Trowbridge-Reitz Sample, near normal, slope_x

Percentage Accurate: 99.0% → 99.0%

Time: 7.7s

Alternatives: 7

Speedup: N/A×

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, N/A× speedup

Math

\[\begin{array}{l} \\ \sqrt{-1 \cdot \frac{-1 \cdot u1}{\frac{-1 \cdot \left(1 - {u1}^{3}\right)}{-1 \cdot \left(1 + \mathsf{fma}\left(u1, u1, u1\right)\right)}}} \cdot \sin \left(\mathsf{fma}\left(-6.28318530718, u2, 0.5 \cdot \pi\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sqrt
   (*
    -1.0
    (/
     (* -1.0 u1)
     (/ (* -1.0 (- 1.0 (pow u1 3.0))) (* -1.0 (+ 1.0 (fma u1 u1 u1)))))))
  (sin (fma -6.28318530718 u2 (* 0.5 PI)))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((-1.0f * ((-1.0f * u1) / ((-1.0f * (1.0f - powf(u1, 3.0f))) / (-1.0f * (1.0f + fmaf(u1, u1, u1))))))) * sinf(fmaf(-6.28318530718f, u2, (0.5f * ((float) M_PI))));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(Float32(-1.0) * Float32(Float32(Float32(-1.0) * u1) / Float32(Float32(Float32(-1.0) * Float32(Float32(1.0) - (u1 ^ Float32(3.0)))) / Float32(Float32(-1.0) * Float32(Float32(1.0) + fma(u1, u1, u1))))))) * sin(fma(Float32(-6.28318530718), u2, Float32(Float32(0.5) * Float32(pi)))))
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

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

   2. flip3-- N/A

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

   3. lower-/.f32 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. metadata-eval N/A

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

   5. lower--.f32 N/A

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

   6. lower-pow.f32 N/A

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

   7. metadata-eval N/A

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

   8. lower-+.f32 N/A

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

   9. lower-fma.f32 N/A

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

   10. lower-*.f32 98.9

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

4. Applied rewrites 98.9%

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

   1. lift-*.f32 N/A

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

   2. lift-cos.f32 N/A

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

   3. cos-neg-rev N/A

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

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

      \[\leadsto \sqrt{\frac{u1}{\frac{1 - {u1}^{3}}{1 + \mathsf{fma}\left(u1, u1, 1 \cdot u1\right)}}} \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\frac{314159265359}{50000000000} \cdot u2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

   5. lower-sin.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{\frac{1 - {u1}^{3}}{1 + \mathsf{fma}\left(u1, u1, 1 \cdot u1\right)}}} \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\frac{314159265359}{50000000000} \cdot u2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

   6. lift-/.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{\frac{1 - {u1}^{3}}{1 + \mathsf{fma}\left(u1, u1, 1 \cdot u1\right)}}} \cdot \sin \left(\left(\mathsf{neg}\left(\frac{314159265359}{50000000000} \cdot u2\right)\right) + \color{blue}{\frac{\mathsf{PI}\left(\right)}{2}}\right) \]

   7. lift-PI.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{\frac{1 - {u1}^{3}}{1 + \mathsf{fma}\left(u1, u1, 1 \cdot u1\right)}}} \cdot \sin \left(\left(\mathsf{neg}\left(\frac{314159265359}{50000000000} \cdot u2\right)\right) + \frac{\color{blue}{\pi}}{2}\right) \]

   8. lower-+.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{\frac{1 - {u1}^{3}}{1 + \mathsf{fma}\left(u1, u1, 1 \cdot u1\right)}}} \cdot \sin \color{blue}{\left(\left(\mathsf{neg}\left(\frac{314159265359}{50000000000} \cdot u2\right)\right) + \frac{\pi}{2}\right)} \]

   9. lower-neg.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{\frac{1 - {u1}^{3}}{1 + \mathsf{fma}\left(u1, u1, 1 \cdot u1\right)}}} \cdot \sin \left(\color{blue}{\left(-\frac{314159265359}{50000000000} \cdot u2\right)} + \frac{\pi}{2}\right) \]

   10. *-commutative N/A

       \[\leadsto \sqrt{\frac{u1}{\frac{1 - {u1}^{3}}{1 + \mathsf{fma}\left(u1, u1, 1 \cdot u1\right)}}} \cdot \sin \left(\left(-\color{blue}{u2 \cdot \frac{314159265359}{50000000000}}\right) + \frac{\pi}{2}\right) \]

   11. lower-*.f32 98.9

       \[\leadsto \sqrt{\frac{u1}{\frac{1 - {u1}^{3}}{1 + \mathsf{fma}\left(u1, u1, 1 \cdot u1\right)}}} \cdot \sin \left(\left(-\color{blue}{u2 \cdot 6.28318530718}\right) + \frac{\pi}{2}\right) \]

6. Applied rewrites 98.9%

   \[\leadsto \sqrt{\frac{u1}{\frac{1 - {u1}^{3}}{1 + \mathsf{fma}\left(u1, u1, 1 \cdot u1\right)}}} \cdot \color{blue}{\sin \left(\left(-u2 \cdot 6.28318530718\right) + \frac{\pi}{2}\right)} \]

