Trowbridge-Reitz Sample, near normal, slope_y

Percentage Accurate: 98.4% → 98.3%

Time: 55.0s

Alternatives: 12

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

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.3% accurate, N/A× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.3%

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

   1. lift--.f32 N/A

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

   2. flip3-- N/A

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

   3. lower-/.f32 N/A

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

   4. metadata-eval N/A

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

   5. lower--.f32 N/A

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

   6. lower-pow.f32 N/A

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

   7. metadata-eval N/A

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

   8. lower-+.f32 N/A

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

   9. lower-fma.f32 N/A

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

   10. lower-*.f32 98.3

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

4. Applied rewrites 98.3%

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

5. Taylor expanded in u2 around inf

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

7. Applied rewrites 98.4%

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

Alternative 2: 98.2% accurate, N/A× speedup

Math

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

FPCore

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

C

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

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 ** 3.0e0) * (((1.0e0 + (1.0e0 / u1)) / u1) - (-1.0e0))) / (1.0e0 - (u1 ** 3.0e0)))) * sin((u2 * 6.28318530718e0))
end function

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((((u1 ^ single(3.0)) * (((single(1.0) + (single(1.0) / u1)) / u1) - single(-1.0))) / (single(1.0) - (u1 ^ single(3.0))))) * sin((u2 * single(6.28318530718)));
end

Derivation

1. Initial program 98.3%

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

   1. lift--.f32 N/A

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

   2. flip3-- N/A

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

   3. lower-/.f32 N/A

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

   4. metadata-eval N/A

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

   5. lower--.f32 N/A

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

   6. lower-pow.f32 N/A

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

   7. metadata-eval N/A

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

   8. lower-+.f32 N/A

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

   9. lower-fma.f32 N/A

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

   10. lower-*.f32 98.3

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

4. Applied rewrites 98.3%

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

5. Taylor expanded in u2 around inf

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

7. Applied rewrites 98.4%

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

8. Taylor expanded in u1 around -inf

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lift-pow.f32 N/A

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

   4. lower--.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. lower-+.f32 N/A

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

   8. lower-/.f32 98.4

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

10. Applied rewrites 98.4%

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

11. Final simplification 98.4%

    \[\leadsto \sqrt{\frac{{u1}^{3} \cdot \left(\frac{1 + \frac{1}{u1}}{u1} - -1\right)}{1 - {u1}^{3}}} \cdot \sin \left(u2 \cdot 6.28318530718\right) \]
12. Add Preprocessing

Alternative 3: 98.2% accurate, N/A× speedup

Math

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

FPCore

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

C

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

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 ** 3.0e0) * (((1.0e0 / u1) + ((1.0e0 / u1) / u1)) - (-1.0e0))) / (1.0e0 - (u1 ** 3.0e0)))) * sin((u2 * 6.28318530718e0))
end function

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((((u1 ^ single(3.0)) * (((single(1.0) / u1) + ((single(1.0) / u1) / u1)) - single(-1.0))) / (single(1.0) - (u1 ^ single(3.0))))) * sin((u2 * single(6.28318530718)));
end

Derivation

1. Initial program 98.3%

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

   1. lift--.f32 N/A

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

   2. flip3-- N/A

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

   3. lower-/.f32 N/A

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

   4. metadata-eval N/A

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

   5. lower--.f32 N/A

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

   6. lower-pow.f32 N/A

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

   7. metadata-eval N/A

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

   8. lower-+.f32 N/A

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

   9. lower-fma.f32 N/A

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

   10. lower-*.f32 98.3

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

4. Applied rewrites 98.3%

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

5. Taylor expanded in u2 around inf

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

7. Applied rewrites 98.4%

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

8. Taylor expanded in u1 around -inf

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lift-pow.f32 N/A

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

   4. lower--.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. lower-+.f32 N/A

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

   8. lower-/.f32 98.4

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

10. Applied rewrites 98.4%

    \[\leadsto \sqrt{\frac{-1 \cdot \left({u1}^{3} \cdot \left(-1 \cdot \frac{1 + \frac{1}{u1}}{u1} - 1\right)\right)}{1 - {u1}^{3}}} \cdot \sin \left(u2 \cdot 6.28318530718\right) \]
11. Step-by-step derivation

    1. lift-/.f32 N/A

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

    2. lift-+.f32 N/A

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

    3. lift-/.f32 N/A

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

    4. div-add N/A

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

    5. lower-+.f32 N/A

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

    6. lift-/.f32 N/A

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

    7. lower-/.f32 N/A

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

    8. lift-/.f32 98.3

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

12. Applied rewrites 98.3%

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

13. Final simplification 98.3%

    \[\leadsto \sqrt{\frac{{u1}^{3} \cdot \left(\left(\frac{1}{u1} + \frac{\frac{1}{u1}}{u1}\right) - -1\right)}{1 - {u1}^{3}}} \cdot \sin \left(u2 \cdot 6.28318530718\right) \]
14. Add Preprocessing

