Trowbridge-Reitz Sample, near normal, slope_x

Percentage Accurate: 99.0% → 99.0%

Time: 4.7s

Alternatives: 16

Speedup: 1.0×

Specification

\[\left(\left(cosTheta\_i > 0.9999 \land cosTheta\_i \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u2 \land u2 \leq 1\right)\]

Math

\[\begin{array}{l} \\ \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (/ u1 (- 1.0 u1))) (cos (* 6.28318530718 u2))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - u1))) * cosf((6.28318530718f * u2));
}

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(4) function code(costheta_i, u1, u2)
use fmin_fmax_functions
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = sqrt((u1 / (1.0e0 - u1))) * cos((6.28318530718e0 * u2))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * cos(Float32(Float32(6.28318530718) * u2)))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((u1 / (single(1.0) - u1))) * cos((single(6.28318530718) * u2));
end

Initial Program: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (/ u1 (- 1.0 u1))) (cos (* 6.28318530718 u2))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - u1))) * cosf((6.28318530718f * u2));
}

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(4) function code(costheta_i, u1, u2)
use fmin_fmax_functions
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = sqrt((u1 / (1.0e0 - u1))) * cos((6.28318530718e0 * u2))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * cos(Float32(Float32(6.28318530718) * u2)))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((u1 / (single(1.0) - u1))) * cos((single(6.28318530718) * u2));
end

Alternative 1: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (/ u1 (- 1.0 u1))) (cos (* 6.28318530718 u2))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - u1))) * cosf((6.28318530718f * u2));
}

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(4) function code(costheta_i, u1, u2)
use fmin_fmax_functions
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = sqrt((u1 / (1.0e0 - u1))) * cos((6.28318530718e0 * u2))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * cos(Float32(Float32(6.28318530718) * u2)))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((u1 / (single(1.0) - u1))) * cos((single(6.28318530718) * u2));
end

Derivation

1. Initial program 99.0%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
2. Add Preprocessing

Alternative 2: 97.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \leq 0.10999999940395355:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-85.45681720672748 \cdot u2, u2, 64.93939402268539\right) \cdot u2, u2, -19.739208802181317\right) \cdot u2, u2, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(u1, u1, u1\right)} \cdot \cos \left(u2 \cdot -6.28318530718\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u2 0.10999999940395355)
   (*
    (sqrt (/ u1 (- 1.0 u1)))
    (fma
     (*
      (fma
       (* (fma (* -85.45681720672748 u2) u2 64.93939402268539) u2)
       u2
       -19.739208802181317)
      u2)
     u2
     1.0))
   (* (sqrt (fma u1 u1 u1)) (cos (* u2 -6.28318530718)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (u2 <= 0.10999999940395355f) {
		tmp = sqrtf((u1 / (1.0f - u1))) * fmaf((fmaf((fmaf((-85.45681720672748f * u2), u2, 64.93939402268539f) * u2), u2, -19.739208802181317f) * u2), u2, 1.0f);
	} else {
		tmp = sqrtf(fmaf(u1, u1, u1)) * cosf((u2 * -6.28318530718f));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (u2 <= Float32(0.10999999940395355))
		tmp = Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * fma(Float32(fma(Float32(fma(Float32(Float32(-85.45681720672748) * u2), u2, Float32(64.93939402268539)) * u2), u2, Float32(-19.739208802181317)) * u2), u2, Float32(1.0)));
	else
		tmp = Float32(sqrt(fma(u1, u1, u1)) * cos(Float32(u2 * Float32(-6.28318530718))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u2 < 0.109999999

   1. Initial program 99.0%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]

   2. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\left(1 + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)\right)} \]
   3. Step-by-step derivation

      1. lower-+.f32 N/A

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + \color{blue}{{u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)}\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + {u2}^{2} \cdot \color{blue}{\left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)}\right) \]

      3. lower-pow.f32 N/A

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + {u2}^{2} \cdot \left(\color{blue}{{u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right)} - \frac{98696044010906577398881}{5000000000000000000000}\right)\right) \]

      4. lower--.f32 N/A

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \color{blue}{\frac{98696044010906577398881}{5000000000000000000000}}\right)\right) \]

      5. lower-*.f32 N/A

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)\right) \]

      6. lower-pow.f32 N/A

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)\right) \]

      7. lower-+.f32 N/A

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)\right) \]

      8. lower-*.f32 N/A

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)\right) \]

      9. lower-pow.f32 93.3

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(64.93939402268539 + -85.45681720672748 \cdot {u2}^{2}\right) - 19.739208802181317\right)\right) \]

   4. Applied rewrites 93.3%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\left(1 + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(64.93939402268539 + -85.45681720672748 \cdot {u2}^{2}\right) - 19.739208802181317\right)\right)} \]
   5. Step-by-step derivation

      1. lift-+.f32 N/A

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + \color{blue}{{u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left({u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right) + \color{blue}{1}\right) \]

      3. lift-*.f32 N/A

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left({u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right) + 1\right) \]

      4. *-commutative N/A

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(\left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right) \cdot {u2}^{2} + 1\right) \]

      5. lift-pow.f32 N/A

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(\left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right) \cdot {u2}^{2} + 1\right) \]

      6. unpow2 N/A

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(\left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right) \cdot \left(u2 \cdot u2\right) + 1\right) \]

      7. associate-*r* N/A

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(\left(\left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right) \cdot u2\right) \cdot u2 + 1\right) \]

      8. lower-fma.f32 N/A

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right) \cdot u2, \color{blue}{u2}, 1\right) \]

   6. Applied rewrites 93.3%

      \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-85.45681720672748, u2 \cdot u2, 64.93939402268539\right), u2 \cdot u2, -19.739208802181317\right) \cdot u2, u2, 1\right)} \]
   7. Step-by-step derivation

      1. lift-fma.f32 N/A

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000}, u2 \cdot u2, \frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}\right) \cdot \left(u2 \cdot u2\right) + \frac{-98696044010906577398881}{5000000000000000000000}\right) \cdot u2, u2, 1\right) \]

      2. lift-*.f32 N/A

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000}, u2 \cdot u2, \frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}\right) \cdot \left(u2 \cdot u2\right) + \frac{-98696044010906577398881}{5000000000000000000000}\right) \cdot u2, u2, 1\right) \]

      3. associate-*r* N/A

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\left(\left(\mathsf{fma}\left(\frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000}, u2 \cdot u2, \frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}\right) \cdot u2\right) \cdot u2 + \frac{-98696044010906577398881}{5000000000000000000000}\right) \cdot u2, u2, 1\right) \]

      4. lower-fma.f32 N/A

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000}, u2 \cdot u2, \frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}\right) \cdot u2, u2, \frac{-98696044010906577398881}{5000000000000000000000}\right) \cdot u2, u2, 1\right) \]

      5. lower-*.f32 93.3

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-85.45681720672748, u2 \cdot u2, 64.93939402268539\right) \cdot u2, u2, -19.739208802181317\right) \cdot u2, u2, 1\right) \]

      6. lift-fma.f32 N/A

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(\frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot \left(u2 \cdot u2\right) + \frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}\right) \cdot u2, u2, \frac{-98696044010906577398881}{5000000000000000000000}\right) \cdot u2, u2, 1\right) \]

      7. *-commutative N/A

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(\left(u2 \cdot u2\right) \cdot \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} + \frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}\right) \cdot u2, u2, \frac{-98696044010906577398881}{5000000000000000000000}\right) \cdot u2, u2, 1\right) \]

      8. lower-fma.f32 93.3

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(u2 \cdot u2, -85.45681720672748, 64.93939402268539\right) \cdot u2, u2, -19.739208802181317\right) \cdot u2, u2, 1\right) \]

   8. Applied rewrites 93.3%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(u2 \cdot u2, -85.45681720672748, 64.93939402268539\right) \cdot u2, u2, -19.739208802181317\right) \cdot u2, u2, 1\right) \]
   9. Step-by-step derivation

      1. lift-fma.f32 N/A

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(\left(u2 \cdot u2\right) \cdot \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} + \frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}\right) \cdot u2, u2, \frac{-98696044010906577398881}{5000000000000000000000}\right) \cdot u2, u2, 1\right) \]

      2. lift-*.f32 N/A

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(\left(u2 \cdot u2\right) \cdot \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} + \frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}\right) \cdot u2, u2, \frac{-98696044010906577398881}{5000000000000000000000}\right) \cdot u2, u2, 1\right) \]

      3. associate-*l* N/A

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(u2 \cdot \left(u2 \cdot \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000}\right) + \frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}\right) \cdot u2, u2, \frac{-98696044010906577398881}{5000000000000000000000}\right) \cdot u2, u2, 1\right) \]