7. Taylor expanded in u2 around 0

   \[\leadsto \sqrt{\frac{u1}{\frac{1 - {u1}^{3}}{1 + \mathsf{fma}\left(u1, u1, 1 \cdot u1\right)}}} \cdot \sin \color{blue}{\left(\frac{-314159265359}{50000000000} \cdot u2 + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)} \]
8. Step-by-step derivation

   1. lower-fma.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{\frac{1 - {u1}^{3}}{1 + \mathsf{fma}\left(u1, u1, 1 \cdot u1\right)}}} \cdot \sin \left(\mathsf{fma}\left(\frac{-314159265359}{50000000000}, \color{blue}{u2}, \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right) \]

   2. lower-*.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{\frac{1 - {u1}^{3}}{1 + \mathsf{fma}\left(u1, u1, 1 \cdot u1\right)}}} \cdot \sin \left(\mathsf{fma}\left(\frac{-314159265359}{50000000000}, u2, \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right) \]

   3. lift-PI.f32 99.1

      \[\leadsto \sqrt{\frac{u1}{\frac{1 - {u1}^{3}}{1 + \mathsf{fma}\left(u1, u1, 1 \cdot u1\right)}}} \cdot \sin \left(\mathsf{fma}\left(-6.28318530718, u2, 0.5 \cdot \pi\right)\right) \]

9. Applied rewrites 99.1%

   \[\leadsto \sqrt{\frac{u1}{\frac{1 - {u1}^{3}}{1 + \mathsf{fma}\left(u1, u1, 1 \cdot u1\right)}}} \cdot \sin \color{blue}{\left(\mathsf{fma}\left(-6.28318530718, u2, 0.5 \cdot \pi\right)\right)} \]

10. Final simplification 99.1%

    \[\leadsto \sqrt{-1 \cdot \frac{-1 \cdot u1}{\frac{-1 \cdot \left(1 - {u1}^{3}\right)}{-1 \cdot \left(1 + \mathsf{fma}\left(u1, u1, u1\right)\right)}}} \cdot \sin \left(\mathsf{fma}\left(-6.28318530718, u2, 0.5 \cdot \pi\right)\right) \]
11. Add Preprocessing

Alternative 2: 99.0% accurate, N/A× 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 3: 98.8% accurate, N/A× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((-1.0f * ((-1.0f * u1) / (((1.0f / 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(((-1.0e0) * (((-1.0e0) * u1) / (((1.0e0 / u1) - 1.0e0) * u1)))) * cos((6.28318530718e0 * u2))
end function

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((single(-1.0) * ((single(-1.0) * u1) / (((single(1.0) / 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. Taylor expanded in u1 around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. lower-/.f32 98.8

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

5. Applied rewrites 98.8%

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

6. Final simplification 98.8%

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

Alternative 4: 98.2% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sqrt{u1}} \cdot 0.5\\ t_1 := \sin \left(\mathsf{fma}\left(u2, 6.28318530718, \frac{\pi}{2}\right)\right)\\ t_2 := \sqrt{\frac{-1 \cdot \left(u1 + \left(\left(1 + u1\right) \cdot u1\right) \cdot u1\right)}{-1 \cdot \left(1 - {u1}^{3}\right)}}\\ \mathbf{if}\;u2 \leq 0.10999999940395355:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(-85.45681720672748 \cdot t\_2\right) \cdot \left(u2 \cdot u2\right) - -64.93939402268539 \cdot t\_2, u2 \cdot u2, -19.739208802181317 \cdot t\_2\right), u2 \cdot u2, t\_2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0 \cdot t\_1 - \left(-1 \cdot u1\right) \cdot \mathsf{fma}\left(t\_0, t\_1, \left(0.5 \cdot \sqrt{u1}\right) \cdot \left(\left(1 - 0.25 \cdot \frac{1}{u1}\right) \cdot t\_1\right)\right), u1 \cdot u1, \sqrt{u1} \cdot t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* (/ 1.0 (sqrt u1)) 0.5))
        (t_1 (sin (fma u2 6.28318530718 (/ PI 2.0))))
        (t_2
         (sqrt
          (/
           (* -1.0 (+ u1 (* (* (+ 1.0 u1) u1) u1)))
           (* -1.0 (- 1.0 (pow u1 3.0)))))))
   (if (<= u2 0.10999999940395355)
     (fma
      (fma
       (- (* (* -85.45681720672748 t_2) (* u2 u2)) (* -64.93939402268539 t_2))
       (* u2 u2)
       (* -19.739208802181317 t_2))
      (* u2 u2)
      t_2)
     (fma
      (-
       (* t_0 t_1)
       (*
        (* -1.0 u1)
        (fma
         t_0
         t_1
         (* (* 0.5 (sqrt u1)) (* (- 1.0 (* 0.25 (/ 1.0 u1))) t_1)))))
      (* u1 u1)
      (* (sqrt u1) t_1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = (1.0f / sqrtf(u1)) * 0.5f;
	float t_1 = sinf(fmaf(u2, 6.28318530718f, (((float) M_PI) / 2.0f)));
	float t_2 = sqrtf(((-1.0f * (u1 + (((1.0f + u1) * u1) * u1))) / (-1.0f * (1.0f - powf(u1, 3.0f)))));
	float tmp;
	if (u2 <= 0.10999999940395355f) {
		tmp = fmaf(fmaf((((-85.45681720672748f * t_2) * (u2 * u2)) - (-64.93939402268539f * t_2)), (u2 * u2), (-19.739208802181317f * t_2)), (u2 * u2), t_2);
	} else {
		tmp = fmaf(((t_0 * t_1) - ((-1.0f * u1) * fmaf(t_0, t_1, ((0.5f * sqrtf(u1)) * ((1.0f - (0.25f * (1.0f / u1))) * t_1))))), (u1 * u1), (sqrtf(u1) * t_1));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(Float32(Float32(1.0) / sqrt(u1)) * Float32(0.5))
	t_1 = sin(fma(u2, Float32(6.28318530718), Float32(Float32(pi) / Float32(2.0))))
	t_2 = sqrt(Float32(Float32(Float32(-1.0) * Float32(u1 + Float32(Float32(Float32(Float32(1.0) + u1) * u1) * u1))) / Float32(Float32(-1.0) * Float32(Float32(1.0) - (u1 ^ Float32(3.0))))))
	tmp = Float32(0.0)
	if (u2 <= Float32(0.10999999940395355))
		tmp = fma(fma(Float32(Float32(Float32(Float32(-85.45681720672748) * t_2) * Float32(u2 * u2)) - Float32(Float32(-64.93939402268539) * t_2)), Float32(u2 * u2), Float32(Float32(-19.739208802181317) * t_2)), Float32(u2 * u2), t_2);
	else
		tmp = fma(Float32(Float32(t_0 * t_1) - Float32(Float32(Float32(-1.0) * u1) * fma(t_0, t_1, Float32(Float32(Float32(0.5) * sqrt(u1)) * Float32(Float32(Float32(1.0) - Float32(Float32(0.25) * Float32(Float32(1.0) / u1))) * t_1))))), Float32(u1 * u1), Float32(sqrt(u1) * t_1));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u2 < 0.109999999