Alternative 4: 97.7% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(u2 \cdot 6.28318530718\right)\\ t_1 := \sqrt{\frac{u1}{1 - u1}}\\ t_2 := \frac{1}{\sqrt{u1}}\\ \mathbf{if}\;u2 \leq 0.11999999731779099:\\ \;\;\;\;\mathsf{fma}\left(t\_1 \cdot 6.28318530718, u2, \left(\left(u2 \cdot u2\right) \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -76.70585975309672, t\_1, 81.6052492761019 \cdot t\_1\right), -41.341702240407926 \cdot t\_1\right)\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(t\_2 \cdot 0.5, t\_0, \left(0.5 \cdot \mathsf{fma}\left(\left(1 - \frac{0.25}{u1}\right) \cdot t\_0, \sqrt{u1}, t\_2 \cdot t\_0\right)\right) \cdot u1\right), u1 \cdot u1, \sqrt{u1} \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sin (* u2 6.28318530718)))
        (t_1 (sqrt (/ u1 (- 1.0 u1))))
        (t_2 (/ 1.0 (sqrt u1))))
   (if (<= u2 0.11999999731779099)
     (fma
      (* t_1 6.28318530718)
      u2
      (*
       (*
        (* u2 u2)
        (fma
         (* u2 u2)
         (fma (* (* u2 u2) -76.70585975309672) t_1 (* 81.6052492761019 t_1))
         (* -41.341702240407926 t_1)))
       u2))
     (fma
      (fma
       (* t_2 0.5)
       t_0
       (* (* 0.5 (fma (* (- 1.0 (/ 0.25 u1)) t_0) (sqrt u1) (* t_2 t_0))) u1))
      (* u1 u1)
      (* (sqrt u1) t_0)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sinf((u2 * 6.28318530718f));
	float t_1 = sqrtf((u1 / (1.0f - u1)));
	float t_2 = 1.0f / sqrtf(u1);
	float tmp;
	if (u2 <= 0.11999999731779099f) {
		tmp = fmaf((t_1 * 6.28318530718f), u2, (((u2 * u2) * fmaf((u2 * u2), fmaf(((u2 * u2) * -76.70585975309672f), t_1, (81.6052492761019f * t_1)), (-41.341702240407926f * t_1))) * u2));
	} else {
		tmp = fmaf(fmaf((t_2 * 0.5f), t_0, ((0.5f * fmaf(((1.0f - (0.25f / u1)) * t_0), sqrtf(u1), (t_2 * t_0))) * u1)), (u1 * u1), (sqrtf(u1) * t_0));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sin(Float32(u2 * Float32(6.28318530718)))
	t_1 = sqrt(Float32(u1 / Float32(Float32(1.0) - u1)))
	t_2 = Float32(Float32(1.0) / sqrt(u1))
	tmp = Float32(0.0)
	if (u2 <= Float32(0.11999999731779099))
		tmp = fma(Float32(t_1 * Float32(6.28318530718)), u2, Float32(Float32(Float32(u2 * u2) * fma(Float32(u2 * u2), fma(Float32(Float32(u2 * u2) * Float32(-76.70585975309672)), t_1, Float32(Float32(81.6052492761019) * t_1)), Float32(Float32(-41.341702240407926) * t_1))) * u2));
	else
		tmp = fma(fma(Float32(t_2 * Float32(0.5)), t_0, Float32(Float32(Float32(0.5) * fma(Float32(Float32(Float32(1.0) - Float32(Float32(0.25) / u1)) * t_0), sqrt(u1), Float32(t_2 * t_0))) * u1)), Float32(u1 * u1), Float32(sqrt(u1) * t_0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u2 < 0.119999997

   1. Initial program 98.5%

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

   3. Taylor expanded in u2 around 0

      \[\leadsto \color{blue}{u2 \cdot \left(\frac{314159265359}{50000000000} \cdot \sqrt{\frac{u1}{1 - u1}} + {u2}^{2} \cdot \left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot \sqrt{\frac{u1}{1 - u1}} + {u2}^{2} \cdot \left(\frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot {u2}^{2}\right) + \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot \sqrt{\frac{u1}{1 - u1}}\right)\right)\right)} \]

   5. Applied rewrites 98.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-76.70585975309672, \left(u2 \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}, 81.6052492761019 \cdot \sqrt{\frac{u1}{1 - u1}}\right), u2 \cdot u2, -41.341702240407926 \cdot \sqrt{\frac{u1}{1 - u1}}\right), u2 \cdot u2, \sqrt{\frac{u1}{1 - u1}} \cdot 6.28318530718\right) \cdot u2} \]

   7. Applied rewrites 98.5%

      \[\leadsto \mathsf{fma}\left(\sqrt{\frac{u1}{1 - u1}} \cdot 6.28318530718, \color{blue}{u2}, \left(\left(u2 \cdot u2\right) \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -76.70585975309672, \sqrt{\frac{u1}{1 - u1}}, 81.6052492761019 \cdot \sqrt{\frac{u1}{1 - u1}}\right), -41.341702240407926 \cdot \sqrt{\frac{u1}{1 - u1}}\right)\right) \cdot u2\right) \]

   if 0.119999997 < u2

   1. Initial program 97.3%

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

   3. Taylor expanded in u1 around 0

      \[\leadsto \color{blue}{\sqrt{u1} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) + {u1}^{2} \cdot \left(\frac{1}{2} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right)\right) + u1 \cdot \left(\frac{1}{2} \cdot \left(\sqrt{u1} \cdot \left(\sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \left(1 - \frac{1}{4} \cdot \frac{1}{u1}\right)\right)\right) + \frac{1}{2} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