      4. *-commutative N/A

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(\left(u2 \cdot \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000}\right) \cdot u2 + \frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}\right) \cdot u2, u2, \frac{-98696044010906577398881}{5000000000000000000000}\right) \cdot u2, u2, 1\right) \]

      5. lower-fma.f32 N/A

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(u2 \cdot \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000}, u2, \frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}\right) \cdot u2, u2, \frac{-98696044010906577398881}{5000000000000000000000}\right) \cdot u2, u2, 1\right) \]

      6. *-commutative N/A

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot u2, u2, \frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}\right) \cdot u2, u2, \frac{-98696044010906577398881}{5000000000000000000000}\right) \cdot u2, u2, 1\right) \]

      7. lower-*.f32 93.3

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-85.45681720672748 \cdot u2, u2, 64.93939402268539\right) \cdot u2, u2, -19.739208802181317\right) \cdot u2, u2, 1\right) \]

   10. Applied rewrites 93.3%

       \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-85.45681720672748 \cdot u2, u2, 64.93939402268539\right) \cdot u2, u2, -19.739208802181317\right) \cdot u2, u2, 1\right) \]

   if 0.109999999 < u2

   1. Initial program 99.0%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]

   2. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1\right)}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]
   3. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \sqrt{u1 \cdot \color{blue}{\left(1 + u1\right)}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      2. lower-+.f32 86.7

         \[\leadsto \sqrt{u1 \cdot \left(1 + \color{blue}{u1}\right)} \cdot \cos \left(6.28318530718 \cdot u2\right) \]

   4. Applied rewrites 86.7%

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1\right)}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
   5. Step-by-step derivation

      1. lift-*.f32 N/A

         \[\leadsto \sqrt{u1 \cdot \color{blue}{\left(1 + u1\right)}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      2. lift-+.f32 N/A

         \[\leadsto \sqrt{u1 \cdot \left(1 + \color{blue}{u1}\right)} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \sqrt{1 \cdot u1 + \color{blue}{u1 \cdot u1}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      4. *-lft-identity N/A

         \[\leadsto \sqrt{u1 + \color{blue}{u1} \cdot u1} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      5. +-commutative N/A

         \[\leadsto \sqrt{u1 \cdot u1 + \color{blue}{u1}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      6. lower-fma.f32 86.8

         \[\leadsto \sqrt{\mathsf{fma}\left(u1, \color{blue}{u1}, u1\right)} \cdot \cos \left(6.28318530718 \cdot u2\right) \]

      7. lower-fma.f32 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(u1, \color{blue}{u1}, u1\right)} \cdot \mathsf{Rewrite=>}\left(lift-cos.f32, \cos \left(\frac{314159265359}{50000000000} \cdot u2\right)\right) \]

      8. lower-fma.f32 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(u1, \color{blue}{u1}, u1\right)} \cdot \mathsf{Rewrite=>}\left(cos-neg-rev, \cos \left(\mathsf{neg}\left(\frac{314159265359}{50000000000} \cdot u2\right)\right)\right) \]

      9. lower-fma.f32 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(u1, \color{blue}{u1}, u1\right)} \cdot \mathsf{Rewrite=>}\left(cos-neg-rev, \cos \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{314159265359}{50000000000} \cdot u2\right)\right)\right)\right)\right) \]

      10. lower-fma.f32 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(u1, \color{blue}{u1}, u1\right)} \cdot \cos \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\mathsf{Rewrite=>}\left(lift-*.f32, \left(\frac{314159265359}{50000000000} \cdot u2\right)\right)\right)\right)\right)\right) \]

      11. lower-fma.f32 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(u1, \color{blue}{u1}, u1\right)} \cdot \cos \left(\mathsf{neg}\left(\mathsf{Rewrite=>}\left(distribute-lft-neg-in, \left(\left(\mathsf{neg}\left(\frac{314159265359}{50000000000}\right)\right) \cdot u2\right)\right)\right)\right) \]

      12. lower-fma.f32 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(u1, \color{blue}{u1}, u1\right)} \cdot \cos \left(\mathsf{neg}\left(\mathsf{Rewrite=>}\left(metadata-eval, \frac{-314159265359}{50000000000}\right) \cdot u2\right)\right) \]

      13. lower-fma.f32 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(u1, \color{blue}{u1}, u1\right)} \cdot \mathsf{Rewrite<=}\left(cos-neg-rev, \cos \left(\frac{-314159265359}{50000000000} \cdot u2\right)\right) \]

      14. lower-fma.f32 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(u1, \color{blue}{u1}, u1\right)} \cdot \mathsf{Rewrite=>}\left(lower-cos.f32, \cos \left(\frac{-314159265359}{50000000000} \cdot u2\right)\right) \]

      15. lower-fma.f32 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(u1, \color{blue}{u1}, u1\right)} \cdot \cos \mathsf{Rewrite=>}\left(*-commutative, \left(u2 \cdot \frac{-314159265359}{50000000000}\right)\right) \]

      16. lower-fma.f32 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(u1, \color{blue}{u1}, u1\right)} \cdot \cos \mathsf{Rewrite=>}\left(lower-*.f32, \left(u2 \cdot \frac{-314159265359}{50000000000}\right)\right) \]

   6. Applied rewrites 86.8%

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(u1, u1, u1\right)} \cdot \cos \left(u2 \cdot -6.28318530718\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 96.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \leq 0.3199999928474426:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(\mathsf{fma}\left(u2 \cdot u2, -85.45681720672748, 64.93939402268539\right) \cdot u2\right) \cdot u2, u2, -19.739208802181317 \cdot u2\right), u2, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \cos \left(6.28318530718 \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u2 0.3199999928474426)
   (*
    (sqrt (/ u1 (- 1.0 u1)))
    (fma
     (fma
      (* (* (fma (* u2 u2) -85.45681720672748 64.93939402268539) u2) u2)
      u2
      (* -19.739208802181317 u2))
     u2
     1.0))
   (* (sqrt u1) (cos (* 6.28318530718 u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (u2 <= 0.3199999928474426f) {
		tmp = sqrtf((u1 / (1.0f - u1))) * fmaf(fmaf(((fmaf((u2 * u2), -85.45681720672748f, 64.93939402268539f) * u2) * u2), u2, (-19.739208802181317f * u2)), u2, 1.0f);
	} else {
		tmp = sqrtf(u1) * cosf((6.28318530718f * u2));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (u2 <= Float32(0.3199999928474426))
		tmp = Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * fma(fma(Float32(Float32(fma(Float32(u2 * u2), Float32(-85.45681720672748), Float32(64.93939402268539)) * u2) * u2), u2, Float32(Float32(-19.739208802181317) * u2)), u2, Float32(1.0)));
	else
		tmp = Float32(sqrt(u1) * cos(Float32(Float32(6.28318530718) * u2)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u2 < 0.319999993

   1. Initial program 99.0%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]

   2. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\left(1 + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)\right)} \]
   3. Step-by-step derivation

      1. lower-+.f32 N/A

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + \color{blue}{{u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)}\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + {u2}^{2} \cdot \color{blue}{\left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)}\right) \]

      3. lower-pow.f32 N/A

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + {u2}^{2} \cdot \left(\color{blue}{{u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right)} - \frac{98696044010906577398881}{5000000000000000000000}\right)\right) \]

      4. lower--.f32 N/A

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \color{blue}{\frac{98696044010906577398881}{5000000000000000000000}}\right)\right) \]

      5. lower-*.f32 N/A

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)\right) \]

      6. lower-pow.f32 N/A

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)\right) \]

      7. lower-+.f32 N/A

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)\right) \]

      8. lower-*.f32 N/A

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)\right) \]

      9. lower-pow.f32 93.3

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(64.93939402268539 + -85.45681720672748 \cdot {u2}^{2}\right) - 19.739208802181317\right)\right) \]

   4. Applied rewrites 93.3%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\left(1 + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(64.93939402268539 + -85.45681720672748 \cdot {u2}^{2}\right) - 19.739208802181317\right)\right)} \]
   5. Step-by-step derivation

      1. lift-+.f32 N/A

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + \color{blue}{{u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left({u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right) + \color{blue}{1}\right) \]

      3. lift-*.f32 N/A

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left({u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right) + 1\right) \]

      4. *-commutative N/A

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(\left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right) \cdot {u2}^{2} + 1\right) \]

      5. lift-pow.f32 N/A

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(\left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right) \cdot {u2}^{2} + 1\right) \]

      6. unpow2 N/A

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(\left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right) \cdot \left(u2 \cdot u2\right) + 1\right) \]

      7. associate-*r* N/A

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(\left(\left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right) \cdot u2\right) \cdot u2 + 1\right) \]

      8. lower-fma.f32 N/A

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right) \cdot u2, \color{blue}{u2}, 1\right) \]

   6. Applied rewrites 93.3%

      \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-85.45681720672748, u2 \cdot u2, 64.93939402268539\right), u2 \cdot u2, -19.739208802181317\right) \cdot u2, u2, 1\right)} \]
   7. Step-by-step derivation