   1. Initial program 99.3%

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

      1. lift--.f32 N/A

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

      2. flip3-- N/A

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

      3. lower-/.f32 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. metadata-eval N/A

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

      5. lower--.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. metadata-eval N/A

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

      8. lower-+.f32 N/A

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

      9. lower-fma.f32 N/A

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

      10. lower-*.f32 99.2

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in u2 around 0

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

   7. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\left(-85.45681720672748 \cdot \sqrt{\frac{u1 + \left(\left(1 + u1\right) \cdot u1\right) \cdot u1}{1 - {u1}^{3}}}\right) \cdot \left(u2 \cdot u2\right) - -64.93939402268539 \cdot \sqrt{\frac{u1 + \left(\left(1 + u1\right) \cdot u1\right) \cdot u1}{1 - {u1}^{3}}}, u2 \cdot u2, -19.739208802181317 \cdot \sqrt{\frac{u1 + \left(\left(1 + u1\right) \cdot u1\right) \cdot u1}{1 - {u1}^{3}}}\right), u2 \cdot u2, \sqrt{\frac{u1 + \left(\left(1 + u1\right) \cdot u1\right) \cdot u1}{1 - {u1}^{3}}}\right)} \]

   if 0.109999999 < u2

   1. Initial program 95.9%

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

   3. Taylor expanded in u1 around 0

      \[\leadsto \color{blue}{\sqrt{u1} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) + {u1}^{2} \cdot \left(\frac{1}{2} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right)\right) + u1 \cdot \left(\frac{1}{2} \cdot \left(\sqrt{u1} \cdot \left(\cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \left(1 - \frac{1}{4} \cdot \frac{1}{u1}\right)\right)\right) + \frac{1}{2} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right)\right)\right)\right)} \]