   5. Applied rewrites 91.7%

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

Alternative 5: 93.9% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{u1}{1 - u1}}\\ \mathsf{fma}\left(t\_0 \cdot 6.28318530718, u2, \left(\left(u2 \cdot u2\right) \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -76.70585975309672, t\_0, 81.6052492761019 \cdot t\_0\right), -41.341702240407926 \cdot t\_0\right)\right) \cdot u2\right) \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (/ u1 (- 1.0 u1)))))
   (fma
    (* t_0 6.28318530718)
    u2
    (*
     (*
      (* u2 u2)
      (fma
       (* u2 u2)
       (fma (* (* u2 u2) -76.70585975309672) t_0 (* 81.6052492761019 t_0))
       (* -41.341702240407926 t_0)))
     u2))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sqrtf((u1 / (1.0f - u1)));
	return fmaf((t_0 * 6.28318530718f), u2, (((u2 * u2) * fmaf((u2 * u2), fmaf(((u2 * u2) * -76.70585975309672f), t_0, (81.6052492761019f * t_0)), (-41.341702240407926f * t_0))) * u2));
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sqrt(Float32(u1 / Float32(Float32(1.0) - u1)))
	return fma(Float32(t_0 * Float32(6.28318530718)), u2, Float32(Float32(Float32(u2 * u2) * fma(Float32(u2 * u2), fma(Float32(Float32(u2 * u2) * Float32(-76.70585975309672)), t_0, Float32(Float32(81.6052492761019) * t_0)), Float32(Float32(-41.341702240407926) * t_0))) * u2))
end

Derivation

1. Initial program 98.3%

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

3. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{u2 \cdot \left(\frac{314159265359}{50000000000} \cdot \sqrt{\frac{u1}{1 - u1}} + {u2}^{2} \cdot \left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot \sqrt{\frac{u1}{1 - u1}} + {u2}^{2} \cdot \left(\frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot {u2}^{2}\right) + \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot \sqrt{\frac{u1}{1 - u1}}\right)\right)\right)} \]

5. Applied rewrites 94.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-76.70585975309672, \left(u2 \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}, 81.6052492761019 \cdot \sqrt{\frac{u1}{1 - u1}}\right), u2 \cdot u2, -41.341702240407926 \cdot \sqrt{\frac{u1}{1 - u1}}\right), u2 \cdot u2, \sqrt{\frac{u1}{1 - u1}} \cdot 6.28318530718\right) \cdot u2} \]

7. Applied rewrites 94.8%

   \[\leadsto \mathsf{fma}\left(\sqrt{\frac{u1}{1 - u1}} \cdot 6.28318530718, \color{blue}{u2}, \left(\left(u2 \cdot u2\right) \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -76.70585975309672, \sqrt{\frac{u1}{1 - u1}}, 81.6052492761019 \cdot \sqrt{\frac{u1}{1 - u1}}\right), -41.341702240407926 \cdot \sqrt{\frac{u1}{1 - u1}}\right)\right) \cdot u2\right) \]
8. Add Preprocessing

Alternative 6: 93.9% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{u1}{1 - u1}}\\ \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-76.70585975309672, \left(u2 \cdot u2\right) \cdot t\_0, 81.6052492761019 \cdot t\_0\right), u2 \cdot u2, -41.341702240407926 \cdot t\_0\right), u2 \cdot u2, t\_0 \cdot 6.28318530718\right) \cdot u2 \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (/ u1 (- 1.0 u1)))))
   (*
    (fma
     (fma
      (fma -76.70585975309672 (* (* u2 u2) t_0) (* 81.6052492761019 t_0))
      (* u2 u2)
      (* -41.341702240407926 t_0))
     (* u2 u2)
     (* t_0 6.28318530718))
    u2)))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sqrtf((u1 / (1.0f - u1)));
	return fmaf(fmaf(fmaf(-76.70585975309672f, ((u2 * u2) * t_0), (81.6052492761019f * t_0)), (u2 * u2), (-41.341702240407926f * t_0)), (u2 * u2), (t_0 * 6.28318530718f)) * u2;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sqrt(Float32(u1 / Float32(Float32(1.0) - u1)))
	return Float32(fma(fma(fma(Float32(-76.70585975309672), Float32(Float32(u2 * u2) * t_0), Float32(Float32(81.6052492761019) * t_0)), Float32(u2 * u2), Float32(Float32(-41.341702240407926) * t_0)), Float32(u2 * u2), Float32(t_0 * Float32(6.28318530718))) * u2)
end

Derivation

1. Initial program 98.3%

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

3. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{u2 \cdot \left(\frac{314159265359}{50000000000} \cdot \sqrt{\frac{u1}{1 - u1}} + {u2}^{2} \cdot \left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot \sqrt{\frac{u1}{1 - u1}} + {u2}^{2} \cdot \left(\frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot {u2}^{2}\right) + \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot \sqrt{\frac{u1}{1 - u1}}\right)\right)\right)} \]

