Trowbridge-Reitz Sample, near normal, slope_x

Percentage Accurate: 99.0% → 99.0%

Time: 9.1s

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

\[\begin{array}{l} \\ \sin \left(\mathsf{fma}\left(0.5, \mathsf{PI}\left(\right), -6.28318530718 \cdot u2\right)\right) \cdot \sqrt{\frac{u1}{1 - u1}} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sin (fma 0.5 (PI) (* -6.28318530718 u2))) (sqrt (/ u1 (- 1.0 u1)))))

Derivation

1. Initial program 98.9%

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

   1. lift-*.f32 N/A

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

   2. lift-sqrt.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. sqrt-div N/A

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

   5. associate-*l/ N/A

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

   6. lower-/.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. lower-sqrt.f32 N/A

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

   9. lift-cos.f32 N/A

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

   10. cos-neg-rev N/A

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

   11. lower-cos.f32 N/A

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

   12. lift-*.f32 N/A

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

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

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

   14. lower-*.f32 N/A

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

   15. metadata-eval N/A

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

   16. lower-sqrt.f32 98.5

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

4. Applied rewrites 98.5%

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

   1. lift-cos.f32 N/A

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

   2. sin-+PI/2-rev N/A

      \[\leadsto \frac{\sqrt{u1} \cdot \color{blue}{\sin \left(\frac{-314159265359}{50000000000} \cdot u2 + \frac{\mathsf{PI}\left(\right)}{2}\right)}}{\sqrt{1 - u1}} \]

   3. lower-sin.f32 N/A

      \[\leadsto \frac{\sqrt{u1} \cdot \color{blue}{\sin \left(\frac{-314159265359}{50000000000} \cdot u2 + \frac{\mathsf{PI}\left(\right)}{2}\right)}}{\sqrt{1 - u1}} \]

   4. lift-*.f32 N/A

      \[\leadsto \frac{\sqrt{u1} \cdot \sin \left(\color{blue}{\frac{-314159265359}{50000000000} \cdot u2} + \frac{\mathsf{PI}\left(\right)}{2}\right)}{\sqrt{1 - u1}} \]

   5. metadata-eval N/A

      \[\leadsto \frac{\sqrt{u1} \cdot \sin \left(\color{blue}{\left(\mathsf{neg}\left(\frac{314159265359}{50000000000}\right)\right)} \cdot u2 + \frac{\mathsf{PI}\left(\right)}{2}\right)}{\sqrt{1 - u1}} \]

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

      \[\leadsto \frac{\sqrt{u1} \cdot \sin \left(\color{blue}{\left(\mathsf{neg}\left(\frac{314159265359}{50000000000} \cdot u2\right)\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right)}{\sqrt{1 - u1}} \]

   7. lift-*.f32 N/A

      \[\leadsto \frac{\sqrt{u1} \cdot \sin \left(\left(\mathsf{neg}\left(\color{blue}{\frac{314159265359}{50000000000} \cdot u2}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}{\sqrt{1 - u1}} \]

   8. lift-*.f32 N/A

      \[\leadsto \frac{\sqrt{u1} \cdot \sin \left(\left(\mathsf{neg}\left(\color{blue}{\frac{314159265359}{50000000000} \cdot u2}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}{\sqrt{1 - u1}} \]

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

      \[\leadsto \frac{\sqrt{u1} \cdot \sin \left(\color{blue}{\left(\mathsf{neg}\left(\frac{314159265359}{50000000000}\right)\right) \cdot u2} + \frac{\mathsf{PI}\left(\right)}{2}\right)}{\sqrt{1 - u1}} \]

   10. metadata-eval N/A

       \[\leadsto \frac{\sqrt{u1} \cdot \sin \left(\color{blue}{\frac{-314159265359}{50000000000}} \cdot u2 + \frac{\mathsf{PI}\left(\right)}{2}\right)}{\sqrt{1 - u1}} \]

   11. lower-fma.f32 N/A

       \[\leadsto \frac{\sqrt{u1} \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\frac{-314159265359}{50000000000}, u2, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}}{\sqrt{1 - u1}} \]

   12. lower-/.f32 N/A

       \[\leadsto \frac{\sqrt{u1} \cdot \sin \left(\mathsf{fma}\left(\frac{-314159265359}{50000000000}, u2, \color{blue}{\frac{\mathsf{PI}\left(\right)}{2}}\right)\right)}{\sqrt{1 - u1}} \]