   5. Applied rewrites 86.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\frac{1}{\sqrt{u1}} \cdot 0.5\right) \cdot \sin \left(\mathsf{fma}\left(u2, 6.28318530718, \frac{\pi}{2}\right)\right) - \left(-1 \cdot u1\right) \cdot \mathsf{fma}\left(\frac{1}{\sqrt{u1}} \cdot 0.5, \sin \left(\mathsf{fma}\left(u2, 6.28318530718, \frac{\pi}{2}\right)\right), \left(0.5 \cdot \sqrt{u1}\right) \cdot \left(\left(1 - 0.25 \cdot \frac{1}{u1}\right) \cdot \sin \left(\mathsf{fma}\left(u2, 6.28318530718, \frac{\pi}{2}\right)\right)\right)\right), u1 \cdot u1, \sqrt{u1} \cdot \sin \left(\mathsf{fma}\left(u2, 6.28318530718, \frac{\pi}{2}\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \leq 0.10999999940395355:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(-85.45681720672748 \cdot \sqrt{\frac{-1 \cdot \left(u1 + \left(\left(1 + u1\right) \cdot u1\right) \cdot u1\right)}{-1 \cdot \left(1 - {u1}^{3}\right)}}\right) \cdot \left(u2 \cdot u2\right) - -64.93939402268539 \cdot \sqrt{\frac{-1 \cdot \left(u1 + \left(\left(1 + u1\right) \cdot u1\right) \cdot u1\right)}{-1 \cdot \left(1 - {u1}^{3}\right)}}, u2 \cdot u2, -19.739208802181317 \cdot \sqrt{\frac{-1 \cdot \left(u1 + \left(\left(1 + u1\right) \cdot u1\right) \cdot u1\right)}{-1 \cdot \left(1 - {u1}^{3}\right)}}\right), u2 \cdot u2, \sqrt{\frac{-1 \cdot \left(u1 + \left(\left(1 + u1\right) \cdot u1\right) \cdot u1\right)}{-1 \cdot \left(1 - {u1}^{3}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{1}{\sqrt{u1}} \cdot 0.5\right) \cdot \sin \left(\mathsf{fma}\left(u2, 6.28318530718, \frac{\pi}{2}\right)\right) - \left(-1 \cdot u1\right) \cdot \mathsf{fma}\left(\frac{1}{\sqrt{u1}} \cdot 0.5, \sin \left(\mathsf{fma}\left(u2, 6.28318530718, \frac{\pi}{2}\right)\right), \left(0.5 \cdot \sqrt{u1}\right) \cdot \left(\left(1 - 0.25 \cdot \frac{1}{u1}\right) \cdot \sin \left(\mathsf{fma}\left(u2, 6.28318530718, \frac{\pi}{2}\right)\right)\right)\right), u1 \cdot u1, \sqrt{u1} \cdot \sin \left(\mathsf{fma}\left(u2, 6.28318530718, \frac{\pi}{2}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.2% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sqrt{u1}} \cdot 0.5\\ t_1 := {\left(\frac{u1}{1 - u1}\right)}^{0.5}\\ t_2 := \sin \left(\mathsf{fma}\left(u2, 6.28318530718, \frac{\pi}{2}\right)\right)\\ \mathbf{if}\;u2 \leq 0.10999999940395355:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(-85.45681720672748 \cdot t\_1\right) \cdot \left(u2 \cdot u2\right) - -64.93939402268539 \cdot t\_1, u2 \cdot u2, -19.739208802181317 \cdot t\_1\right), u2 \cdot u2, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0 \cdot t\_2 - \left(-1 \cdot u1\right) \cdot \mathsf{fma}\left(t\_0, t\_2, \left(0.5 \cdot \sqrt{u1}\right) \cdot \left(\left(1 - 0.25 \cdot \frac{1}{u1}\right) \cdot t\_2\right)\right), u1 \cdot u1, \sqrt{u1} \cdot t\_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* (/ 1.0 (sqrt u1)) 0.5))
        (t_1 (pow (/ u1 (- 1.0 u1)) 0.5))
        (t_2 (sin (fma u2 6.28318530718 (/ PI 2.0)))))
   (if (<= u2 0.10999999940395355)
     (fma
      (fma
       (- (* (* -85.45681720672748 t_1) (* u2 u2)) (* -64.93939402268539 t_1))
       (* u2 u2)
       (* -19.739208802181317 t_1))
      (* u2 u2)
      t_1)
     (fma
      (-
       (* t_0 t_2)
       (*
        (* -1.0 u1)
        (fma
         t_0
         t_2
         (* (* 0.5 (sqrt u1)) (* (- 1.0 (* 0.25 (/ 1.0 u1))) t_2)))))
      (* u1 u1)
      (* (sqrt u1) t_2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = (1.0f / sqrtf(u1)) * 0.5f;
	float t_1 = powf((u1 / (1.0f - u1)), 0.5f);
	float t_2 = sinf(fmaf(u2, 6.28318530718f, (((float) M_PI) / 2.0f)));
	float tmp;
	if (u2 <= 0.10999999940395355f) {
		tmp = fmaf(fmaf((((-85.45681720672748f * t_1) * (u2 * u2)) - (-64.93939402268539f * t_1)), (u2 * u2), (-19.739208802181317f * t_1)), (u2 * u2), t_1);
	} else {
		tmp = fmaf(((t_0 * t_2) - ((-1.0f * u1) * fmaf(t_0, t_2, ((0.5f * sqrtf(u1)) * ((1.0f - (0.25f * (1.0f / u1))) * t_2))))), (u1 * u1), (sqrtf(u1) * t_2));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(Float32(Float32(1.0) / sqrt(u1)) * Float32(0.5))
	t_1 = Float32(u1 / Float32(Float32(1.0) - u1)) ^ Float32(0.5)
	t_2 = sin(fma(u2, Float32(6.28318530718), Float32(Float32(pi) / Float32(2.0))))
	tmp = Float32(0.0)
	if (u2 <= Float32(0.10999999940395355))
		tmp = fma(fma(Float32(Float32(Float32(Float32(-85.45681720672748) * t_1) * Float32(u2 * u2)) - Float32(Float32(-64.93939402268539) * t_1)), Float32(u2 * u2), Float32(Float32(-19.739208802181317) * t_1)), Float32(u2 * u2), t_1);
	else
		tmp = fma(Float32(Float32(t_0 * t_2) - Float32(Float32(Float32(-1.0) * u1) * fma(t_0, t_2, Float32(Float32(Float32(0.5) * sqrt(u1)) * Float32(Float32(Float32(1.0) - Float32(Float32(0.25) * Float32(Float32(1.0) / u1))) * t_2))))), Float32(u1 * u1), Float32(sqrt(u1) * t_2));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u2 < 0.109999999