5. Applied rewrites 94.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-76.70585975309672, \left(u2 \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}, 81.6052492761019 \cdot \sqrt{\frac{u1}{1 - u1}}\right), u2 \cdot u2, -41.341702240407926 \cdot \sqrt{\frac{u1}{1 - u1}}\right), u2 \cdot u2, \sqrt{\frac{u1}{1 - u1}} \cdot 6.28318530718\right) \cdot u2} \]
6. Add Preprocessing

Alternative 7: 93.8% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{u1}{1 - u1}}\\ t_1 := \left(u2 \cdot u2\right) \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -76.70585975309672, t\_0, 81.6052492761019 \cdot t\_0\right), -41.341702240407926 \cdot t\_0\right)\\ t_2 := t\_0 \cdot 6.28318530718\\ \frac{t\_1 \cdot t\_1 - t\_2 \cdot t\_2}{t\_1 - t\_2} \cdot u2 \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (/ u1 (- 1.0 u1))))
        (t_1
         (*
          (* u2 u2)
          (fma
           (* u2 u2)
           (fma (* (* u2 u2) -76.70585975309672) t_0 (* 81.6052492761019 t_0))
           (* -41.341702240407926 t_0))))
        (t_2 (* t_0 6.28318530718)))
   (* (/ (- (* t_1 t_1) (* t_2 t_2)) (- t_1 t_2)) u2)))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sqrtf((u1 / (1.0f - u1)));
	float t_1 = (u2 * u2) * fmaf((u2 * u2), fmaf(((u2 * u2) * -76.70585975309672f), t_0, (81.6052492761019f * t_0)), (-41.341702240407926f * t_0));
	float t_2 = t_0 * 6.28318530718f;
	return (((t_1 * t_1) - (t_2 * t_2)) / (t_1 - t_2)) * u2;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sqrt(Float32(u1 / Float32(Float32(1.0) - u1)))
	t_1 = Float32(Float32(u2 * u2) * fma(Float32(u2 * u2), fma(Float32(Float32(u2 * u2) * Float32(-76.70585975309672)), t_0, Float32(Float32(81.6052492761019) * t_0)), Float32(Float32(-41.341702240407926) * t_0)))
	t_2 = Float32(t_0 * Float32(6.28318530718))
	return Float32(Float32(Float32(Float32(t_1 * t_1) - Float32(t_2 * t_2)) / Float32(t_1 - t_2)) * u2)
end

Derivation

1. Initial program 98.3%

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

3. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{u2 \cdot \left(\frac{314159265359}{50000000000} \cdot \sqrt{\frac{u1}{1 - u1}} + {u2}^{2} \cdot \left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot \sqrt{\frac{u1}{1 - u1}} + {u2}^{2} \cdot \left(\frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot {u2}^{2}\right) + \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot \sqrt{\frac{u1}{1 - u1}}\right)\right)\right)} \]

5. Applied rewrites 94.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-76.70585975309672, \left(u2 \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}, 81.6052492761019 \cdot \sqrt{\frac{u1}{1 - u1}}\right), u2 \cdot u2, -41.341702240407926 \cdot \sqrt{\frac{u1}{1 - u1}}\right), u2 \cdot u2, \sqrt{\frac{u1}{1 - u1}} \cdot 6.28318530718\right) \cdot u2} \]

7. Applied rewrites 94.6%

   \[\leadsto \frac{\left(\left(u2 \cdot u2\right) \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -76.70585975309672, \sqrt{\frac{u1}{1 - u1}}, 81.6052492761019 \cdot \sqrt{\frac{u1}{1 - u1}}\right), -41.341702240407926 \cdot \sqrt{\frac{u1}{1 - u1}}\right)\right) \cdot \left(\left(u2 \cdot u2\right) \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -76.70585975309672, \sqrt{\frac{u1}{1 - u1}}, 81.6052492761019 \cdot \sqrt{\frac{u1}{1 - u1}}\right), -41.341702240407926 \cdot \sqrt{\frac{u1}{1 - u1}}\right)\right) - \left(\sqrt{\frac{u1}{1 - u1}} \cdot 6.28318530718\right) \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot 6.28318530718\right)}{\left(u2 \cdot u2\right) \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -76.70585975309672, \sqrt{\frac{u1}{1 - u1}}, 81.6052492761019 \cdot \sqrt{\frac{u1}{1 - u1}}\right), -41.341702240407926 \cdot \sqrt{\frac{u1}{1 - u1}}\right) - \sqrt{\frac{u1}{1 - u1}} \cdot 6.28318530718} \cdot u2 \]
8. Add Preprocessing