   13. lower-PI.f32 98.7

       \[\leadsto \frac{\sqrt{u1} \cdot \sin \left(\mathsf{fma}\left(-6.28318530718, u2, \frac{\color{blue}{\mathsf{PI}\left(\right)}}{2}\right)\right)}{\sqrt{1 - u1}} \]

6. Applied rewrites 98.7%

   \[\leadsto \frac{\sqrt{u1} \cdot \color{blue}{\sin \left(\mathsf{fma}\left(-6.28318530718, u2, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}}{\sqrt{1 - u1}} \]

7. Taylor expanded in u2 around inf

   \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \frac{314159265359}{50000000000} \cdot u2\right)} \]
8. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \frac{314159265359}{50000000000} \cdot u2\right) \cdot \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]

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

      \[\leadsto \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) + \left(\mathsf{neg}\left(\frac{314159265359}{50000000000}\right)\right) \cdot u2\right) \cdot \sqrt{\frac{\color{blue}{u1}}{1 - u1}} \]

   3. metadata-eval N/A

      \[\leadsto \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) + \frac{-314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}} \]

   4. +-commutative N/A

      \[\leadsto \sin \left(\frac{-314159265359}{50000000000} \cdot u2 + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\frac{\color{blue}{u1}}{1 - u1}} \]

   5. lower-*.f32 N/A

      \[\leadsto \sin \left(\frac{-314159265359}{50000000000} \cdot u2 + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]

   6. lower-sin.f32 N/A

      \[\leadsto \sin \left(\frac{-314159265359}{50000000000} \cdot u2 + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{\frac{u1}{1 - u1}}} \]

   7. +-commutative N/A

      \[\leadsto \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) + \frac{-314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{\color{blue}{u1}}{1 - u1}} \]

   8. lower-fma.f32 N/A

      \[\leadsto \sin \left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{PI}\left(\right), \frac{-314159265359}{50000000000} \cdot u2\right)\right) \cdot \sqrt{\frac{\color{blue}{u1}}{1 - u1}} \]

   9. lower-PI.f32 N/A

      \[\leadsto \sin \left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{PI}\left(\right), \frac{-314159265359}{50000000000} \cdot u2\right)\right) \cdot \sqrt{\frac{u1}{1 - u1}} \]

   10. lower-*.f32 N/A

       \[\leadsto \sin \left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{PI}\left(\right), \frac{-314159265359}{50000000000} \cdot u2\right)\right) \cdot \sqrt{\frac{u1}{1 - u1}} \]

   11. lower-sqrt.f32 N/A

       \[\leadsto \sin \left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{PI}\left(\right), \frac{-314159265359}{50000000000} \cdot u2\right)\right) \cdot \sqrt{\frac{u1}{1 - u1}} \]

   12. lower-/.f32 N/A

       \[\leadsto \sin \left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{PI}\left(\right), \frac{-314159265359}{50000000000} \cdot u2\right)\right) \cdot \sqrt{\frac{u1}{1 - u1}} \]

   13. lower--.f32 99.0

       \[\leadsto \sin \left(\mathsf{fma}\left(0.5, \mathsf{PI}\left(\right), -6.28318530718 \cdot u2\right)\right) \cdot \sqrt{\frac{u1}{1 - u1}} \]

9. Applied rewrites 99.0%

   \[\leadsto \color{blue}{\sin \left(\mathsf{fma}\left(0.5, \mathsf{PI}\left(\right), -6.28318530718 \cdot u2\right)\right) \cdot \sqrt{\frac{u1}{1 - u1}}} \]
10. Add Preprocessing

Alternative 2: 83.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{u1}{1 - u1}}\\ \mathbf{if}\;t\_0 \cdot \cos \left(6.28318530718 \cdot u2\right) \leq 0.0006000000284984708:\\ \;\;\;\;\mathsf{fma}\left(u2 \cdot u2, -19.739208802181317, 1\right) \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (/ u1 (- 1.0 u1)))))
   (if (<= (* t_0 (cos (* 6.28318530718 u2))) 0.0006000000284984708)
     (* (fma (* u2 u2) -19.739208802181317 1.0) (sqrt u1))
     t_0)))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sqrtf((u1 / (1.0f - u1)));
	float tmp;
	if ((t_0 * cosf((6.28318530718f * u2))) <= 0.0006000000284984708f) {
		tmp = fmaf((u2 * u2), -19.739208802181317f, 1.0f) * sqrtf(u1);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sqrt(Float32(u1 / Float32(Float32(1.0) - u1)))
	tmp = Float32(0.0)
	if (Float32(t_0 * cos(Float32(Float32(6.28318530718) * u2))) <= Float32(0.0006000000284984708))
		tmp = Float32(fma(Float32(u2 * u2), Float32(-19.739208802181317), Float32(1.0)) * sqrt(u1));
	else
		tmp = t_0;
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (sqrt.f32 (/.f32 u1 (-.f32 #s(literal 1 binary32) u1))) (cos.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2))) < 6.00000028e-4