   1. Initial program 99.3%

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

   3. Taylor expanded in u2 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

   5. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\left(-85.45681720672748 \cdot {\left(\frac{u1}{1 - u1}\right)}^{0.5}\right) \cdot \left(u2 \cdot u2\right) - -64.93939402268539 \cdot {\left(\frac{u1}{1 - u1}\right)}^{0.5}, u2 \cdot u2, -19.739208802181317 \cdot {\left(\frac{u1}{1 - u1}\right)}^{0.5}\right), u2 \cdot u2, {\left(\frac{u1}{1 - u1}\right)}^{0.5}\right)} \]

   if 0.109999999 < u2

   1. Initial program 95.9%

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

   3. Taylor expanded in u1 around 0

      \[\leadsto \color{blue}{\sqrt{u1} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) + {u1}^{2} \cdot \left(\frac{1}{2} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right)\right) + u1 \cdot \left(\frac{1}{2} \cdot \left(\sqrt{u1} \cdot \left(\cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \left(1 - \frac{1}{4} \cdot \frac{1}{u1}\right)\right)\right) + \frac{1}{2} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right)\right)\right)\right)} \]

   5. Applied rewrites 86.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\frac{1}{\sqrt{u1}} \cdot 0.5\right) \cdot \sin \left(\mathsf{fma}\left(u2, 6.28318530718, \frac{\pi}{2}\right)\right) - \left(-1 \cdot u1\right) \cdot \mathsf{fma}\left(\frac{1}{\sqrt{u1}} \cdot 0.5, \sin \left(\mathsf{fma}\left(u2, 6.28318530718, \frac{\pi}{2}\right)\right), \left(0.5 \cdot \sqrt{u1}\right) \cdot \left(\left(1 - 0.25 \cdot \frac{1}{u1}\right) \cdot \sin \left(\mathsf{fma}\left(u2, 6.28318530718, \frac{\pi}{2}\right)\right)\right)\right), u1 \cdot u1, \sqrt{u1} \cdot \sin \left(\mathsf{fma}\left(u2, 6.28318530718, \frac{\pi}{2}\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \leq 0.10999999940395355:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(-85.45681720672748 \cdot {\left(\frac{u1}{1 - u1}\right)}^{0.5}\right) \cdot \left(u2 \cdot u2\right) - -64.93939402268539 \cdot {\left(\frac{u1}{1 - u1}\right)}^{0.5}, u2 \cdot u2, -19.739208802181317 \cdot {\left(\frac{u1}{1 - u1}\right)}^{0.5}\right), u2 \cdot u2, {\left(\frac{u1}{1 - u1}\right)}^{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{1}{\sqrt{u1}} \cdot 0.5\right) \cdot \sin \left(\mathsf{fma}\left(u2, 6.28318530718, \frac{\pi}{2}\right)\right) - \left(-1 \cdot u1\right) \cdot \mathsf{fma}\left(\frac{1}{\sqrt{u1}} \cdot 0.5, \sin \left(\mathsf{fma}\left(u2, 6.28318530718, \frac{\pi}{2}\right)\right), \left(0.5 \cdot \sqrt{u1}\right) \cdot \left(\left(1 - 0.25 \cdot \frac{1}{u1}\right) \cdot \sin \left(\mathsf{fma}\left(u2, 6.28318530718, \frac{\pi}{2}\right)\right)\right)\right), u1 \cdot u1, \sqrt{u1} \cdot \sin \left(\mathsf{fma}\left(u2, 6.28318530718, \frac{\pi}{2}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.1% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sqrt{u1}} \cdot 0.5\\ t_1 := \sin \left(\mathsf{fma}\left(u2, 6.28318530718, \frac{\pi}{2}\right)\right)\\ \mathsf{fma}\left(t\_0 \cdot t\_1 - \left(-1 \cdot u1\right) \cdot \mathsf{fma}\left(t\_0, t\_1, \left(0.5 \cdot \sqrt{u1}\right) \cdot \left(\left(1 - 0.25 \cdot \frac{1}{u1}\right) \cdot t\_1\right)\right), u1 \cdot u1, \sqrt{u1} \cdot t\_1\right) \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* (/ 1.0 (sqrt u1)) 0.5))
        (t_1 (sin (fma u2 6.28318530718 (/ PI 2.0)))))
   (fma
    (-
     (* t_0 t_1)
     (*
      (* -1.0 u1)
      (fma t_0 t_1 (* (* 0.5 (sqrt u1)) (* (- 1.0 (* 0.25 (/ 1.0 u1))) t_1)))))
    (* u1 u1)
    (* (sqrt u1) t_1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = (1.0f / sqrtf(u1)) * 0.5f;
	float t_1 = sinf(fmaf(u2, 6.28318530718f, (((float) M_PI) / 2.0f)));
	return fmaf(((t_0 * t_1) - ((-1.0f * u1) * fmaf(t_0, t_1, ((0.5f * sqrtf(u1)) * ((1.0f - (0.25f * (1.0f / u1))) * t_1))))), (u1 * u1), (sqrtf(u1) * t_1));
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(Float32(Float32(1.0) / sqrt(u1)) * Float32(0.5))
	t_1 = sin(fma(u2, Float32(6.28318530718), Float32(Float32(pi) / Float32(2.0))))
	return fma(Float32(Float32(t_0 * t_1) - Float32(Float32(Float32(-1.0) * u1) * fma(t_0, t_1, Float32(Float32(Float32(0.5) * sqrt(u1)) * Float32(Float32(Float32(1.0) - Float32(Float32(0.25) * Float32(Float32(1.0) / u1))) * t_1))))), Float32(u1 * u1), Float32(sqrt(u1) * t_1))
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 u1 around 0

   \[\leadsto \color{blue}{\sqrt{u1} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) + {u1}^{2} \cdot \left(\frac{1}{2} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right)\right) + u1 \cdot \left(\frac{1}{2} \cdot \left(\sqrt{u1} \cdot \left(\cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \left(1 - \frac{1}{4} \cdot \frac{1}{u1}\right)\right)\right) + \frac{1}{2} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right)\right)\right)\right)} \]