Alternative 8: 93.6% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{u1}{1 - u1}}\\ t_1 := \left(u2 \cdot u2\right) \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -76.70585975309672, t\_0, 81.6052492761019 \cdot t\_0\right), -41.341702240407926 \cdot t\_0\right)\\ t_2 := t\_0 \cdot 6.28318530718\\ \frac{{t\_2}^{3} + {t\_1}^{3}}{\mathsf{fma}\left(t\_2, t\_2, t\_1 \cdot t\_1 - t\_2 \cdot t\_1\right)} \cdot u2 \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (/ u1 (- 1.0 u1))))
        (t_1
         (*
          (* u2 u2)
          (fma
           (* u2 u2)
           (fma (* (* u2 u2) -76.70585975309672) t_0 (* 81.6052492761019 t_0))
           (* -41.341702240407926 t_0))))
        (t_2 (* t_0 6.28318530718)))
   (*
    (/
     (+ (pow t_2 3.0) (pow t_1 3.0))
     (fma t_2 t_2 (- (* t_1 t_1) (* t_2 t_1))))
    u2)))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sqrtf((u1 / (1.0f - u1)));
	float t_1 = (u2 * u2) * fmaf((u2 * u2), fmaf(((u2 * u2) * -76.70585975309672f), t_0, (81.6052492761019f * t_0)), (-41.341702240407926f * t_0));
	float t_2 = t_0 * 6.28318530718f;
	return ((powf(t_2, 3.0f) + powf(t_1, 3.0f)) / fmaf(t_2, t_2, ((t_1 * t_1) - (t_2 * t_1)))) * u2;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sqrt(Float32(u1 / Float32(Float32(1.0) - u1)))
	t_1 = Float32(Float32(u2 * u2) * fma(Float32(u2 * u2), fma(Float32(Float32(u2 * u2) * Float32(-76.70585975309672)), t_0, Float32(Float32(81.6052492761019) * t_0)), Float32(Float32(-41.341702240407926) * t_0)))
	t_2 = Float32(t_0 * Float32(6.28318530718))
	return Float32(Float32(Float32((t_2 ^ Float32(3.0)) + (t_1 ^ Float32(3.0))) / fma(t_2, t_2, Float32(Float32(t_1 * t_1) - Float32(t_2 * t_1)))) * u2)
end

Derivation

1. Initial program 98.3%

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

3. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{u2 \cdot \left(\frac{314159265359}{50000000000} \cdot \sqrt{\frac{u1}{1 - u1}} + {u2}^{2} \cdot \left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot \sqrt{\frac{u1}{1 - u1}} + {u2}^{2} \cdot \left(\frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot {u2}^{2}\right) + \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot \sqrt{\frac{u1}{1 - u1}}\right)\right)\right)} \]

5. Applied rewrites 94.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-76.70585975309672, \left(u2 \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}, 81.6052492761019 \cdot \sqrt{\frac{u1}{1 - u1}}\right), u2 \cdot u2, -41.341702240407926 \cdot \sqrt{\frac{u1}{1 - u1}}\right), u2 \cdot u2, \sqrt{\frac{u1}{1 - u1}} \cdot 6.28318530718\right) \cdot u2} \]

7. Applied rewrites 94.4%

   \[\leadsto \frac{{\left(\sqrt{\frac{u1}{1 - u1}} \cdot 6.28318530718\right)}^{3} + {\left(\left(u2 \cdot u2\right) \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -76.70585975309672, \sqrt{\frac{u1}{1 - u1}}, 81.6052492761019 \cdot \sqrt{\frac{u1}{1 - u1}}\right), -41.341702240407926 \cdot \sqrt{\frac{u1}{1 - u1}}\right)\right)}^{3}}{\mathsf{fma}\left(\sqrt{\frac{u1}{1 - u1}} \cdot 6.28318530718, \sqrt{\frac{u1}{1 - u1}} \cdot 6.28318530718, \left(\left(u2 \cdot u2\right) \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -76.70585975309672, \sqrt{\frac{u1}{1 - u1}}, 81.6052492761019 \cdot \sqrt{\frac{u1}{1 - u1}}\right), -41.341702240407926 \cdot \sqrt{\frac{u1}{1 - u1}}\right)\right) \cdot \left(\left(u2 \cdot u2\right) \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -76.70585975309672, \sqrt{\frac{u1}{1 - u1}}, 81.6052492761019 \cdot \sqrt{\frac{u1}{1 - u1}}\right), -41.341702240407926 \cdot \sqrt{\frac{u1}{1 - u1}}\right)\right) - \left(\sqrt{\frac{u1}{1 - u1}} \cdot 6.28318530718\right) \cdot \left(\left(u2 \cdot u2\right) \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -76.70585975309672, \sqrt{\frac{u1}{1 - u1}}, 81.6052492761019 \cdot \sqrt{\frac{u1}{1 - u1}}\right), -41.341702240407926 \cdot \sqrt{\frac{u1}{1 - u1}}\right)\right)\right)} \cdot u2 \]
8. Add Preprocessing