   1. Initial program 98.5%

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

      1. lift-*.f32 N/A

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

      2. lift-sqrt.f32 N/A

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

      3. lift-/.f32 N/A

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

      4. sqrt-div N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f32 N/A

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

      7. lower-*.f32 N/A

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

      8. lower-sqrt.f32 N/A

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

      9. lift-cos.f32 N/A

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

      10. cos-neg-rev N/A

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

      11. lower-cos.f32 N/A

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

      12. lift-*.f32 N/A

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

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

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

      14. lower-*.f32 N/A

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

      15. metadata-eval N/A

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

      16. lower-sqrt.f32 98.2

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

   4. Applied rewrites 98.2%

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

   5. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. distribute-rgt1-in N/A

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

      4. +-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f32 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f32 N/A

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

      11. lower-sqrt.f32 N/A

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

      12. lower-/.f32 N/A

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

      13. lower--.f32 82.7

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

   7. Applied rewrites 82.7%

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

   8. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f32 N/A

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

      8. lower-sqrt.f32 82.4

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

   10. Applied rewrites 82.4%

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

   if 6.00000028e-4 < (*.f32 (sqrt.f32 (/.f32 u1 (-.f32 #s(literal 1 binary32) u1))) (cos.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2)))

   1. Initial program 99.1%

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

   3. Taylor expanded in u2 around 0

      \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]
   4. 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 84.3

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

   5. Applied rewrites 84.3%

      \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \leq 0.0006000000284984708:\\ \;\;\;\;\mathsf{fma}\left(u2 \cdot u2, -19.739208802181317, 1\right) \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{u1}{1 - u1}}\\ \mathbf{if}\;u2 \leq 0.14000000059604645:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(u2 \cdot u2, -19.739208802181317, 1\right), t\_0, \left(\left(u2 \cdot u2\right) \cdot \left(t\_0 \cdot \mathsf{fma}\left(u2 \cdot u2, -85.45681720672748, 64.93939402268539\right)\right)\right) \cdot \left(u2 \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1 \cdot \left(\mathsf{fma}\left(u1, u1, u1\right) + 1\right)} \cdot \cos \left(6.28318530718 \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (/ u1 (- 1.0 u1)))))
   (if (<= u2 0.14000000059604645)
     (fma
      (fma (* u2 u2) -19.739208802181317 1.0)
      t_0
      (*
       (*
        (* u2 u2)
        (* t_0 (fma (* u2 u2) -85.45681720672748 64.93939402268539)))
       (* u2 u2)))
     (* (sqrt (* u1 (+ (fma u1 u1 u1) 1.0))) (cos (* 6.28318530718 u2))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sqrtf((u1 / (1.0f - u1)));
	float tmp;
	if (u2 <= 0.14000000059604645f) {
		tmp = fmaf(fmaf((u2 * u2), -19.739208802181317f, 1.0f), t_0, (((u2 * u2) * (t_0 * fmaf((u2 * u2), -85.45681720672748f, 64.93939402268539f))) * (u2 * u2)));
	} else {
		tmp = sqrtf((u1 * (fmaf(u1, u1, u1) + 1.0f))) * cosf((6.28318530718f * u2));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sqrt(Float32(u1 / Float32(Float32(1.0) - u1)))
	tmp = Float32(0.0)
	if (u2 <= Float32(0.14000000059604645))
		tmp = fma(fma(Float32(u2 * u2), Float32(-19.739208802181317), Float32(1.0)), t_0, Float32(Float32(Float32(u2 * u2) * Float32(t_0 * fma(Float32(u2 * u2), Float32(-85.45681720672748), Float32(64.93939402268539)))) * Float32(u2 * u2)));
	else
		tmp = Float32(sqrt(Float32(u1 * Float32(fma(u1, u1, u1) + Float32(1.0)))) * cos(Float32(Float32(6.28318530718) * u2)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u2 < 0.140000001