5. Applied rewrites 92.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\frac{1}{\sqrt{u1}} \cdot 0.5\right) \cdot \sin \left(\mathsf{fma}\left(u2, 6.28318530718, \frac{\pi}{2}\right)\right) - \left(-1 \cdot u1\right) \cdot \mathsf{fma}\left(\frac{1}{\sqrt{u1}} \cdot 0.5, \sin \left(\mathsf{fma}\left(u2, 6.28318530718, \frac{\pi}{2}\right)\right), \left(0.5 \cdot \sqrt{u1}\right) \cdot \left(\left(1 - 0.25 \cdot \frac{1}{u1}\right) \cdot \sin \left(\mathsf{fma}\left(u2, 6.28318530718, \frac{\pi}{2}\right)\right)\right)\right), u1 \cdot u1, \sqrt{u1} \cdot \sin \left(\mathsf{fma}\left(u2, 6.28318530718, \frac{\pi}{2}\right)\right)\right)} \]

6. Final simplification 92.7%

   \[\leadsto \mathsf{fma}\left(\left(\frac{1}{\sqrt{u1}} \cdot 0.5\right) \cdot \sin \left(\mathsf{fma}\left(u2, 6.28318530718, \frac{\pi}{2}\right)\right) - \left(-1 \cdot u1\right) \cdot \mathsf{fma}\left(\frac{1}{\sqrt{u1}} \cdot 0.5, \sin \left(\mathsf{fma}\left(u2, 6.28318530718, \frac{\pi}{2}\right)\right), \left(0.5 \cdot \sqrt{u1}\right) \cdot \left(\left(1 - 0.25 \cdot \frac{1}{u1}\right) \cdot \sin \left(\mathsf{fma}\left(u2, 6.28318530718, \frac{\pi}{2}\right)\right)\right)\right), u1 \cdot u1, \sqrt{u1} \cdot \sin \left(\mathsf{fma}\left(u2, 6.28318530718, \frac{\pi}{2}\right)\right)\right) \]
7. Add Preprocessing

Alternative 7: 91.6% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\mathsf{fma}\left(u2, 6.28318530718, \frac{\pi}{2}\right)\right)\\ t_1 := \left(-1 \cdot \sqrt{u1}\right) \cdot t\_0\\ t_2 := \left(\frac{1}{\sqrt{u1}} \cdot 0.5\right) \cdot t\_0\\ t_3 := t\_2 - \left(-1 \cdot u1\right) \cdot \mathsf{fma}\left(\left(1 - \frac{1}{u1} \cdot 0.25\right) \cdot t\_0, \sqrt{u1} \cdot 0.5, t\_2\right)\\ t_4 := t\_3 \cdot \left(\left(-1 \cdot u1\right) \cdot u1\right)\\ \frac{\mathsf{fma}\left({t\_3}^{3}, {\left(-1 \cdot {u1}^{3}\right)}^{2}, {\left(\sqrt{u1} \cdot t\_0\right)}^{3}\right)}{t\_4 \cdot t\_4 + \left(t\_1 \cdot t\_1 - t\_4 \cdot t\_1\right)} \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sin (fma u2 6.28318530718 (/ PI 2.0))))
        (t_1 (* (* -1.0 (sqrt u1)) t_0))
        (t_2 (* (* (/ 1.0 (sqrt u1)) 0.5) t_0))
        (t_3
         (-
          t_2
          (*
           (* -1.0 u1)
           (fma (* (- 1.0 (* (/ 1.0 u1) 0.25)) t_0) (* (sqrt u1) 0.5) t_2))))
        (t_4 (* t_3 (* (* -1.0 u1) u1))))
   (/
    (fma
     (pow t_3 3.0)
     (pow (* -1.0 (pow u1 3.0)) 2.0)
     (pow (* (sqrt u1) t_0) 3.0))
    (+ (* t_4 t_4) (- (* t_1 t_1) (* t_4 t_1))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sinf(fmaf(u2, 6.28318530718f, (((float) M_PI) / 2.0f)));
	float t_1 = (-1.0f * sqrtf(u1)) * t_0;
	float t_2 = ((1.0f / sqrtf(u1)) * 0.5f) * t_0;
	float t_3 = t_2 - ((-1.0f * u1) * fmaf(((1.0f - ((1.0f / u1) * 0.25f)) * t_0), (sqrtf(u1) * 0.5f), t_2));
	float t_4 = t_3 * ((-1.0f * u1) * u1);
	return fmaf(powf(t_3, 3.0f), powf((-1.0f * powf(u1, 3.0f)), 2.0f), powf((sqrtf(u1) * t_0), 3.0f)) / ((t_4 * t_4) + ((t_1 * t_1) - (t_4 * t_1)));
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sin(fma(u2, Float32(6.28318530718), Float32(Float32(pi) / Float32(2.0))))
	t_1 = Float32(Float32(Float32(-1.0) * sqrt(u1)) * t_0)
	t_2 = Float32(Float32(Float32(Float32(1.0) / sqrt(u1)) * Float32(0.5)) * t_0)
	t_3 = Float32(t_2 - Float32(Float32(Float32(-1.0) * u1) * fma(Float32(Float32(Float32(1.0) - Float32(Float32(Float32(1.0) / u1) * Float32(0.25))) * t_0), Float32(sqrt(u1) * Float32(0.5)), t_2)))
	t_4 = Float32(t_3 * Float32(Float32(Float32(-1.0) * u1) * u1))
	return Float32(fma((t_3 ^ Float32(3.0)), (Float32(Float32(-1.0) * (u1 ^ Float32(3.0))) ^ Float32(2.0)), (Float32(sqrt(u1) * t_0) ^ Float32(3.0))) / Float32(Float32(t_4 * t_4) + Float32(Float32(t_1 * t_1) - Float32(t_4 * t_1))))
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 u1 around 0