Alternative 9: 92.6% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{u1}{1 - u1}\\ t_1 := \sqrt{t\_0}\\ \frac{{u2}^{4} \cdot \mathsf{fma}\left(1709.1363441345495, t\_0, \left(u2 \cdot u2\right) \cdot \mathsf{fma}\left(-6747.399833653739, t\_0, \left(u2 \cdot u2\right) \cdot \mathsf{fma}\left(6342.301628014029, t\_0, 6659.416709414731 \cdot t\_0\right)\right)\right) - 39.47841760436263 \cdot t\_0}{\left(u2 \cdot u2\right) \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -76.70585975309672, t\_1, 81.6052492761019 \cdot t\_1\right), -41.341702240407926 \cdot t\_1\right) - t\_1 \cdot 6.28318530718} \cdot u2 \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (/ u1 (- 1.0 u1))) (t_1 (sqrt t_0)))
   (*
    (/
     (-
      (*
       (pow u2 4.0)
       (fma
        1709.1363441345495
        t_0
        (*
         (* u2 u2)
         (fma
          -6747.399833653739
          t_0
          (*
           (* u2 u2)
           (fma 6342.301628014029 t_0 (* 6659.416709414731 t_0)))))))
      (* 39.47841760436263 t_0))
     (-
      (*
       (* u2 u2)
       (fma
        (* u2 u2)
        (fma (* (* u2 u2) -76.70585975309672) t_1 (* 81.6052492761019 t_1))
        (* -41.341702240407926 t_1)))
      (* t_1 6.28318530718)))
    u2)))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = u1 / (1.0f - u1);
	float t_1 = sqrtf(t_0);
	return (((powf(u2, 4.0f) * fmaf(1709.1363441345495f, t_0, ((u2 * u2) * fmaf(-6747.399833653739f, t_0, ((u2 * u2) * fmaf(6342.301628014029f, t_0, (6659.416709414731f * t_0))))))) - (39.47841760436263f * t_0)) / (((u2 * u2) * fmaf((u2 * u2), fmaf(((u2 * u2) * -76.70585975309672f), t_1, (81.6052492761019f * t_1)), (-41.341702240407926f * t_1))) - (t_1 * 6.28318530718f))) * u2;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(u1 / Float32(Float32(1.0) - u1))
	t_1 = sqrt(t_0)
	return Float32(Float32(Float32(Float32((u2 ^ Float32(4.0)) * fma(Float32(1709.1363441345495), t_0, Float32(Float32(u2 * u2) * fma(Float32(-6747.399833653739), t_0, Float32(Float32(u2 * u2) * fma(Float32(6342.301628014029), t_0, Float32(Float32(6659.416709414731) * t_0))))))) - Float32(Float32(39.47841760436263) * t_0)) / Float32(Float32(Float32(u2 * u2) * fma(Float32(u2 * u2), fma(Float32(Float32(u2 * u2) * Float32(-76.70585975309672)), t_1, Float32(Float32(81.6052492761019) * t_1)), Float32(Float32(-41.341702240407926) * t_1))) - Float32(t_1 * Float32(6.28318530718)))) * u2)
end

Derivation

1. Initial program 98.3%

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

3. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{u2 \cdot \left(\frac{314159265359}{50000000000} \cdot \sqrt{\frac{u1}{1 - u1}} + {u2}^{2} \cdot \left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot \sqrt{\frac{u1}{1 - u1}} + {u2}^{2} \cdot \left(\frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot {u2}^{2}\right) + \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot \sqrt{\frac{u1}{1 - u1}}\right)\right)\right)} \]

5. Applied rewrites 94.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-76.70585975309672, \left(u2 \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}, 81.6052492761019 \cdot \sqrt{\frac{u1}{1 - u1}}\right), u2 \cdot u2, -41.341702240407926 \cdot \sqrt{\frac{u1}{1 - u1}}\right), u2 \cdot u2, \sqrt{\frac{u1}{1 - u1}} \cdot 6.28318530718\right) \cdot u2} \]

7. Applied rewrites 94.6%

   \[\leadsto \frac{\left(\left(u2 \cdot u2\right) \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -76.70585975309672, \sqrt{\frac{u1}{1 - u1}}, 81.6052492761019 \cdot \sqrt{\frac{u1}{1 - u1}}\right), -41.341702240407926 \cdot \sqrt{\frac{u1}{1 - u1}}\right)\right) \cdot \left(\left(u2 \cdot u2\right) \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -76.70585975309672, \sqrt{\frac{u1}{1 - u1}}, 81.6052492761019 \cdot \sqrt{\frac{u1}{1 - u1}}\right), -41.341702240407926 \cdot \sqrt{\frac{u1}{1 - u1}}\right)\right) - \left(\sqrt{\frac{u1}{1 - u1}} \cdot 6.28318530718\right) \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot 6.28318530718\right)}{\left(u2 \cdot u2\right) \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -76.70585975309672, \sqrt{\frac{u1}{1 - u1}}, 81.6052492761019 \cdot \sqrt{\frac{u1}{1 - u1}}\right), -41.341702240407926 \cdot \sqrt{\frac{u1}{1 - u1}}\right) - \sqrt{\frac{u1}{1 - u1}} \cdot 6.28318530718} \cdot u2 \]