   1. Initial program 99.4%

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

      1. lift-*.f32 N/A

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

      2. lift-sqrt.f32 N/A

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

      3. lift-/.f32 N/A

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

      4. sqrt-div N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f32 N/A

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

      7. lower-*.f32 N/A

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

      8. lower-sqrt.f32 N/A

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

      9. lift-cos.f32 N/A

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

      10. cos-neg-rev N/A

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

      11. lower-cos.f32 N/A

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

      12. lift-*.f32 N/A

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

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

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

      14. lower-*.f32 N/A

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

      15. metadata-eval N/A

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

      16. lower-sqrt.f32 98.9

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

   4. Applied rewrites 98.9%

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

   5. Taylor expanded in u2 around 0

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

   7. Applied rewrites 99.1%

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

   if 0.140000001 < u2

   1. Initial program 94.5%

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

      1. lift-/.f32 N/A

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

      2. lift--.f32 N/A

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

      3. flip3-- N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. metadata-eval N/A

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

      8. lower--.f32 N/A

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

      9. lower-pow.f32 N/A

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

      12. lower-+.f32 N/A

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

      13. *-lft-identity N/A

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

      14. lower-fma.f32 94.5

          \[\leadsto \sqrt{\frac{u1}{1 - {u1}^{3}} \cdot \left(\color{blue}{\mathsf{fma}\left(u1, u1, u1\right)} + 1\right)} \cdot \cos \left(6.28318530718 \cdot u2\right) \]

   4. Applied rewrites 94.5%

      \[\leadsto \sqrt{\color{blue}{\frac{u1}{1 - {u1}^{3}} \cdot \left(\mathsf{fma}\left(u1, u1, u1\right) + 1\right)}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]

   5. Taylor expanded in u1 around 0

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

   7. Applied rewrites 90.2%

      \[\leadsto \sqrt{\color{blue}{u1} \cdot \left(\mathsf{fma}\left(u1, u1, u1\right) + 1\right)} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \leq 0.14000000059604645:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(u2 \cdot u2, -19.739208802181317, 1\right), \sqrt{\frac{u1}{1 - u1}}, \left(\left(u2 \cdot u2\right) \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(u2 \cdot u2, -85.45681720672748, 64.93939402268539\right)\right)\right) \cdot \left(u2 \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1 \cdot \left(\mathsf{fma}\left(u1, u1, u1\right) + 1\right)} \cdot \cos \left(6.28318530718 \cdot u2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{u1}{1 - u1}}\\ \mathbf{if}\;u2 \leq 0.14000000059604645:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(u2 \cdot u2, -19.739208802181317, 1\right), t\_0, \left(\left(u2 \cdot u2\right) \cdot \left(t\_0 \cdot \mathsf{fma}\left(u2 \cdot u2, -85.45681720672748, 64.93939402268539\right)\right)\right) \cdot \left(u2 \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(u1, u1, u1\right), u1, u1\right)} \cdot \cos \left(6.28318530718 \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (/ u1 (- 1.0 u1)))))
   (if (<= u2 0.14000000059604645)
     (fma
      (fma (* u2 u2) -19.739208802181317 1.0)
      t_0
      (*
       (*
        (* u2 u2)
        (* t_0 (fma (* u2 u2) -85.45681720672748 64.93939402268539)))
       (* u2 u2)))
     (* (sqrt (fma (fma u1 u1 u1) u1 u1)) (cos (* 6.28318530718 u2))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sqrtf((u1 / (1.0f - u1)));
	float tmp;
	if (u2 <= 0.14000000059604645f) {
		tmp = fmaf(fmaf((u2 * u2), -19.739208802181317f, 1.0f), t_0, (((u2 * u2) * (t_0 * fmaf((u2 * u2), -85.45681720672748f, 64.93939402268539f))) * (u2 * u2)));
	} else {
		tmp = sqrtf(fmaf(fmaf(u1, u1, u1), u1, u1)) * cosf((6.28318530718f * u2));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sqrt(Float32(u1 / Float32(Float32(1.0) - u1)))
	tmp = Float32(0.0)
	if (u2 <= Float32(0.14000000059604645))
		tmp = fma(fma(Float32(u2 * u2), Float32(-19.739208802181317), Float32(1.0)), t_0, Float32(Float32(Float32(u2 * u2) * Float32(t_0 * fma(Float32(u2 * u2), Float32(-85.45681720672748), Float32(64.93939402268539)))) * Float32(u2 * u2)));
	else
		tmp = Float32(sqrt(fma(fma(u1, u1, u1), u1, u1)) * cos(Float32(Float32(6.28318530718) * u2)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u2 < 0.140000001