   \[\leadsto \color{blue}{\sqrt{u1} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) + {u1}^{2} \cdot \left(\frac{1}{2} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right)\right) + u1 \cdot \left(\frac{1}{2} \cdot \left(\sqrt{u1} \cdot \left(\cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \left(1 - \frac{1}{4} \cdot \frac{1}{u1}\right)\right)\right) + \frac{1}{2} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right)\right)\right)\right)} \]

5. Applied rewrites 92.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\frac{1}{\sqrt{u1}} \cdot 0.5\right) \cdot \sin \left(\mathsf{fma}\left(u2, 6.28318530718, \frac{\pi}{2}\right)\right) - \left(-1 \cdot u1\right) \cdot \mathsf{fma}\left(\frac{1}{\sqrt{u1}} \cdot 0.5, \sin \left(\mathsf{fma}\left(u2, 6.28318530718, \frac{\pi}{2}\right)\right), \left(0.5 \cdot \sqrt{u1}\right) \cdot \left(\left(1 - 0.25 \cdot \frac{1}{u1}\right) \cdot \sin \left(\mathsf{fma}\left(u2, 6.28318530718, \frac{\pi}{2}\right)\right)\right)\right), u1 \cdot u1, \sqrt{u1} \cdot \sin \left(\mathsf{fma}\left(u2, 6.28318530718, \frac{\pi}{2}\right)\right)\right)} \]

7. Applied rewrites 92.3%

   \[\leadsto \frac{\mathsf{fma}\left({\left(\left(\frac{1}{\sqrt{u1}} \cdot 0.5\right) \cdot \sin \left(\mathsf{fma}\left(u2, 6.28318530718, \frac{\pi}{2}\right)\right) - \left(-1 \cdot u1\right) \cdot \mathsf{fma}\left(\left(1 - \frac{1}{u1} \cdot 0.25\right) \cdot \sin \left(\mathsf{fma}\left(u2, 6.28318530718, \frac{\pi}{2}\right)\right), \sqrt{u1} \cdot 0.5, \left(\frac{1}{\sqrt{u1}} \cdot 0.5\right) \cdot \sin \left(\mathsf{fma}\left(u2, 6.28318530718, \frac{\pi}{2}\right)\right)\right)\right)}^{3}, {\left({u1}^{3} \cdot -1\right)}^{2}, {\left(\sqrt{u1} \cdot \sin \left(\mathsf{fma}\left(u2, 6.28318530718, \frac{\pi}{2}\right)\right)\right)}^{3}\right)}{\color{blue}{\left(\left(\left(\frac{1}{\sqrt{u1}} \cdot 0.5\right) \cdot \sin \left(\mathsf{fma}\left(u2, 6.28318530718, \frac{\pi}{2}\right)\right) - \left(-1 \cdot u1\right) \cdot \mathsf{fma}\left(\left(1 - \frac{1}{u1} \cdot 0.25\right) \cdot \sin \left(\mathsf{fma}\left(u2, 6.28318530718, \frac{\pi}{2}\right)\right), \sqrt{u1} \cdot 0.5, \left(\frac{1}{\sqrt{u1}} \cdot 0.5\right) \cdot \sin \left(\mathsf{fma}\left(u2, 6.28318530718, \frac{\pi}{2}\right)\right)\right)\right) \cdot \left(u1 \cdot u1\right)\right) \cdot \left(\left(\left(\frac{1}{\sqrt{u1}} \cdot 0.5\right) \cdot \sin \left(\mathsf{fma}\left(u2, 6.28318530718, \frac{\pi}{2}\right)\right) - \left(-1 \cdot u1\right) \cdot \mathsf{fma}\left(\left(1 - \frac{1}{u1} \cdot 0.25\right) \cdot \sin \left(\mathsf{fma}\left(u2, 6.28318530718, \frac{\pi}{2}\right)\right), \sqrt{u1} \cdot 0.5, \left(\frac{1}{\sqrt{u1}} \cdot 0.5\right) \cdot \sin \left(\mathsf{fma}\left(u2, 6.28318530718, \frac{\pi}{2}\right)\right)\right)\right) \cdot \left(u1 \cdot u1\right)\right) + \left(\left(\sqrt{u1} \cdot \sin \left(\mathsf{fma}\left(u2, 6.28318530718, \frac{\pi}{2}\right)\right)\right) \cdot \left(\sqrt{u1} \cdot \sin \left(\mathsf{fma}\left(u2, 6.28318530718, \frac{\pi}{2}\right)\right)\right) - \left(\left(\left(\frac{1}{\sqrt{u1}} \cdot 0.5\right) \cdot \sin \left(\mathsf{fma}\left(u2, 6.28318530718, \frac{\pi}{2}\right)\right) - \left(-1 \cdot u1\right) \cdot \mathsf{fma}\left(\left(1 - \frac{1}{u1} \cdot 0.25\right) \cdot \sin \left(\mathsf{fma}\left(u2, 6.28318530718, \frac{\pi}{2}\right)\right), \sqrt{u1} \cdot 0.5, \left(\frac{1}{\sqrt{u1}} \cdot 0.5\right) \cdot \sin \left(\mathsf{fma}\left(u2, 6.28318530718, \frac{\pi}{2}\right)\right)\right)\right) \cdot \left(u1 \cdot u1\right)\right) \cdot \left(\sqrt{u1} \cdot \sin \left(\mathsf{fma}\left(u2, 6.28318530718, \frac{\pi}{2}\right)\right)\right)\right)}} \]