      1. lift-*.f32 N/A

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000}, u2 \cdot u2, \frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}\right), u2 \cdot u2, \frac{-98696044010906577398881}{5000000000000000000000}\right) \cdot u2, u2, 1\right) \]

      2. *-commutative N/A

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(u2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000}, u2 \cdot u2, \frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}\right), u2 \cdot u2, \frac{-98696044010906577398881}{5000000000000000000000}\right), u2, 1\right) \]

      3. lift-fma.f32 N/A

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(u2 \cdot \left(\mathsf{fma}\left(\frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000}, u2 \cdot u2, \frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}\right) \cdot \left(u2 \cdot u2\right) + \frac{-98696044010906577398881}{5000000000000000000000}\right), u2, 1\right) \]

      4. distribute-rgt-in N/A

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000}, u2 \cdot u2, \frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}\right) \cdot \left(u2 \cdot u2\right)\right) \cdot u2 + \frac{-98696044010906577398881}{5000000000000000000000} \cdot u2, u2, 1\right) \]

      5. lower-fma.f32 N/A

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000}, u2 \cdot u2, \frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}\right) \cdot \left(u2 \cdot u2\right), u2, \frac{-98696044010906577398881}{5000000000000000000000} \cdot u2\right), u2, 1\right) \]

      6. lift-*.f32 N/A

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000}, u2 \cdot u2, \frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}\right) \cdot \left(u2 \cdot u2\right), u2, \frac{-98696044010906577398881}{5000000000000000000000} \cdot u2\right), u2, 1\right) \]

      7. associate-*r* N/A

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000}, u2 \cdot u2, \frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}\right) \cdot u2\right) \cdot u2, u2, \frac{-98696044010906577398881}{5000000000000000000000} \cdot u2\right), u2, 1\right) \]

      8. lower-*.f32 N/A

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000}, u2 \cdot u2, \frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}\right) \cdot u2\right) \cdot u2, u2, \frac{-98696044010906577398881}{5000000000000000000000} \cdot u2\right), u2, 1\right) \]

      9. lower-*.f32 N/A

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000}, u2 \cdot u2, \frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}\right) \cdot u2\right) \cdot u2, u2, \frac{-98696044010906577398881}{5000000000000000000000} \cdot u2\right), u2, 1\right) \]

      10. lift-fma.f32 N/A

          \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(\left(\frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot \left(u2 \cdot u2\right) + \frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}\right) \cdot u2\right) \cdot u2, u2, \frac{-98696044010906577398881}{5000000000000000000000} \cdot u2\right), u2, 1\right) \]

      11. *-commutative N/A

          \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(\left(\left(u2 \cdot u2\right) \cdot \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} + \frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}\right) \cdot u2\right) \cdot u2, u2, \frac{-98696044010906577398881}{5000000000000000000000} \cdot u2\right), u2, 1\right) \]

      12. lower-fma.f32 N/A

          \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(\mathsf{fma}\left(u2 \cdot u2, \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000}, \frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}\right) \cdot u2\right) \cdot u2, u2, \frac{-98696044010906577398881}{5000000000000000000000} \cdot u2\right), u2, 1\right) \]

      13. lower-*.f32 93.3

          \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(\mathsf{fma}\left(u2 \cdot u2, -85.45681720672748, 64.93939402268539\right) \cdot u2\right) \cdot u2, u2, -19.739208802181317 \cdot u2\right), u2, 1\right) \]

   8. Applied rewrites 93.3%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(\mathsf{fma}\left(u2 \cdot u2, -85.45681720672748, 64.93939402268539\right) \cdot u2\right) \cdot u2, u2, -19.739208802181317 \cdot u2\right), u2, 1\right) \]

   if 0.319999993 < u2

   1. Initial program 99.0%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]

   2. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\color{blue}{u1}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]
   3. Step-by-step derivation

   4. Applied rewrites 74.6%

      \[\leadsto \sqrt{\color{blue}{u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 93.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(\mathsf{fma}\left(u2 \cdot u2, -85.45681720672748, 64.93939402268539\right) \cdot u2\right) \cdot u2, u2, -19.739208802181317 \cdot u2\right), u2, 1\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sqrt (/ u1 (- 1.0 u1)))
  (fma
   (fma
    (* (* (fma (* u2 u2) -85.45681720672748 64.93939402268539) u2) u2)
    u2
    (* -19.739208802181317 u2))
   u2
   1.0)))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - u1))) * fmaf(fmaf(((fmaf((u2 * u2), -85.45681720672748f, 64.93939402268539f) * u2) * u2), u2, (-19.739208802181317f * u2)), u2, 1.0f);
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * fma(fma(Float32(Float32(fma(Float32(u2 * u2), Float32(-85.45681720672748), Float32(64.93939402268539)) * u2) * u2), u2, Float32(Float32(-19.739208802181317) * u2)), u2, Float32(1.0)))
end

Derivation

1. Initial program 99.0%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]

2. Taylor expanded in u2 around 0

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\left(1 + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)\right)} \]
3. Step-by-step derivation

   1. lower-+.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + \color{blue}{{u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)}\right) \]

   2. lower-*.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + {u2}^{2} \cdot \color{blue}{\left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)}\right) \]

   3. lower-pow.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + {u2}^{2} \cdot \left(\color{blue}{{u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right)} - \frac{98696044010906577398881}{5000000000000000000000}\right)\right) \]

   4. lower--.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \color{blue}{\frac{98696044010906577398881}{5000000000000000000000}}\right)\right) \]

   5. lower-*.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)\right) \]

   6. lower-pow.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)\right) \]

   7. lower-+.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)\right) \]

   8. lower-*.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)\right) \]

   9. lower-pow.f32 93.3

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(64.93939402268539 + -85.45681720672748 \cdot {u2}^{2}\right) - 19.739208802181317\right)\right) \]

4. Applied rewrites 93.3%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\left(1 + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(64.93939402268539 + -85.45681720672748 \cdot {u2}^{2}\right) - 19.739208802181317\right)\right)} \]
5. Step-by-step derivation

   1. lift-+.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + \color{blue}{{u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)}\right) \]

   2. +-commutative N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left({u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right) + \color{blue}{1}\right) \]

   3. lift-*.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left({u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right) + 1\right) \]

   4. *-commutative N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(\left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right) \cdot {u2}^{2} + 1\right) \]

   5. lift-pow.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(\left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right) \cdot {u2}^{2} + 1\right) \]

   6. unpow2 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(\left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right) \cdot \left(u2 \cdot u2\right) + 1\right) \]

   7. associate-*r* N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(\left(\left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right) \cdot u2\right) \cdot u2 + 1\right) \]

   8. lower-fma.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right) \cdot u2, \color{blue}{u2}, 1\right) \]

6. Applied rewrites 93.3%

   \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-85.45681720672748, u2 \cdot u2, 64.93939402268539\right), u2 \cdot u2, -19.739208802181317\right) \cdot u2, u2, 1\right)} \]
7. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000}, u2 \cdot u2, \frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}\right), u2 \cdot u2, \frac{-98696044010906577398881}{5000000000000000000000}\right) \cdot u2, u2, 1\right) \]

   2. *-commutative N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(u2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000}, u2 \cdot u2, \frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}\right), u2 \cdot u2, \frac{-98696044010906577398881}{5000000000000000000000}\right), u2, 1\right) \]

   3. lift-fma.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(u2 \cdot \left(\mathsf{fma}\left(\frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000}, u2 \cdot u2, \frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}\right) \cdot \left(u2 \cdot u2\right) + \frac{-98696044010906577398881}{5000000000000000000000}\right), u2, 1\right) \]

   4. distribute-rgt-in N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000}, u2 \cdot u2, \frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}\right) \cdot \left(u2 \cdot u2\right)\right) \cdot u2 + \frac{-98696044010906577398881}{5000000000000000000000} \cdot u2, u2, 1\right) \]

   5. lower-fma.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000}, u2 \cdot u2, \frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}\right) \cdot \left(u2 \cdot u2\right), u2, \frac{-98696044010906577398881}{5000000000000000000000} \cdot u2\right), u2, 1\right) \]

   6. lift-*.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000}, u2 \cdot u2, \frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}\right) \cdot \left(u2 \cdot u2\right), u2, \frac{-98696044010906577398881}{5000000000000000000000} \cdot u2\right), u2, 1\right) \]

   7. associate-*r* N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000}, u2 \cdot u2, \frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}\right) \cdot u2\right) \cdot u2, u2, \frac{-98696044010906577398881}{5000000000000000000000} \cdot u2\right), u2, 1\right) \]