   1. Initial program 99.4%

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

      1. lift-*.f32 N/A

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

      2. lift-sqrt.f32 N/A

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

      3. lift-/.f32 N/A

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

      4. sqrt-div N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f32 N/A

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

      7. lower-*.f32 N/A

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

      8. lower-sqrt.f32 N/A

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

      9. lift-cos.f32 N/A

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

      10. cos-neg-rev N/A

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

      11. lower-cos.f32 N/A

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

      12. lift-*.f32 N/A

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

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

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

      14. lower-*.f32 N/A

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

      15. metadata-eval N/A

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

      16. lower-sqrt.f32 98.9

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

   4. Applied rewrites 98.9%

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

   5. Taylor expanded in u2 around 0

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

   7. Applied rewrites 99.1%

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

   if 0.140000001 < u2

   1. Initial program 94.5%

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

   3. Taylor expanded in u1 around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f32 N/A

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

      5. +-commutative N/A

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

      6. distribute-rgt-in N/A

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

      7. *-lft-identity N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(u1 \cdot u1 + u1, u1, u1\right)} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      8. lower-fma.f32 89.9

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

   5. Applied rewrites 89.9%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \leq 0.14000000059604645:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(u2 \cdot u2, -19.739208802181317, 1\right), \sqrt{\frac{u1}{1 - u1}}, \left(\left(u2 \cdot u2\right) \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(u2 \cdot u2, -85.45681720672748, 64.93939402268539\right)\right)\right) \cdot \left(u2 \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(u1, u1, u1\right), u1, u1\right)} \cdot \cos \left(6.28318530718 \cdot u2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{u1}{1 - u1}}\\ \mathbf{if}\;u2 \leq 0.14000000059604645:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(u2 \cdot u2, -19.739208802181317, 1\right), t\_0, \left(\left(u2 \cdot u2\right) \cdot \left(t\_0 \cdot \mathsf{fma}\left(u2 \cdot u2, -85.45681720672748, 64.93939402268539\right)\right)\right) \cdot \left(u2 \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1 \cdot \left(1 + u1\right)} \cdot \cos \left(6.28318530718 \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (/ u1 (- 1.0 u1)))))
   (if (<= u2 0.14000000059604645)
     (fma
      (fma (* u2 u2) -19.739208802181317 1.0)
      t_0
      (*
       (*
        (* u2 u2)
        (* t_0 (fma (* u2 u2) -85.45681720672748 64.93939402268539)))
       (* u2 u2)))
     (* (sqrt (* u1 (+ 1.0 u1))) (cos (* 6.28318530718 u2))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sqrtf((u1 / (1.0f - u1)));
	float tmp;
	if (u2 <= 0.14000000059604645f) {
		tmp = fmaf(fmaf((u2 * u2), -19.739208802181317f, 1.0f), t_0, (((u2 * u2) * (t_0 * fmaf((u2 * u2), -85.45681720672748f, 64.93939402268539f))) * (u2 * u2)));
	} else {
		tmp = sqrtf((u1 * (1.0f + u1))) * cosf((6.28318530718f * u2));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sqrt(Float32(u1 / Float32(Float32(1.0) - u1)))
	tmp = Float32(0.0)
	if (u2 <= Float32(0.14000000059604645))
		tmp = fma(fma(Float32(u2 * u2), Float32(-19.739208802181317), Float32(1.0)), t_0, Float32(Float32(Float32(u2 * u2) * Float32(t_0 * fma(Float32(u2 * u2), Float32(-85.45681720672748), Float32(64.93939402268539)))) * Float32(u2 * u2)));
	else
		tmp = Float32(sqrt(Float32(u1 * Float32(Float32(1.0) + u1))) * cos(Float32(Float32(6.28318530718) * u2)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u2 < 0.140000001