8. Taylor expanded in u2 around 0

   \[\leadsto \frac{{u2}^{4} \cdot \left(\frac{961389193575684075633145058384385882649239799132134631991269883031841}{562500000000000000000000000000000000000000000000000000000000000000} \cdot \frac{u1}{1 - u1} + {u2}^{2} \cdot \left(\frac{-94885310160755698508969199161917078090991542041945444570644759847389875187381489531880769921}{14062500000000000000000000000000000000000000000000000000000000000000000000000000000000000} \cdot \frac{u1}{1 - u1} + {u2}^{2} \cdot \left(\frac{9364804747614465472577070281338582601863864447718755728585928828509634295353730111062330319448960021928803803858401}{1476562500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000} \cdot \frac{u1}{1 - u1} + \frac{9364804747614465472577070281338582601863864447718755728585928828509634295353730111062330319448960021928803803858401}{1406250000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000} \cdot \frac{u1}{1 - u1}\right)\right)\right) - \frac{98696044010906577398881}{2500000000000000000000} \cdot \frac{u1}{1 - u1}}{\left(u2 \cdot u2\right) \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000}, \sqrt{\frac{u1}{1 - u1}}, \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot \sqrt{\frac{u1}{1 - u1}}\right), \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot \sqrt{\frac{u1}{1 - u1}}\right) - \sqrt{\frac{u1}{1 - u1}} \cdot \frac{314159265359}{50000000000}} \cdot u2 \]

10. Applied rewrites 93.6%

    \[\leadsto \frac{{u2}^{4} \cdot \mathsf{fma}\left(1709.1363441345495, \frac{u1}{1 - u1}, \left(u2 \cdot u2\right) \cdot \mathsf{fma}\left(-6747.399833653739, \frac{u1}{1 - u1}, \left(u2 \cdot u2\right) \cdot \mathsf{fma}\left(6342.301628014029, \frac{u1}{1 - u1}, 6659.416709414731 \cdot \frac{u1}{1 - u1}\right)\right)\right) - 39.47841760436263 \cdot \frac{u1}{1 - u1}}{\left(u2 \cdot u2\right) \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -76.70585975309672, \sqrt{\frac{u1}{1 - u1}}, 81.6052492761019 \cdot \sqrt{\frac{u1}{1 - u1}}\right), -41.341702240407926 \cdot \sqrt{\frac{u1}{1 - u1}}\right) - \sqrt{\frac{u1}{1 - u1}} \cdot 6.28318530718} \cdot u2 \]
11. Add Preprocessing

