac{\frac{-1}{2} \cdot {th}^{2}}{\sqrt{2}} + \frac{\color{blue}{1}}{\sqrt{2}}\right) \]

       4. lift-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot \left(\frac{\frac{-1}{2} \cdot {th}^{2}}{\sqrt{2}} + \frac{1}{\color{blue}{\sqrt{2}}}\right) \]

       5. div-add-rev N/A

          \[\leadsto \mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot \frac{\frac{-1}{2} \cdot {th}^{2} + 1}{\color{blue}{\sqrt{2}}} \]

       6. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot \frac{\frac{-1}{2} \cdot {th}^{2} + 1}{\color{blue}{\sqrt{2}}} \]

       7. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot \frac{{th}^{2} \cdot \frac{-1}{2} + 1}{\sqrt{2}} \]

       8. lower-fma.f64 63.3%

          \[\leadsto \mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot \frac{\mathsf{fma}\left({th}^{2}, -0.5, 1\right)}{\sqrt{\color{blue}{2}}} \]

       9. lift-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot \frac{\mathsf{fma}\left({th}^{2}, \frac{-1}{2}, 1\right)}{\sqrt{2}} \]

       10. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot \frac{\mathsf{fma}\left(th \cdot th, \frac{-1}{2}, 1\right)}{\sqrt{2}} \]

       11. lower-*.f64 63.3%

           \[\leadsto \mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot \frac{\mathsf{fma}\left(th \cdot th, -0.5, 1\right)}{\sqrt{2}} \]

   14. Applied rewrites 63.3%

       \[\leadsto \color{blue}{\mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot \frac{\mathsf{fma}\left(th \cdot th, -0.5, 1\right)}{\sqrt{2}}} \]

   if -2e-132 < (+.f64 (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a1 a1)) (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a2 a2)))

   1. Initial program 99.5%

      \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]

   2. Taylor expanded in th around 0

      \[\leadsto \frac{\color{blue}{1}}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
   3. Step-by-step derivation

   4. Applied rewrites 82.3%

      \[\leadsto \frac{\color{blue}{1}}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]

   5. Taylor expanded in th around 0

      \[\leadsto \frac{1}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\color{blue}{1}}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 66.5%

      \[\leadsto \frac{1}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\color{blue}{1}}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{1}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\sqrt{2}} \cdot \left(a1 \cdot a1\right)} + \frac{1}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{1}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \color{blue}{\frac{1}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)} \]

      4. distribute-lft-out N/A

         \[\leadsto \color{blue}{\frac{1}{\sqrt{2}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)} \]

      5. +-commutative N/A

         \[\leadsto \frac{1}{\sqrt{2}} \cdot \color{blue}{\left(a2 \cdot a2 + a1 \cdot a1\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{1}{\sqrt{2}} \cdot \left(\color{blue}{a2 \cdot a2} + a1 \cdot a1\right) \]

      7. lift-fma.f64 N/A

         \[\leadsto \frac{1}{\sqrt{2}} \cdot \color{blue}{\mathsf{fma}\left(a2, a2, a1 \cdot a1\right)} \]

      8. *-commutative N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot \frac{1}{\sqrt{2}}} \]

      9. lower-*.f64 66.5%

         \[\leadsto \color{blue}{\mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot \frac{1}{\sqrt{2}}} \]

   9. Applied rewrites 66.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot \frac{1}{\sqrt{2}}} \]

   10. Evaluated real constant 66.5%

       \[\leadsto \mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot \frac{1}{\color{blue}{1.4142135623730951}} \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot \frac{1}{\frac{6369051672525773}{4503599627370496}}} \]

       2. lift-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot \color{blue}{\frac{1}{\frac{6369051672525773}{4503599627370496}}} \]

       3. associate-*r/ N/A

          \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot 1}{\frac{6369051672525773}{4503599627370496}}} \]

       4. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot 1}{\frac{6369051672525773}{4503599627370496}}} \]

       5. *-commutative N/A

          \[\leadsto \frac{\color{blue}{1 \cdot \mathsf{fma}\left(a2, a2, a1 \cdot a1\right)}}{\frac{6369051672525773}{4503599627370496}} \]

       6. lower-*.f64 66.5%

          \[\leadsto \frac{\color{blue}{1 \cdot \mathsf{fma}\left(a2, a2, a1 \cdot a1\right)}}{1.4142135623730951} \]

   12. Applied rewrites 66.5%

       \[\leadsto \color{blue}{\frac{1 \cdot \mathsf{fma}\left(a2, a2, a1 \cdot a1\right)}{1.4142135623730951}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 66.5% accurate, 5.9× speedup

Math

\[\frac{1 \cdot \mathsf{fma}\left(a2, a2, a1 \cdot a1\right)}{1.4142135623730951} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (/ (* 1.0 (fma a2 a2 (* a1 a1))) 1.4142135623730951))

C

double code(double a1, double a2, double th) {
	return (1.0 * fma(a2, a2, (a1 * a1))) / 1.4142135623730951;
}

Julia

function code(a1, a2, th)
	return Float64(Float64(1.0 * fma(a2, a2, Float64(a1 * a1))) / 1.4142135623730951)
end

Wolfram

code[a1_, a2_, th_] := N[(N[(1.0 * N[(a2 * a2 + N[(a1 * a1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 1.4142135623730951), $MachinePrecision]

Derivation

1. Initial program 99.5%

   \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]

2. Taylor expanded in th around 0

   \[\leadsto \frac{\color{blue}{1}}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
3. Step-by-step derivation

4. Applied rewrites 82.3%

   \[\leadsto \frac{\color{blue}{1}}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]