   1. Initial program 99.4%

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

      1. lift-*.f32 N/A

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

      2. lift-sqrt.f32 N/A

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

      3. lift-/.f32 N/A

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

      4. sqrt-div N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f32 N/A

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

      7. lower-*.f32 N/A

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

      8. lower-sqrt.f32 N/A

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

      9. lift-cos.f32 N/A

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

      10. cos-neg-rev N/A

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

      11. lower-cos.f32 N/A

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

      12. lift-*.f32 N/A

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

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

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

      14. lower-*.f32 N/A

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

      15. metadata-eval N/A

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

      16. lower-sqrt.f32 98.9

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

   4. Applied rewrites 98.9%

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

   5. Taylor expanded in u2 around 0

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

   7. Applied rewrites 99.1%

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

   if 0.140000001 < u2

   1. Initial program 94.5%

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

      1. lift-/.f32 N/A

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

      2. lift--.f32 N/A

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

      3. flip-- N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. metadata-eval N/A

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

      8. lower--.f32 N/A

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

      9. lower-*.f32 N/A

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

      10. lower-+.f32 94.7

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

   4. Applied rewrites 94.7%

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

   5. Taylor expanded in u1 around 0

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

   7. Applied rewrites 83.7%

      \[\leadsto \sqrt{\color{blue}{u1} \cdot \left(1 + u1\right)} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \leq 0.14000000059604645:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(u2 \cdot u2, -19.739208802181317, 1\right), \sqrt{\frac{u1}{1 - u1}}, \left(\left(u2 \cdot u2\right) \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(u2 \cdot u2, -85.45681720672748, 64.93939402268539\right)\right)\right) \cdot \left(u2 \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1 \cdot \left(1 + u1\right)} \cdot \cos \left(6.28318530718 \cdot u2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 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 98.9%

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

Alternative 7: 97.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{u1}{1 - u1}}\\ \mathbf{if}\;u2 \leq 0.14000000059604645:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(u2 \cdot u2, -19.739208802181317, 1\right), t\_0, \left(\left(u2 \cdot u2\right) \cdot \left(t\_0 \cdot \mathsf{fma}\left(u2 \cdot u2, -85.45681720672748, 64.93939402268539\right)\right)\right) \cdot \left(u2 \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(u1, u1, u1\right)} \cdot \cos \left(6.28318530718 \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (/ u1 (- 1.0 u1)))))
   (if (<= u2 0.14000000059604645)
     (fma
      (fma (* u2 u2) -19.739208802181317 1.0)
      t_0
      (*
       (*
        (* u2 u2)
        (* t_0 (fma (* u2 u2) -85.45681720672748 64.93939402268539)))
       (* u2 u2)))
     (* (sqrt (fma u1 u1 u1)) (cos (* 6.28318530718 u2))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sqrtf((u1 / (1.0f - u1)));
	float tmp;
	if (u2 <= 0.14000000059604645f) {
		tmp = fmaf(fmaf((u2 * u2), -19.739208802181317f, 1.0f), t_0, (((u2 * u2) * (t_0 * fmaf((u2 * u2), -85.45681720672748f, 64.93939402268539f))) * (u2 * u2)));
	} else {
		tmp = sqrtf(fmaf(u1, u1, u1)) * cosf((6.28318530718f * u2));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sqrt(Float32(u1 / Float32(Float32(1.0) - u1)))
	tmp = Float32(0.0)
	if (u2 <= Float32(0.14000000059604645))
		tmp = fma(fma(Float32(u2 * u2), Float32(-19.739208802181317), Float32(1.0)), t_0, Float32(Float32(Float32(u2 * u2) * Float32(t_0 * fma(Float32(u2 * u2), Float32(-85.45681720672748), Float32(64.93939402268539)))) * Float32(u2 * u2)));
	else
		tmp = Float32(sqrt(fma(u1, u1, u1)) * cos(Float32(Float32(6.28318530718) * u2)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u2 < 0.140000001