   8. lower-*.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000}, u2 \cdot u2, \frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}\right) \cdot u2\right) \cdot u2, u2, \frac{-98696044010906577398881}{5000000000000000000000} \cdot u2\right), u2, 1\right) \]

   9. lower-*.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000}, u2 \cdot u2, \frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}\right) \cdot u2\right) \cdot u2, u2, \frac{-98696044010906577398881}{5000000000000000000000} \cdot u2\right), u2, 1\right) \]

   10. lift-fma.f32 N/A

       \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(\left(\frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot \left(u2 \cdot u2\right) + \frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}\right) \cdot u2\right) \cdot u2, u2, \frac{-98696044010906577398881}{5000000000000000000000} \cdot u2\right), u2, 1\right) \]

   11. *-commutative N/A

       \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(\left(\left(u2 \cdot u2\right) \cdot \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} + \frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}\right) \cdot u2\right) \cdot u2, u2, \frac{-98696044010906577398881}{5000000000000000000000} \cdot u2\right), u2, 1\right) \]

   12. lower-fma.f32 N/A

       \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(\mathsf{fma}\left(u2 \cdot u2, \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000}, \frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}\right) \cdot u2\right) \cdot u2, u2, \frac{-98696044010906577398881}{5000000000000000000000} \cdot u2\right), u2, 1\right) \]

   13. lower-*.f32 93.3

       \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(\mathsf{fma}\left(u2 \cdot u2, -85.45681720672748, 64.93939402268539\right) \cdot u2\right) \cdot u2, u2, -19.739208802181317 \cdot u2\right), u2, 1\right) \]

8. Applied rewrites 93.3%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(\mathsf{fma}\left(u2 \cdot u2, -85.45681720672748, 64.93939402268539\right) \cdot u2\right) \cdot u2, u2, -19.739208802181317 \cdot u2\right), u2, 1\right) \]
9. Add Preprocessing

Alternative 5: 93.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-85.45681720672748, u2 \cdot u2, 64.93939402268539\right), u2 \cdot u2, -19.739208802181317\right), u2 \cdot u2, 1\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sqrt (/ u1 (- 1.0 u1)))
  (fma
   (fma
    (fma -85.45681720672748 (* u2 u2) 64.93939402268539)
    (* u2 u2)
    -19.739208802181317)
   (* u2 u2)
   1.0)))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - u1))) * fmaf(fmaf(fmaf(-85.45681720672748f, (u2 * u2), 64.93939402268539f), (u2 * u2), -19.739208802181317f), (u2 * u2), 1.0f);
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * fma(fma(fma(Float32(-85.45681720672748), Float32(u2 * u2), Float32(64.93939402268539)), Float32(u2 * u2), Float32(-19.739208802181317)), Float32(u2 * u2), Float32(1.0)))
end

Derivation

1. Initial program 99.0%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]

2. Taylor expanded in u2 around 0

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\left(1 + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)\right)} \]
3. Step-by-step derivation

   1. lower-+.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + \color{blue}{{u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)}\right) \]

   2. lower-*.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + {u2}^{2} \cdot \color{blue}{\left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)}\right) \]

   3. lower-pow.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + {u2}^{2} \cdot \left(\color{blue}{{u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right)} - \frac{98696044010906577398881}{5000000000000000000000}\right)\right) \]

   4. lower--.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \color{blue}{\frac{98696044010906577398881}{5000000000000000000000}}\right)\right) \]

   5. lower-*.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)\right) \]

   6. lower-pow.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)\right) \]

   7. lower-+.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)\right) \]

   8. lower-*.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)\right) \]

   9. lower-pow.f32 93.3

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(64.93939402268539 + -85.45681720672748 \cdot {u2}^{2}\right) - 19.739208802181317\right)\right) \]

4. Applied rewrites 93.3%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\left(1 + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(64.93939402268539 + -85.45681720672748 \cdot {u2}^{2}\right) - 19.739208802181317\right)\right)} \]
5. Step-by-step derivation

   1. lift-+.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + \color{blue}{{u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)}\right) \]

   2. +-commutative N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left({u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right) + \color{blue}{1}\right) \]

   3. lift-*.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left({u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right) + 1\right) \]

   4. *-commutative N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(\left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right) \cdot {u2}^{2} + 1\right) \]

   5. lower-fma.f32 93.3

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left({u2}^{2} \cdot \left(64.93939402268539 + -85.45681720672748 \cdot {u2}^{2}\right) - 19.739208802181317, \color{blue}{{u2}^{2}}, 1\right) \]

6. Applied rewrites 93.3%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-85.45681720672748, u2 \cdot u2, 64.93939402268539\right), u2 \cdot u2, -19.739208802181317\right), \color{blue}{u2 \cdot u2}, 1\right) \]
7. Add Preprocessing

Alternative 6: 93.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-85.45681720672748, u2 \cdot u2, 64.93939402268539\right) \cdot u2, u2, -19.739208802181317\right), u2 \cdot u2, 1\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sqrt (/ u1 (- 1.0 u1)))
  (fma
   (fma
    (* (fma -85.45681720672748 (* u2 u2) 64.93939402268539) u2)
    u2
    -19.739208802181317)
   (* u2 u2)
   1.0)))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - u1))) * fmaf(fmaf((fmaf(-85.45681720672748f, (u2 * u2), 64.93939402268539f) * u2), u2, -19.739208802181317f), (u2 * u2), 1.0f);
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * fma(fma(Float32(fma(Float32(-85.45681720672748), Float32(u2 * u2), Float32(64.93939402268539)) * u2), u2, Float32(-19.739208802181317)), Float32(u2 * u2), Float32(1.0)))
end

Derivation

1. Initial program 99.0%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]

2. Taylor expanded in u2 around 0

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\left(1 + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)\right)} \]
3. Step-by-step derivation

   1. lower-+.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + \color{blue}{{u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)}\right) \]

   2. lower-*.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + {u2}^{2} \cdot \color{blue}{\left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)}\right) \]

   3. lower-pow.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + {u2}^{2} \cdot \left(\color{blue}{{u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right)} - \frac{98696044010906577398881}{5000000000000000000000}\right)\right) \]

   4. lower--.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \color{blue}{\frac{98696044010906577398881}{5000000000000000000000}}\right)\right) \]

   5. lower-*.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)\right) \]

   6. lower-pow.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)\right) \]

   7. lower-+.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)\right) \]

   8. lower-*.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)\right) \]

   9. lower-pow.f32 93.3

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(64.93939402268539 + -85.45681720672748 \cdot {u2}^{2}\right) - 19.739208802181317\right)\right) \]

4. Applied rewrites 93.3%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\left(1 + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(64.93939402268539 + -85.45681720672748 \cdot {u2}^{2}\right) - 19.739208802181317\right)\right)} \]
5. Step-by-step derivation

   1. lift-+.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + \color{blue}{{u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)}\right) \]

   2. +-commutative N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left({u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right) + \color{blue}{1}\right) \]

   3. lift-*.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left({u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right) + 1\right) \]

   4. *-commutative N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(\left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right) \cdot {u2}^{2} + 1\right) \]

   5. lift-pow.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(\left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right) \cdot {u2}^{2} + 1\right) \]

   6. unpow2 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(\left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right) \cdot \left(u2 \cdot u2\right) + 1\right) \]

   7. associate-*r* N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(\left(\left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right) \cdot u2\right) \cdot u2 + 1\right) \]

   8. lower-fma.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right) \cdot u2, \color{blue}{u2}, 1\right) \]

6. Applied rewrites 93.3%

   \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-85.45681720672748, u2 \cdot u2, 64.93939402268539\right), u2 \cdot u2, -19.739208802181317\right) \cdot u2, u2, 1\right)} \]
7. Step-by-step derivation

   1. lift-fma.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000}, u2 \cdot u2, \frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}\right) \cdot \left(u2 \cdot u2\right) + \frac{-98696044010906577398881}{5000000000000000000000}\right) \cdot u2, u2, 1\right) \]

   2. lift-*.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000}, u2 \cdot u2, \frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}\right) \cdot \left(u2 \cdot u2\right) + \frac{-98696044010906577398881}{5000000000000000000000}\right) \cdot u2, u2, 1\right) \]

   3. associate-*r* N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\left(\left(\mathsf{fma}\left(\frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000}, u2 \cdot u2, \frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}\right) \cdot u2\right) \cdot u2 + \frac{-98696044010906577398881}{5000000000000000000000}\right) \cdot u2, u2, 1\right) \]

   4. lower-fma.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000}, u2 \cdot u2, \frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}\right) \cdot u2, u2, \frac{-98696044010906577398881}{5000000000000000000000}\right) \cdot u2, u2, 1\right) \]

   5. lower-*.f32 93.3

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-85.45681720672748, u2 \cdot u2, 64.93939402268539\right) \cdot u2, u2, -19.739208802181317\right) \cdot u2, u2, 1\right) \]