5. Taylor expanded in th around 0

   \[\leadsto \frac{1}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\color{blue}{1}}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
6. Step-by-step derivation

7. Applied rewrites 66.5%

   \[\leadsto \frac{1}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\color{blue}{1}}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
8. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{1}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{\sqrt{2}} \cdot \left(a1 \cdot a1\right)} + \frac{1}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{1}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \color{blue}{\frac{1}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)} \]

   4. distribute-lft-out N/A

      \[\leadsto \color{blue}{\frac{1}{\sqrt{2}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)} \]

   5. +-commutative N/A

      \[\leadsto \frac{1}{\sqrt{2}} \cdot \color{blue}{\left(a2 \cdot a2 + a1 \cdot a1\right)} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{1}{\sqrt{2}} \cdot \left(\color{blue}{a2 \cdot a2} + a1 \cdot a1\right) \]

   7. lift-fma.f64 N/A

      \[\leadsto \frac{1}{\sqrt{2}} \cdot \color{blue}{\mathsf{fma}\left(a2, a2, a1 \cdot a1\right)} \]

   8. *-commutative N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot \frac{1}{\sqrt{2}}} \]

   9. lower-*.f64 66.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot \frac{1}{\sqrt{2}}} \]

9. Applied rewrites 66.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot \frac{1}{\sqrt{2}}} \]

10. Evaluated real constant 66.5%

    \[\leadsto \mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot \frac{1}{\color{blue}{1.4142135623730951}} \]
11. Step-by-step derivation

    1. lift-*.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot \frac{1}{\frac{6369051672525773}{4503599627370496}}} \]

    2. lift-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot \color{blue}{\frac{1}{\frac{6369051672525773}{4503599627370496}}} \]

    3. associate-*r/ N/A

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot 1}{\frac{6369051672525773}{4503599627370496}}} \]

    4. lower-/.f64 N/A

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot 1}{\frac{6369051672525773}{4503599627370496}}} \]

    5. *-commutative N/A

       \[\leadsto \frac{\color{blue}{1 \cdot \mathsf{fma}\left(a2, a2, a1 \cdot a1\right)}}{\frac{6369051672525773}{4503599627370496}} \]

    6. lower-*.f64 66.5%

       \[\leadsto \frac{\color{blue}{1 \cdot \mathsf{fma}\left(a2, a2, a1 \cdot a1\right)}}{1.4142135623730951} \]

12. Applied rewrites 66.5%

    \[\leadsto \color{blue}{\frac{1 \cdot \mathsf{fma}\left(a2, a2, a1 \cdot a1\right)}{1.4142135623730951}} \]
13. Add Preprocessing

Alternative 8: 66.5% accurate, 7.5× speedup

Math

\[\mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot 0.7071067811865475 \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (* (fma a2 a2 (* a1 a1)) 0.7071067811865475))

C

double code(double a1, double a2, double th) {
	return fma(a2, a2, (a1 * a1)) * 0.7071067811865475;
}

Julia

function code(a1, a2, th)
	return Float64(fma(a2, a2, Float64(a1 * a1)) * 0.7071067811865475)
end

Wolfram

code[a1_, a2_, th_] := N[(N[(a2 * a2 + N[(a1 * a1), $MachinePrecision]), $MachinePrecision] * 0.7071067811865475), $MachinePrecision]

Derivation

1. Initial program 99.5%

   \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]

2. Taylor expanded in th around 0

   \[\leadsto \frac{\color{blue}{1}}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
3. Step-by-step derivation

4. Applied rewrites 82.3%

   \[\leadsto \frac{\color{blue}{1}}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]

5. Taylor expanded in th around 0

   \[\leadsto \frac{1}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\color{blue}{1}}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
6. Step-by-step derivation

7. Applied rewrites 66.5%

   \[\leadsto \frac{1}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\color{blue}{1}}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
8. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{1}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{\sqrt{2}} \cdot \left(a1 \cdot a1\right)} + \frac{1}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{1}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \color{blue}{\frac{1}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)} \]

   4. distribute-lft-out N/A

      \[\leadsto \color{blue}{\frac{1}{\sqrt{2}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)} \]

   5. +-commutative N/A

      \[\leadsto \frac{1}{\sqrt{2}} \cdot \color{blue}{\left(a2 \cdot a2 + a1 \cdot a1\right)} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{1}{\sqrt{2}} \cdot \left(\color{blue}{a2 \cdot a2} + a1 \cdot a1\right) \]

   7. lift-fma.f64 N/A

      \[\leadsto \frac{1}{\sqrt{2}} \cdot \color{blue}{\mathsf{fma}\left(a2, a2, a1 \cdot a1\right)} \]

   8. *-commutative N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot \frac{1}{\sqrt{2}}} \]

   9. lower-*.f64 66.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot \frac{1}{\sqrt{2}}} \]

9. Applied rewrites 66.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot \frac{1}{\sqrt{2}}} \]

10. Evaluated real constant 66.5%

    \[\leadsto \mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot \frac{1}{\color{blue}{1.4142135623730951}} \]

11. Taylor expanded in th around 0

    \[\leadsto \mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot \color{blue}{\frac{4503599627370496}{6369051672525773}} \]
12. Step-by-step derivation

13. Applied rewrites 66.5%

    \[\leadsto \mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot \color{blue}{0.7071067811865475} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2025187 
(FPCore (a1 a2 th)
  :name "Migdal et al, Equation (64)"
  :precision binary64
  (+ (* (/ (cos th) (sqrt 2.0)) (* a1 a1)) (* (/ (cos th) (sqrt 2.0)) (* a2 a2))))