   1. Initial program 99.4%

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

      1. lift-*.f32 N/A

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

      2. lift-sqrt.f32 N/A

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

      3. lift-/.f32 N/A

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

      4. sqrt-div N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f32 N/A

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

      7. lower-*.f32 N/A

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

      8. lower-sqrt.f32 N/A

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

      9. lift-cos.f32 N/A

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

      10. cos-neg-rev N/A

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

      11. lower-cos.f32 N/A

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

      12. lift-*.f32 N/A

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

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

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

      14. lower-*.f32 N/A

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

      15. metadata-eval N/A

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

      16. lower-sqrt.f32 98.9

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

   4. Applied rewrites 98.9%

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

   5. Taylor expanded in u2 around 0

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

   7. Applied rewrites 99.1%

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

   if 0.140000001 < u2

   1. Initial program 94.5%

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

   3. Taylor expanded in u1 around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f32 83.6

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

   5. Applied rewrites 83.6%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \leq 0.14000000059604645:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(u2 \cdot u2, -19.739208802181317, 1\right), \sqrt{\frac{u1}{1 - u1}}, \left(\left(u2 \cdot u2\right) \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(u2 \cdot u2, -85.45681720672748, 64.93939402268539\right)\right)\right) \cdot \left(u2 \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(u1, u1, u1\right)} \cdot \cos \left(6.28318530718 \cdot u2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 93.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{u1}{1 - u1}}\\ \mathsf{fma}\left(\mathsf{fma}\left(u2 \cdot u2, -19.739208802181317, 1\right), t\_0, \left(\left(u2 \cdot u2\right) \cdot \left(t\_0 \cdot \mathsf{fma}\left(u2 \cdot u2, -85.45681720672748, 64.93939402268539\right)\right)\right) \cdot \left(u2 \cdot u2\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (/ u1 (- 1.0 u1)))))
   (fma
    (fma (* u2 u2) -19.739208802181317 1.0)
    t_0
    (*
     (* (* u2 u2) (* t_0 (fma (* u2 u2) -85.45681720672748 64.93939402268539)))
     (* u2 u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sqrtf((u1 / (1.0f - u1)));
	return fmaf(fmaf((u2 * u2), -19.739208802181317f, 1.0f), t_0, (((u2 * u2) * (t_0 * fmaf((u2 * u2), -85.45681720672748f, 64.93939402268539f))) * (u2 * u2)));
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sqrt(Float32(u1 / Float32(Float32(1.0) - u1)))
	return fma(fma(Float32(u2 * u2), Float32(-19.739208802181317), Float32(1.0)), t_0, Float32(Float32(Float32(u2 * u2) * Float32(t_0 * fma(Float32(u2 * u2), Float32(-85.45681720672748), Float32(64.93939402268539)))) * Float32(u2 * u2)))
end

Derivation

1. Initial program 98.9%

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

   1. lift-*.f32 N/A

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

   2. lift-sqrt.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. sqrt-div N/A

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

   5. associate-*l/ N/A

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

   6. lower-/.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. lower-sqrt.f32 N/A

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

   9. lift-cos.f32 N/A

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

   10. cos-neg-rev N/A

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

   11. lower-cos.f32 N/A

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

   12. lift-*.f32 N/A

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

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

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

   14. lower-*.f32 N/A

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

   15. metadata-eval N/A

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

   16. lower-sqrt.f32 98.5

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

4. Applied rewrites 98.5%

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

5. Taylor expanded in u2 around 0

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

7. Applied rewrites 93.3%

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

8. Final simplification 93.3%

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

Alternative 9: 93.1% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

   1. lift-*.f32 N/A

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

   2. lift-sqrt.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. sqrt-div N/A

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

   5. associate-*l/ N/A

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

   6. lower-/.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. lower-sqrt.f32 N/A

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

   9. lift-cos.f32 N/A

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

   10. cos-neg-rev N/A

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

   11. lower-cos.f32 N/A

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

   12. lift-*.f32 N/A

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

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

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

   14. lower-*.f32 N/A

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

   15. metadata-eval N/A

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

   16. lower-sqrt.f32 98.5

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

4. Applied rewrites 98.5%

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

5. Taylor expanded in u2 around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. lower--.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. +-commutative N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f32 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f32 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f32 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f32 92.9

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

7. Applied rewrites 92.9%

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

Alternative 10: 91.4% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

   1. lift-*.f32 N/A

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

   2. lift-sqrt.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. sqrt-div N/A

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

   5. associate-*l/ N/A

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

   6. lower-/.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. lower-sqrt.f32 N/A

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

   9. lift-cos.f32 N/A

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

   10. cos-neg-rev N/A

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

   11. lower-cos.f32 N/A

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

   12. lift-*.f32 N/A

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

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

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

   14. lower-*.f32 N/A

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

   15. metadata-eval N/A

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

   16. lower-sqrt.f32 98.5

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

4. Applied rewrites 98.5%

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

5. Taylor expanded in u2 around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. lower--.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 91.3