   6. lift-fma.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(\frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot \left(u2 \cdot u2\right) + \frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}\right) \cdot u2, u2, \frac{-98696044010906577398881}{5000000000000000000000}\right) \cdot u2, u2, 1\right) \]

   7. *-commutative N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(\left(u2 \cdot u2\right) \cdot \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} + \frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}\right) \cdot u2, u2, \frac{-98696044010906577398881}{5000000000000000000000}\right) \cdot u2, u2, 1\right) \]

   8. lower-fma.f32 93.3

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(u2 \cdot u2, -85.45681720672748, 64.93939402268539\right) \cdot u2, u2, -19.739208802181317\right) \cdot u2, u2, 1\right) \]

8. Applied rewrites 93.3%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(u2 \cdot u2, -85.45681720672748, 64.93939402268539\right) \cdot u2, u2, -19.739208802181317\right) \cdot u2, u2, 1\right) \]
9. Step-by-step derivation

   1. lift-fma.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(u2 \cdot u2, \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000}, \frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}\right) \cdot u2, u2, \frac{-98696044010906577398881}{5000000000000000000000}\right) \cdot u2\right) \cdot u2 + \color{blue}{1}\right) \]

   2. lift-*.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(u2 \cdot u2, \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000}, \frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}\right) \cdot u2, u2, \frac{-98696044010906577398881}{5000000000000000000000}\right) \cdot u2\right) \cdot u2 + 1\right) \]

   3. associate-*l* N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(u2 \cdot u2, \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000}, \frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}\right) \cdot u2, u2, \frac{-98696044010906577398881}{5000000000000000000000}\right) \cdot \left(u2 \cdot u2\right) + 1\right) \]

   4. lift-*.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(u2 \cdot u2, \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000}, \frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}\right) \cdot u2, u2, \frac{-98696044010906577398881}{5000000000000000000000}\right) \cdot \left(u2 \cdot u2\right) + 1\right) \]

   5. lower-fma.f32 93.3

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(u2 \cdot u2, -85.45681720672748, 64.93939402268539\right) \cdot u2, u2, -19.739208802181317\right), \color{blue}{u2 \cdot u2}, 1\right) \]

   6. lift-fma.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(\left(u2 \cdot u2\right) \cdot \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} + \frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}\right) \cdot u2, u2, \frac{-98696044010906577398881}{5000000000000000000000}\right), u2 \cdot u2, 1\right) \]

   7. *-commutative N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(\frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot \left(u2 \cdot u2\right) + \frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}\right) \cdot u2, u2, \frac{-98696044010906577398881}{5000000000000000000000}\right), u2 \cdot u2, 1\right) \]

   8. lower-fma.f32 93.3

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-85.45681720672748, u2 \cdot u2, 64.93939402268539\right) \cdot u2, u2, -19.739208802181317\right), u2 \cdot u2, 1\right) \]

10. Applied rewrites 93.3%

    \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-85.45681720672748, u2 \cdot u2, 64.93939402268539\right) \cdot u2, u2, -19.739208802181317\right), \color{blue}{u2 \cdot u2}, 1\right) \]
11. Add Preprocessing

Alternative 7: 93.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-85.45681720672748 \cdot u2, u2, 64.93939402268539\right), u2 \cdot u2, -19.739208802181317\right) \cdot u2, u2, 1\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sqrt (/ u1 (- 1.0 u1)))
  (fma
   (*
    (fma
     (fma (* -85.45681720672748 u2) u2 64.93939402268539)
     (* u2 u2)
     -19.739208802181317)
    u2)
   u2
   1.0)))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - u1))) * fmaf((fmaf(fmaf((-85.45681720672748f * u2), u2, 64.93939402268539f), (u2 * u2), -19.739208802181317f) * u2), u2, 1.0f);
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * fma(Float32(fma(fma(Float32(Float32(-85.45681720672748) * u2), u2, Float32(64.93939402268539)), Float32(u2 * u2), Float32(-19.739208802181317)) * u2), u2, Float32(1.0)))
end

Derivation

1. Initial program 99.0%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]

2. Taylor expanded in u2 around 0

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\left(1 + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)\right)} \]
3. Step-by-step derivation

   1. lower-+.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + \color{blue}{{u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)}\right) \]

   2. lower-*.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + {u2}^{2} \cdot \color{blue}{\left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)}\right) \]

   3. lower-pow.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + {u2}^{2} \cdot \left(\color{blue}{{u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right)} - \frac{98696044010906577398881}{5000000000000000000000}\right)\right) \]

   4. lower--.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \color{blue}{\frac{98696044010906577398881}{5000000000000000000000}}\right)\right) \]

   5. lower-*.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)\right) \]

   6. lower-pow.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)\right) \]

   7. lower-+.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)\right) \]

   8. lower-*.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)\right) \]

   9. lower-pow.f32 93.3

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(64.93939402268539 + -85.45681720672748 \cdot {u2}^{2}\right) - 19.739208802181317\right)\right) \]

4. Applied rewrites 93.3%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\left(1 + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(64.93939402268539 + -85.45681720672748 \cdot {u2}^{2}\right) - 19.739208802181317\right)\right)} \]
5. Step-by-step derivation

   1. lift-+.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + \color{blue}{{u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)}\right) \]

   2. +-commutative N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left({u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right) + \color{blue}{1}\right) \]

   3. lift-*.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left({u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right) + 1\right) \]

   4. *-commutative N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(\left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right) \cdot {u2}^{2} + 1\right) \]

   5. lift-pow.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(\left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right) \cdot {u2}^{2} + 1\right) \]

   6. unpow2 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(\left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right) \cdot \left(u2 \cdot u2\right) + 1\right) \]

   7. associate-*r* N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(\left(\left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right) \cdot u2\right) \cdot u2 + 1\right) \]

   8. lower-fma.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right) \cdot u2, \color{blue}{u2}, 1\right) \]

6. Applied rewrites 93.3%

   \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-85.45681720672748, u2 \cdot u2, 64.93939402268539\right), u2 \cdot u2, -19.739208802181317\right) \cdot u2, u2, 1\right)} \]
7. Step-by-step derivation

   1. lift-fma.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot \left(u2 \cdot u2\right) + \frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}, u2 \cdot u2, \frac{-98696044010906577398881}{5000000000000000000000}\right) \cdot u2, u2, 1\right) \]

   2. lift-*.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot \left(u2 \cdot u2\right) + \frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}, u2 \cdot u2, \frac{-98696044010906577398881}{5000000000000000000000}\right) \cdot u2, u2, 1\right) \]

   3. associate-*r* N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(\frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot u2\right) \cdot u2 + \frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}, u2 \cdot u2, \frac{-98696044010906577398881}{5000000000000000000000}\right) \cdot u2, u2, 1\right) \]

   4. lower-fma.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot u2, u2, \frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}\right), u2 \cdot u2, \frac{-98696044010906577398881}{5000000000000000000000}\right) \cdot u2, u2, 1\right) \]

   5. lower-*.f32 93.3

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-85.45681720672748 \cdot u2, u2, 64.93939402268539\right), u2 \cdot u2, -19.739208802181317\right) \cdot u2, u2, 1\right) \]

8. Applied rewrites 93.3%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-85.45681720672748 \cdot u2, u2, 64.93939402268539\right), u2 \cdot u2, -19.739208802181317\right) \cdot u2, u2, 1\right) \]
9. Add Preprocessing

Alternative 8: 93.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-85.45681720672748, u2 \cdot u2, 64.93939402268539\right), u2 \cdot u2, -19.739208802181317\right) \cdot u2, u2, 1\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sqrt (/ u1 (- 1.0 u1)))
  (fma
   (*
    (fma
     (fma -85.45681720672748 (* u2 u2) 64.93939402268539)
     (* u2 u2)
     -19.739208802181317)
    u2)
   u2
   1.0)))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - u1))) * fmaf((fmaf(fmaf(-85.45681720672748f, (u2 * u2), 64.93939402268539f), (u2 * u2), -19.739208802181317f) * u2), u2, 1.0f);
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * fma(Float32(fma(fma(Float32(-85.45681720672748), Float32(u2 * u2), Float32(64.93939402268539)), Float32(u2 * u2), Float32(-19.739208802181317)) * u2), u2, Float32(1.0)))
end

Derivation

1. Initial program 99.0%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]

2. Taylor expanded in u2 around 0

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\left(1 + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)\right)} \]
3. Step-by-step derivation

   1. lower-+.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + \color{blue}{{u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)}\right) \]

   2. lower-*.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + {u2}^{2} \cdot \color{blue}{\left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)}\right) \]

   3. lower-pow.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + {u2}^{2} \cdot \left(\color{blue}{{u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right)} - \frac{98696044010906577398881}{5000000000000000000000}\right)\right) \]