8. Final simplification 92.3%

   \[\leadsto \frac{\mathsf{fma}\left({\left(\left(\frac{1}{\sqrt{u1}} \cdot 0.5\right) \cdot \sin \left(\mathsf{fma}\left(u2, 6.28318530718, \frac{\pi}{2}\right)\right) - \left(-1 \cdot u1\right) \cdot \mathsf{fma}\left(\left(1 - \frac{1}{u1} \cdot 0.25\right) \cdot \sin \left(\mathsf{fma}\left(u2, 6.28318530718, \frac{\pi}{2}\right)\right), \sqrt{u1} \cdot 0.5, \left(\frac{1}{\sqrt{u1}} \cdot 0.5\right) \cdot \sin \left(\mathsf{fma}\left(u2, 6.28318530718, \frac{\pi}{2}\right)\right)\right)\right)}^{3}, {\left(-1 \cdot {u1}^{3}\right)}^{2}, {\left(\sqrt{u1} \cdot \sin \left(\mathsf{fma}\left(u2, 6.28318530718, \frac{\pi}{2}\right)\right)\right)}^{3}\right)}{\left(\left(\left(\frac{1}{\sqrt{u1}} \cdot 0.5\right) \cdot \sin \left(\mathsf{fma}\left(u2, 6.28318530718, \frac{\pi}{2}\right)\right) - \left(-1 \cdot u1\right) \cdot \mathsf{fma}\left(\left(1 - \frac{1}{u1} \cdot 0.25\right) \cdot \sin \left(\mathsf{fma}\left(u2, 6.28318530718, \frac{\pi}{2}\right)\right), \sqrt{u1} \cdot 0.5, \left(\frac{1}{\sqrt{u1}} \cdot 0.5\right) \cdot \sin \left(\mathsf{fma}\left(u2, 6.28318530718, \frac{\pi}{2}\right)\right)\right)\right) \cdot \left(\left(-1 \cdot u1\right) \cdot u1\right)\right) \cdot \left(\left(\left(\frac{1}{\sqrt{u1}} \cdot 0.5\right) \cdot \sin \left(\mathsf{fma}\left(u2, 6.28318530718, \frac{\pi}{2}\right)\right) - \left(-1 \cdot u1\right) \cdot \mathsf{fma}\left(\left(1 - \frac{1}{u1} \cdot 0.25\right) \cdot \sin \left(\mathsf{fma}\left(u2, 6.28318530718, \frac{\pi}{2}\right)\right), \sqrt{u1} \cdot 0.5, \left(\frac{1}{\sqrt{u1}} \cdot 0.5\right) \cdot \sin \left(\mathsf{fma}\left(u2, 6.28318530718, \frac{\pi}{2}\right)\right)\right)\right) \cdot \left(\left(-1 \cdot u1\right) \cdot u1\right)\right) + \left(\left(\left(-1 \cdot \sqrt{u1}\right) \cdot \sin \left(\mathsf{fma}\left(u2, 6.28318530718, \frac{\pi}{2}\right)\right)\right) \cdot \left(\left(-1 \cdot \sqrt{u1}\right) \cdot \sin \left(\mathsf{fma}\left(u2, 6.28318530718, \frac{\pi}{2}\right)\right)\right) - \left(\left(\left(\frac{1}{\sqrt{u1}} \cdot 0.5\right) \cdot \sin \left(\mathsf{fma}\left(u2, 6.28318530718, \frac{\pi}{2}\right)\right) - \left(-1 \cdot u1\right) \cdot \mathsf{fma}\left(\left(1 - \frac{1}{u1} \cdot 0.25\right) \cdot \sin \left(\mathsf{fma}\left(u2, 6.28318530718, \frac{\pi}{2}\right)\right), \sqrt{u1} \cdot 0.5, \left(\frac{1}{\sqrt{u1}} \cdot 0.5\right) \cdot \sin \left(\mathsf{fma}\left(u2, 6.28318530718, \frac{\pi}{2}\right)\right)\right)\right) \cdot \left(\left(-1 \cdot u1\right) \cdot u1\right)\right) \cdot \left(\left(-1 \cdot \sqrt{u1}\right) \cdot \sin \left(\mathsf{fma}\left(u2, 6.28318530718, \frac{\pi}{2}\right)\right)\right)\right)} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025064 
(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))))