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

7. Applied rewrites 91.3%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. lift-sqrt.f32 N/A

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

   6. lift-sqrt.f32 N/A

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

   7. sqrt-div N/A

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

   8. lift-/.f32 N/A

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

   9. lift-sqrt.f32 N/A

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

   10. lower-*.f32 91.7

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

9. Applied rewrites 91.7%

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

Alternative 11: 88.3% accurate, 3.1× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return ((-19.739208802181317f * (u2 * u2)) - -1.0f) * 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 = (((-19.739208802181317e0) * (u2 * u2)) - (-1.0e0)) * sqrt((u1 / (1.0e0 - u1)))
end function

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = ((single(-19.739208802181317) * (u2 * u2)) - single(-1.0)) * sqrt((u1 / (single(1.0) - u1)));
end

Derivation

1. Initial program 98.9%

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

   1. lift-*.f32 N/A

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

   2. lift-sqrt.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. sqrt-div N/A

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

   5. associate-*l/ N/A

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

   6. lower-/.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. lower-sqrt.f32 N/A

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

   9. lift-cos.f32 N/A

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

   10. cos-neg-rev N/A

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

   11. lower-cos.f32 N/A

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

   12. lift-*.f32 N/A

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

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

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

   14. lower-*.f32 N/A

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

   15. metadata-eval N/A

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

   16. lower-sqrt.f32 98.5

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

4. Applied rewrites 98.5%

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

5. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. associate-*l* N/A

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

   3. distribute-rgt1-in N/A

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

   4. +-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. lower-sqrt.f32 N/A

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

   12. lower-/.f32 N/A

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

   13. lower--.f32 88.0

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

7. Applied rewrites 88.0%

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

   1. lift-fma.f32 N/A

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

   2. metadata-eval N/A

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

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

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

   6. metadata-eval N/A

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

   7. lower--.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. metadata-eval 88.0

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

9. Applied rewrites 88.0%

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

10. Final simplification 88.0%

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

Alternative 12: 88.3% accurate, 3.3× speedup

Math

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

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(Float32(-19.739208802181317) * u2), u2, Float32(1.0)) * sqrt(Float32(u1 / Float32(Float32(1.0) - u1))))
end

Derivation

1. Initial program 98.9%

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

   1. lift-*.f32 N/A

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

   2. lift-sqrt.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. sqrt-div N/A

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

   5. associate-*l/ N/A

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

   6. lower-/.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. lower-sqrt.f32 N/A

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

   9. lift-cos.f32 N/A

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

   10. cos-neg-rev N/A

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

   11. lower-cos.f32 N/A

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

   12. lift-*.f32 N/A

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

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

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

   14. lower-*.f32 N/A

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

   15. metadata-eval N/A

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

   16. lower-sqrt.f32 98.5

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

4. Applied rewrites 98.5%

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

5. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. associate-*l* N/A

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

   3. distribute-rgt1-in N/A

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

   4. +-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. lower-sqrt.f32 N/A

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

   12. lower-/.f32 N/A

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

   13. lower--.f32 88.0

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

7. Applied rewrites 88.0%

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

   1. lift-fma.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 88.0

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

9. Applied rewrites 88.0%

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

10. Final simplification 88.0%

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

Alternative 13: 79.8% accurate, 5.4× 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 98.9%

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

3. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]
4. 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 79.8

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

5. Applied rewrites 79.8%

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

Alternative 14: 74.1% accurate, 5.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]
4. 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 79.8

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

5. Applied rewrites 79.8%

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

6. Taylor expanded in u1 around 0

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

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. *-lft-identity N/A

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

   4. lower-fma.f32 N/A

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

   5. *-commutative N/A

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

   6. +-commutative N/A

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

   7. distribute-lft1-in N/A

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

   8. lower-fma.f32 74.8

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

8. Applied rewrites 74.8%

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

Alternative 15: 71.2% accurate, 7.9× 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 98.9%

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

3. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]
4. 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 79.8

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

5. Applied rewrites 79.8%

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

6. Taylor expanded in u1 around 0

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

   1. *-commutative N/A

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

   2. +-commutative N/A

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

   3. distribute-lft1-in N/A

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

   4. lower-fma.f32 71.9

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

8. Applied rewrites 71.9%

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

Alternative 16: 62.8% accurate, 12.3× 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 98.9%

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

3. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]
4. 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 79.8

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

5. Applied rewrites 79.8%

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

6. Taylor expanded in u1 around 0

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

8. Applied rewrites 63.1%

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

Reproduce

herbie shell --seed 2025026 
(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))))