   4. lower--.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \color{blue}{\frac{98696044010906577398881}{5000000000000000000000}}\right)\right) \]

   5. lower-*.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)\right) \]

   6. lower-pow.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)\right) \]

   7. lower-+.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)\right) \]

   8. lower-*.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)\right) \]

   9. lower-pow.f32 93.3

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(64.93939402268539 + -85.45681720672748 \cdot {u2}^{2}\right) - 19.739208802181317\right)\right) \]

4. Applied rewrites 93.3%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\left(1 + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(64.93939402268539 + -85.45681720672748 \cdot {u2}^{2}\right) - 19.739208802181317\right)\right)} \]
5. Step-by-step derivation

   1. lift-+.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + \color{blue}{{u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)}\right) \]

   2. +-commutative N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left({u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right) + \color{blue}{1}\right) \]

   3. lift-*.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left({u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right) + 1\right) \]

   4. *-commutative N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(\left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right) \cdot {u2}^{2} + 1\right) \]

   5. lift-pow.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(\left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right) \cdot {u2}^{2} + 1\right) \]

   6. unpow2 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(\left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right) \cdot \left(u2 \cdot u2\right) + 1\right) \]

   7. associate-*r* N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(\left(\left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right) \cdot u2\right) \cdot u2 + 1\right) \]

   8. lower-fma.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right) \cdot u2, \color{blue}{u2}, 1\right) \]

6. Applied rewrites 93.3%

   \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-85.45681720672748, u2 \cdot u2, 64.93939402268539\right), u2 \cdot u2, -19.739208802181317\right) \cdot u2, u2, 1\right)} \]
7. Add Preprocessing

Alternative 9: 93.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-85.45681720672748 \cdot u2, u2, 64.93939402268539\right) \cdot u2, u2, -19.739208802181317\right) \cdot u2, u2, 1\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sqrt (/ u1 (- 1.0 u1)))
  (fma
   (*
    (fma
     (* (fma (* -85.45681720672748 u2) u2 64.93939402268539) u2)
     u2
     -19.739208802181317)
    u2)
   u2
   1.0)))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - u1))) * fmaf((fmaf((fmaf((-85.45681720672748f * u2), u2, 64.93939402268539f) * u2), u2, -19.739208802181317f) * u2), u2, 1.0f);
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * fma(Float32(fma(Float32(fma(Float32(Float32(-85.45681720672748) * u2), u2, Float32(64.93939402268539)) * u2), u2, Float32(-19.739208802181317)) * u2), u2, Float32(1.0)))
end

Derivation

1. Initial program 99.0%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]

2. Taylor expanded in u2 around 0

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\left(1 + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)\right)} \]
3. Step-by-step derivation

   1. lower-+.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + \color{blue}{{u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)}\right) \]

   2. lower-*.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + {u2}^{2} \cdot \color{blue}{\left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)}\right) \]

   3. lower-pow.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + {u2}^{2} \cdot \left(\color{blue}{{u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right)} - \frac{98696044010906577398881}{5000000000000000000000}\right)\right) \]

   4. lower--.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \color{blue}{\frac{98696044010906577398881}{5000000000000000000000}}\right)\right) \]

   5. lower-*.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)\right) \]

   6. lower-pow.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)\right) \]

   7. lower-+.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)\right) \]

   8. lower-*.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)\right) \]

   9. lower-pow.f32 93.3

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(64.93939402268539 + -85.45681720672748 \cdot {u2}^{2}\right) - 19.739208802181317\right)\right) \]

4. Applied rewrites 93.3%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\left(1 + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(64.93939402268539 + -85.45681720672748 \cdot {u2}^{2}\right) - 19.739208802181317\right)\right)} \]
5. Step-by-step derivation

   1. lift-+.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + \color{blue}{{u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)}\right) \]

   2. +-commutative N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left({u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right) + \color{blue}{1}\right) \]

   3. lift-*.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left({u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right) + 1\right) \]

   4. *-commutative N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(\left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right) \cdot {u2}^{2} + 1\right) \]

   5. lift-pow.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(\left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right) \cdot {u2}^{2} + 1\right) \]

   6. unpow2 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(\left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right) \cdot \left(u2 \cdot u2\right) + 1\right) \]

   7. associate-*r* N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(\left(\left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right) \cdot u2\right) \cdot u2 + 1\right) \]

   8. lower-fma.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right) \cdot u2, \color{blue}{u2}, 1\right) \]

6. Applied rewrites 93.3%

   \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-85.45681720672748, u2 \cdot u2, 64.93939402268539\right), u2 \cdot u2, -19.739208802181317\right) \cdot u2, u2, 1\right)} \]
7. Step-by-step derivation

   1. lift-fma.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000}, u2 \cdot u2, \frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}\right) \cdot \left(u2 \cdot u2\right) + \frac{-98696044010906577398881}{5000000000000000000000}\right) \cdot u2, u2, 1\right) \]

   2. lift-*.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000}, u2 \cdot u2, \frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}\right) \cdot \left(u2 \cdot u2\right) + \frac{-98696044010906577398881}{5000000000000000000000}\right) \cdot u2, u2, 1\right) \]

   3. associate-*r* N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\left(\left(\mathsf{fma}\left(\frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000}, u2 \cdot u2, \frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}\right) \cdot u2\right) \cdot u2 + \frac{-98696044010906577398881}{5000000000000000000000}\right) \cdot u2, u2, 1\right) \]

   4. lower-fma.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000}, u2 \cdot u2, \frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}\right) \cdot u2, u2, \frac{-98696044010906577398881}{5000000000000000000000}\right) \cdot u2, u2, 1\right) \]

   5. lower-*.f32 93.3

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-85.45681720672748, u2 \cdot u2, 64.93939402268539\right) \cdot u2, u2, -19.739208802181317\right) \cdot u2, u2, 1\right) \]

   6. lift-fma.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(\frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot \left(u2 \cdot u2\right) + \frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}\right) \cdot u2, u2, \frac{-98696044010906577398881}{5000000000000000000000}\right) \cdot u2, u2, 1\right) \]

   7. *-commutative N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(\left(u2 \cdot u2\right) \cdot \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} + \frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}\right) \cdot u2, u2, \frac{-98696044010906577398881}{5000000000000000000000}\right) \cdot u2, u2, 1\right) \]

   8. lower-fma.f32 93.3

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(u2 \cdot u2, -85.45681720672748, 64.93939402268539\right) \cdot u2, u2, -19.739208802181317\right) \cdot u2, u2, 1\right) \]

8. Applied rewrites 93.3%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(u2 \cdot u2, -85.45681720672748, 64.93939402268539\right) \cdot u2, u2, -19.739208802181317\right) \cdot u2, u2, 1\right) \]
9. Step-by-step derivation

   1. lift-fma.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(\left(u2 \cdot u2\right) \cdot \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} + \frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}\right) \cdot u2, u2, \frac{-98696044010906577398881}{5000000000000000000000}\right) \cdot u2, u2, 1\right) \]

   2. lift-*.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(\left(u2 \cdot u2\right) \cdot \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} + \frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}\right) \cdot u2, u2, \frac{-98696044010906577398881}{5000000000000000000000}\right) \cdot u2, u2, 1\right) \]

   3. associate-*l* N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(u2 \cdot \left(u2 \cdot \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000}\right) + \frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}\right) \cdot u2, u2, \frac{-98696044010906577398881}{5000000000000000000000}\right) \cdot u2, u2, 1\right) \]

   4. *-commutative N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(\left(u2 \cdot \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000}\right) \cdot u2 + \frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}\right) \cdot u2, u2, \frac{-98696044010906577398881}{5000000000000000000000}\right) \cdot u2, u2, 1\right) \]

   5. lower-fma.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(u2 \cdot \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000}, u2, \frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}\right) \cdot u2, u2, \frac{-98696044010906577398881}{5000000000000000000000}\right) \cdot u2, u2, 1\right) \]

   6. *-commutative N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot u2, u2, \frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}\right) \cdot u2, u2, \frac{-98696044010906577398881}{5000000000000000000000}\right) \cdot u2, u2, 1\right) \]

   7. lower-*.f32 93.3

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-85.45681720672748 \cdot u2, u2, 64.93939402268539\right) \cdot u2, u2, -19.739208802181317\right) \cdot u2, u2, 1\right) \]

10. Applied rewrites 93.3%

    \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-85.45681720672748 \cdot u2, u2, 64.93939402268539\right) \cdot u2, u2, -19.739208802181317\right) \cdot u2, u2, 1\right) \]
11. Add Preprocessing

Alternative 10: 91.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(64.93939402268539, u2 \cdot u2, -19.739208802181317\right) \cdot u2, u2, 1\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sqrt (/ u1 (- 1.0 u1)))
  (fma (* (fma 64.93939402268539 (* u2 u2) -19.739208802181317) u2) u2 1.0)))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - u1))) * fmaf((fmaf(64.93939402268539f, (u2 * u2), -19.739208802181317f) * u2), u2, 1.0f);
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * fma(Float32(fma(Float32(64.93939402268539), Float32(u2 * u2), Float32(-19.739208802181317)) * u2), u2, Float32(1.0)))
end

Derivation

1. Initial program 99.0%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]

2. Taylor expanded in u2 around 0

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\left(1 + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)\right)} \]
3. Step-by-step derivation

   1. lower-+.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + \color{blue}{{u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)}\right) \]

   2. lower-*.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + {u2}^{2} \cdot \color{blue}{\left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)}\right) \]

   3. lower-pow.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + {u2}^{2} \cdot \left(\color{blue}{{u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right)} - \frac{98696044010906577398881}{5000000000000000000000}\right)\right) \]

   4. lower--.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \color{blue}{\frac{98696044010906577398881}{5000000000000000000000}}\right)\right) \]

   5. lower-*.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)\right) \]

   6. lower-pow.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)\right) \]

   7. lower-+.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)\right) \]

   8. lower-*.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)\right) \]

   9. lower-pow.f32 93.3

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(64.93939402268539 + -85.45681720672748 \cdot {u2}^{2}\right) - 19.739208802181317\right)\right) \]

4. Applied rewrites 93.3%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\left(1 + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(64.93939402268539 + -85.45681720672748 \cdot {u2}^{2}\right) - 19.739208802181317\right)\right)} \]
5. Step-by-step derivation

   1. lift-+.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + \color{blue}{{u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)}\right) \]

   2. +-commutative N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left({u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right) + \color{blue}{1}\right) \]

   3. lift-*.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left({u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right) + 1\right) \]

   4. *-commutative N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(\left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right) \cdot {u2}^{2} + 1\right) \]

   5. lift-pow.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(\left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right) \cdot {u2}^{2} + 1\right) \]

   6. unpow2 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(\left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right) \cdot \left(u2 \cdot u2\right) + 1\right) \]

   7. associate-*r* N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(\left(\left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right) \cdot u2\right) \cdot u2 + 1\right) \]

   8. lower-fma.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right) \cdot u2, \color{blue}{u2}, 1\right) \]

6. Applied rewrites 93.3%

   \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-85.45681720672748, u2 \cdot u2, 64.93939402268539\right), u2 \cdot u2, -19.739208802181317\right) \cdot u2, u2, 1\right)} \]

7. Taylor expanded in u2 around 0

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}, u2 \cdot u2, \frac{-98696044010906577398881}{5000000000000000000000}\right) \cdot u2, u2, 1\right) \]
8. Step-by-step derivation

9. Applied rewrites 91.3%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(64.93939402268539, u2 \cdot u2, -19.739208802181317\right) \cdot u2, u2, 1\right) \]
10. Add Preprocessing

Alternative 11: 88.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(-19.739208802181317 \cdot u2, u2, 1\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (/ u1 (- 1.0 u1))) (fma (* -19.739208802181317 u2) u2 1.0)))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - u1))) * fmaf((-19.739208802181317f * u2), u2, 1.0f);
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * fma(Float32(Float32(-19.739208802181317) * u2), u2, Float32(1.0)))
end

Derivation

1. Initial program 99.0%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]

2. Taylor expanded in u2 around 0

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\left(1 + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)\right)} \]
3. Step-by-step derivation

   1. lower-+.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + \color{blue}{{u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)}\right) \]

   2. lower-*.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + {u2}^{2} \cdot \color{blue}{\left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)}\right) \]

   3. lower-pow.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + {u2}^{2} \cdot \left(\color{blue}{{u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right)} - \frac{98696044010906577398881}{5000000000000000000000}\right)\right) \]

   4. lower--.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \color{blue}{\frac{98696044010906577398881}{5000000000000000000000}}\right)\right) \]

   5. lower-*.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)\right) \]

   6. lower-pow.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)\right) \]

   7. lower-+.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)\right) \]

   8. lower-*.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)\right) \]

   9. lower-pow.f32 93.3

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(64.93939402268539 + -85.45681720672748 \cdot {u2}^{2}\right) - 19.739208802181317\right)\right) \]

4. Applied rewrites 93.3%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\left(1 + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(64.93939402268539 + -85.45681720672748 \cdot {u2}^{2}\right) - 19.739208802181317\right)\right)} \]
5. Step-by-step derivation

   1. lift-+.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + \color{blue}{{u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)}\right) \]

   2. +-commutative N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left({u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right) + \color{blue}{1}\right) \]

   3. lift-*.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left({u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right) + 1\right) \]

   4. *-commutative N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(\left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right) \cdot {u2}^{2} + 1\right) \]

   5. lift-pow.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(\left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right) \cdot {u2}^{2} + 1\right) \]

   6. unpow2 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(\left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right) \cdot \left(u2 \cdot u2\right) + 1\right) \]

   7. associate-*r* N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(\left(\left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right) \cdot u2\right) \cdot u2 + 1\right) \]

   8. lower-fma.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right) \cdot u2, \color{blue}{u2}, 1\right) \]

6. Applied rewrites 93.3%

   \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-85.45681720672748, u2 \cdot u2, 64.93939402268539\right), u2 \cdot u2, -19.739208802181317\right) \cdot u2, u2, 1\right)} \]

7. Taylor expanded in u2 around 0

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\frac{-98696044010906577398881}{5000000000000000000000} \cdot u2, u2, 1\right) \]
8. Step-by-step derivation

9. Applied rewrites 88.2%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(-19.739208802181317 \cdot u2, u2, 1\right) \]
10. Add Preprocessing

Alternative 12: 88.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-19.739208802181317, u2 \cdot u2, 1\right) \cdot \sqrt{\frac{u1}{1 - u1}} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (fma -19.739208802181317 (* u2 u2) 1.0) (sqrt (/ u1 (- 1.0 u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	return fmaf(-19.739208802181317f, (u2 * u2), 1.0f) * sqrtf((u1 / (1.0f - u1)));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(fma(Float32(-19.739208802181317), Float32(u2 * u2), Float32(1.0)) * sqrt(Float32(u1 / Float32(Float32(1.0) - u1))))
end

Derivation

1. Initial program 99.0%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]

2. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}} + \frac{-98696044010906577398881}{5000000000000000000000} \cdot \left({u2}^{2} \cdot \sqrt{\frac{u1}{1 - u1}}\right)} \]
3. Step-by-step derivation

   1. lower-+.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} + \color{blue}{\frac{-98696044010906577398881}{5000000000000000000000} \cdot \left({u2}^{2} \cdot \sqrt{\frac{u1}{1 - u1}}\right)} \]

   2. lower-sqrt.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} + \color{blue}{\frac{-98696044010906577398881}{5000000000000000000000}} \cdot \left({u2}^{2} \cdot \sqrt{\frac{u1}{1 - u1}}\right) \]

   3. lower-/.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} + \frac{-98696044010906577398881}{5000000000000000000000} \cdot \left({u2}^{2} \cdot \sqrt{\frac{u1}{1 - u1}}\right) \]

   4. lower--.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} + \frac{-98696044010906577398881}{5000000000000000000000} \cdot \left({u2}^{2} \cdot \sqrt{\frac{u1}{1 - u1}}\right) \]

   5. lower-*.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} + \frac{-98696044010906577398881}{5000000000000000000000} \cdot \color{blue}{\left({u2}^{2} \cdot \sqrt{\frac{u1}{1 - u1}}\right)} \]

   6. lower-*.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} + \frac{-98696044010906577398881}{5000000000000000000000} \cdot \left({u2}^{2} \cdot \color{blue}{\sqrt{\frac{u1}{1 - u1}}}\right) \]

   7. lower-pow.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} + \frac{-98696044010906577398881}{5000000000000000000000} \cdot \left({u2}^{2} \cdot \sqrt{\color{blue}{\frac{u1}{1 - u1}}}\right) \]

   8. lower-sqrt.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} + \frac{-98696044010906577398881}{5000000000000000000000} \cdot \left({u2}^{2} \cdot \sqrt{\frac{u1}{1 - u1}}\right) \]

   9. lower-/.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} + \frac{-98696044010906577398881}{5000000000000000000000} \cdot \left({u2}^{2} \cdot \sqrt{\frac{u1}{1 - u1}}\right) \]

   10. lower--.f32 88.2

       \[\leadsto \sqrt{\frac{u1}{1 - u1}} + -19.739208802181317 \cdot \left({u2}^{2} \cdot \sqrt{\frac{u1}{1 - u1}}\right) \]

4. Applied rewrites 88.2%

   \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}} + -19.739208802181317 \cdot \left({u2}^{2} \cdot \sqrt{\frac{u1}{1 - u1}}\right)} \]
5. Step-by-step derivation

   1. lift-+.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} + \color{blue}{\frac{-98696044010906577398881}{5000000000000000000000} \cdot \left({u2}^{2} \cdot \sqrt{\frac{u1}{1 - u1}}\right)} \]

   2. lift-*.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} + \frac{-98696044010906577398881}{5000000000000000000000} \cdot \color{blue}{\left({u2}^{2} \cdot \sqrt{\frac{u1}{1 - u1}}\right)} \]

   3. lift-*.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} + \frac{-98696044010906577398881}{5000000000000000000000} \cdot \left({u2}^{2} \cdot \color{blue}{\sqrt{\frac{u1}{1 - u1}}}\right) \]

   4. lift-pow.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} + \frac{-98696044010906577398881}{5000000000000000000000} \cdot \left({u2}^{2} \cdot \sqrt{\color{blue}{\frac{u1}{1 - u1}}}\right) \]

   5. pow2 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} + \frac{-98696044010906577398881}{5000000000000000000000} \cdot \left(\left(u2 \cdot u2\right) \cdot \sqrt{\color{blue}{\frac{u1}{1 - u1}}}\right) \]

   6. lift-*.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} + \frac{-98696044010906577398881}{5000000000000000000000} \cdot \left(\left(u2 \cdot u2\right) \cdot \sqrt{\color{blue}{\frac{u1}{1 - u1}}}\right) \]

   7. associate-*r* N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} + \left(\frac{-98696044010906577398881}{5000000000000000000000} \cdot \left(u2 \cdot u2\right)\right) \cdot \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]

   8. distribute-rgt1-in N/A

      \[\leadsto \left(\frac{-98696044010906577398881}{5000000000000000000000} \cdot \left(u2 \cdot u2\right) + 1\right) \cdot \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]

   9. lower-*.f32 N/A

      \[\leadsto \left(\frac{-98696044010906577398881}{5000000000000000000000} \cdot \left(u2 \cdot u2\right) + 1\right) \cdot \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]

   10. lower-fma.f32 88.2

       \[\leadsto \mathsf{fma}\left(-19.739208802181317, u2 \cdot u2, 1\right) \cdot \sqrt{\color{blue}{\frac{u1}{1 - u1}}} \]

6. Applied rewrites 88.2%

   \[\leadsto \mathsf{fma}\left(-19.739208802181317, u2 \cdot u2, 1\right) \cdot \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]
7. Add Preprocessing

Alternative 13: 79.6% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{u1}{1 - u1}} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2) :precision binary32 (sqrt (/ u1 (- 1.0 u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - u1)));
}

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(4) function code(costheta_i, u1, u2)
use fmin_fmax_functions
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = sqrt((u1 / (1.0e0 - u1)))
end function

Julia

function code(cosTheta_i, u1, u2)
	return sqrt(Float32(u1 / Float32(Float32(1.0) - u1)))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((u1 / (single(1.0) - u1)));
end

Derivation

1. Initial program 99.0%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]

2. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]
3. Step-by-step derivation

   1. lower-sqrt.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \]

   2. lower-/.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \]

   3. lower--.f32 79.6

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \]

4. Applied rewrites 79.6%

   \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]
5. Add Preprocessing

Alternative 14: 71.2% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \sqrt{u1 \cdot u1 + u1} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2) :precision binary32 (sqrt (+ (* u1 u1) u1)))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(((u1 * u1) + u1));
}

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(4) function code(costheta_i, u1, u2)
use fmin_fmax_functions
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = sqrt(((u1 * u1) + u1))
end function

Julia

function code(cosTheta_i, u1, u2)
	return sqrt(Float32(Float32(u1 * u1) + u1))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt(((u1 * u1) + u1));
end

Derivation

1. Initial program 99.0%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]

2. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]
3. Step-by-step derivation

   1. lower-sqrt.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \]

   2. lower-/.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \]

   3. lower--.f32 79.6

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \]

4. Applied rewrites 79.6%

   \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]

5. Taylor expanded in u1 around 0

   \[\leadsto \sqrt{u1 \cdot \left(1 + u1\right)} \]
6. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto \sqrt{u1 \cdot \left(1 + u1\right)} \]

   2. lower-+.f32 71.1

      \[\leadsto \sqrt{u1 \cdot \left(1 + u1\right)} \]

7. Applied rewrites 71.1%

   \[\leadsto \sqrt{u1 \cdot \left(1 + u1\right)} \]
8. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \sqrt{u1 \cdot \left(1 + u1\right)} \]

   2. lift-+.f32 N/A

      \[\leadsto \sqrt{u1 \cdot \left(1 + u1\right)} \]

   3. +-commutative N/A

      \[\leadsto \sqrt{u1 \cdot \left(u1 + 1\right)} \]

   4. distribute-lft-in N/A

      \[\leadsto \sqrt{u1 \cdot u1 + u1 \cdot 1} \]

   5. unpow2 N/A

      \[\leadsto \sqrt{{u1}^{2} + u1 \cdot 1} \]

   6. *-rgt-identity N/A

      \[\leadsto \sqrt{{u1}^{2} + u1} \]

   7. lower-+.f32 N/A

      \[\leadsto \sqrt{{u1}^{2} + u1} \]

   8. unpow2 N/A

      \[\leadsto \sqrt{u1 \cdot u1 + u1} \]

   9. lower-*.f32 71.2

      \[\leadsto \sqrt{u1 \cdot u1 + u1} \]

9. Applied rewrites 71.2%

   \[\leadsto \sqrt{u1 \cdot u1 + u1} \]
10. Add Preprocessing

Alternative 15: 71.2% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{\mathsf{fma}\left(u1, u1, u1\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2) :precision binary32 (sqrt (fma u1 u1 u1)))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(fmaf(u1, u1, u1));
}

Julia

function code(cosTheta_i, u1, u2)
	return sqrt(fma(u1, u1, u1))
end

Derivation

1. Initial program 99.0%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]

2. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]
3. Step-by-step derivation

   1. lower-sqrt.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \]

   2. lower-/.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \]

   3. lower--.f32 79.6

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \]

4. Applied rewrites 79.6%

   \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]

5. Taylor expanded in u1 around 0

   \[\leadsto \sqrt{u1 \cdot \left(1 + u1\right)} \]
6. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto \sqrt{u1 \cdot \left(1 + u1\right)} \]

   2. lower-+.f32 71.1

      \[\leadsto \sqrt{u1 \cdot \left(1 + u1\right)} \]

7. Applied rewrites 71.1%

   \[\leadsto \sqrt{u1 \cdot \left(1 + u1\right)} \]
8. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \sqrt{u1 \cdot \left(1 + u1\right)} \]

   2. lift-+.f32 N/A

      \[\leadsto \sqrt{u1 \cdot \left(1 + u1\right)} \]

   3. +-commutative N/A

      \[\leadsto \sqrt{u1 \cdot \left(u1 + 1\right)} \]

   4. distribute-lft-in N/A

      \[\leadsto \sqrt{u1 \cdot u1 + u1 \cdot 1} \]

   5. *-rgt-identity N/A

      \[\leadsto \sqrt{u1 \cdot u1 + u1} \]

   6. lower-fma.f32 71.2

      \[\leadsto \sqrt{\mathsf{fma}\left(u1, u1, u1\right)} \]

9. Applied rewrites 71.2%

   \[\leadsto \sqrt{\mathsf{fma}\left(u1, u1, u1\right)} \]
10. Add Preprocessing

Alternative 16: 62.9% accurate, 16.2× speedup

Math

\[\begin{array}{l} \\ \sqrt{u1} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2) :precision binary32 (sqrt u1))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(u1);
}

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(4) function code(costheta_i, u1, u2)
use fmin_fmax_functions
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = sqrt(u1)
end function

Julia

function code(cosTheta_i, u1, u2)
	return sqrt(u1)
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt(u1);
end

Derivation

1. Initial program 99.0%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]

2. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]
3. Step-by-step derivation

   1. lower-sqrt.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \]

   2. lower-/.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \]

   3. lower--.f32 79.6

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \]

4. Applied rewrites 79.6%

   \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]

5. Taylor expanded in u1 around 0

   \[\leadsto \sqrt{u1} \]
6. Step-by-step derivation

7. Applied rewrites 62.9%

   \[\leadsto \sqrt{u1} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025142 
(FPCore (cosTheta_i u1 u2)
  :name "Trowbridge-Reitz Sample, near normal, slope_x"
  :precision binary32
  :pre (and (and (and (> cosTheta_i 0.9999) (<= cosTheta_i 1.0)) (and (<= 2.328306437e-10 u1) (<= u1 1.0))) (and (<= 2.328306437e-10 u2) (<= u2 1.0)))
  (* (sqrt (/ u1 (- 1.0 u1))) (cos (* 6.28318530718 u2))))