Alternative 10: 91.0% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{0.25}{u1}\\ t_1 := \sqrt{\frac{u1}{1 - u1}}\\ t_2 := \frac{1}{\sqrt{u1}}\\ t_3 := 40.80262463805095 \cdot t\_2\\ t_4 := -20.670851120203963 \cdot t\_2\\ t_5 := -38.35292987654836 \cdot t\_2\\ t_6 := \mathsf{fma}\left(\mathsf{fma}\left(t\_5, u2 \cdot u2, t\_3\right), u2 \cdot u2, t\_4\right) \cdot \left(u2 \cdot u2\right)\\ \mathbf{if}\;t\_1 \cdot \sin \left(6.28318530718 \cdot u2\right) \leq 2.499999936844688 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{u1}, 6.28318530718, \mathsf{fma}\left(\mathsf{fma}\left(3.14159265359, t\_2, \mathsf{fma}\left(\mathsf{fma}\left(3.14159265359, t\_2, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(t\_5, \left(u2 \cdot u2\right) \cdot t\_0, t\_3 \cdot t\_0\right), u2 \cdot u2, t\_4 \cdot t\_0\right), u2 \cdot u2, \left(3.14159265359 \cdot t\_2\right) \cdot t\_0\right), u1, t\_6\right)\right), u1, t\_6\right)\right), u1 \cdot u1, \mathsf{fma}\left(\mathsf{fma}\left(-76.70585975309672 \cdot \sqrt{u1}, u2 \cdot u2, \sqrt{u1} \cdot 81.6052492761019\right), u2 \cdot u2, \sqrt{u1} \cdot -41.341702240407926\right) \cdot \left(u2 \cdot u2\right)\right)\right) \cdot u2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-81.6052492761019 \cdot t\_1, {u2}^{-2}, \mathsf{fma}\left(-6.28318530718 \cdot t\_1, {u2}^{-6}, \mathsf{fma}\left({u2}^{-4} \cdot t\_1, 41.341702240407926, 76.70585975309672 \cdot t\_1\right)\right)\right) \cdot \left(-{u2}^{7}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (- 1.0 (/ 0.25 u1)))
        (t_1 (sqrt (/ u1 (- 1.0 u1))))
        (t_2 (/ 1.0 (sqrt u1)))
        (t_3 (* 40.80262463805095 t_2))
        (t_4 (* -20.670851120203963 t_2))
        (t_5 (* -38.35292987654836 t_2))
        (t_6 (* (fma (fma t_5 (* u2 u2) t_3) (* u2 u2) t_4) (* u2 u2))))
   (if (<= (* t_1 (sin (* 6.28318530718 u2))) 2.499999936844688e-6)
     (*
      (fma
       (sqrt u1)
       6.28318530718
       (fma
        (fma
         3.14159265359
         t_2
         (fma
          (fma
           3.14159265359
           t_2
           (fma
            (fma
             (fma
              (fma t_5 (* (* u2 u2) t_0) (* t_3 t_0))
              (* u2 u2)
              (* t_4 t_0))
             (* u2 u2)
             (* (* 3.14159265359 t_2) t_0))
            u1
            t_6))
          u1
          t_6))
        (* u1 u1)
        (*
         (fma
          (fma
           (* -76.70585975309672 (sqrt u1))
           (* u2 u2)
           (* (sqrt u1) 81.6052492761019))
          (* u2 u2)
          (* (sqrt u1) -41.341702240407926))
         (* u2 u2))))
      u2)
     (*
      (fma
       (* -81.6052492761019 t_1)
       (pow u2 -2.0)
       (fma
        (* -6.28318530718 t_1)
        (pow u2 -6.0)
        (fma
         (* (pow u2 -4.0) t_1)
         41.341702240407926
         (* 76.70585975309672 t_1))))
      (- (pow u2 7.0))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = 1.0f - (0.25f / u1);
	float t_1 = sqrtf((u1 / (1.0f - u1)));
	float t_2 = 1.0f / sqrtf(u1);
	float t_3 = 40.80262463805095f * t_2;
	float t_4 = -20.670851120203963f * t_2;
	float t_5 = -38.35292987654836f * t_2;
	float t_6 = fmaf(fmaf(t_5, (u2 * u2), t_3), (u2 * u2), t_4) * (u2 * u2);
	float tmp;
	if ((t_1 * sinf((6.28318530718f * u2))) <= 2.499999936844688e-6f) {
		tmp = fmaf(sqrtf(u1), 6.28318530718f, fmaf(fmaf(3.14159265359f, t_2, fmaf(fmaf(3.14159265359f, t_2, fmaf(fmaf(fmaf(fmaf(t_5, ((u2 * u2) * t_0), (t_3 * t_0)), (u2 * u2), (t_4 * t_0)), (u2 * u2), ((3.14159265359f * t_2) * t_0)), u1, t_6)), u1, t_6)), (u1 * u1), (fmaf(fmaf((-76.70585975309672f * sqrtf(u1)), (u2 * u2), (sqrtf(u1) * 81.6052492761019f)), (u2 * u2), (sqrtf(u1) * -41.341702240407926f)) * (u2 * u2)))) * u2;
	} else {
		tmp = fmaf((-81.6052492761019f * t_1), powf(u2, -2.0f), fmaf((-6.28318530718f * t_1), powf(u2, -6.0f), fmaf((powf(u2, -4.0f) * t_1), 41.341702240407926f, (76.70585975309672f * t_1)))) * -powf(u2, 7.0f);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(Float32(1.0) - Float32(Float32(0.25) / u1))
	t_1 = sqrt(Float32(u1 / Float32(Float32(1.0) - u1)))
	t_2 = Float32(Float32(1.0) / sqrt(u1))
	t_3 = Float32(Float32(40.80262463805095) * t_2)
	t_4 = Float32(Float32(-20.670851120203963) * t_2)
	t_5 = Float32(Float32(-38.35292987654836) * t_2)
	t_6 = Float32(fma(fma(t_5, Float32(u2 * u2), t_3), Float32(u2 * u2), t_4) * Float32(u2 * u2))
	tmp = Float32(0.0)
	if (Float32(t_1 * sin(Float32(Float32(6.28318530718) * u2))) <= Float32(2.499999936844688e-6))
		tmp = Float32(fma(sqrt(u1), Float32(6.28318530718), fma(fma(Float32(3.14159265359), t_2, fma(fma(Float32(3.14159265359), t_2, fma(fma(fma(fma(t_5, Float32(Float32(u2 * u2) * t_0), Float32(t_3 * t_0)), Float32(u2 * u2), Float32(t_4 * t_0)), Float32(u2 * u2), Float32(Float32(Float32(3.14159265359) * t_2) * t_0)), u1, t_6)), u1, t_6)), Float32(u1 * u1), Float32(fma(fma(Float32(Float32(-76.70585975309672) * sqrt(u1)), Float32(u2 * u2), Float32(sqrt(u1) * Float32(81.6052492761019))), Float32(u2 * u2), Float32(sqrt(u1) * Float32(-41.341702240407926))) * Float32(u2 * u2)))) * u2);
	else
		tmp = Float32(fma(Float32(Float32(-81.6052492761019) * t_1), (u2 ^ Float32(-2.0)), fma(Float32(Float32(-6.28318530718) * t_1), (u2 ^ Float32(-6.0)), fma(Float32((u2 ^ Float32(-4.0)) * t_1), Float32(41.341702240407926), Float32(Float32(76.70585975309672) * t_1)))) * Float32(-(u2 ^ Float32(7.0))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (sqrt.f32 (/.f32 u1 (-.f32 #s(literal 1 binary32) u1))) (sin.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2))) < 2.49999994e-6

   1. Initial program 98.2%

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

   3. Taylor expanded in u2 around 0

      \[\leadsto \color{blue}{u2 \cdot \left(\frac{314159265359}{50000000000} \cdot \sqrt{\frac{u1}{1 - u1}} + {u2}^{2} \cdot \left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot \sqrt{\frac{u1}{1 - u1}} + {u2}^{2} \cdot \left(\frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot {u2}^{2}\right) + \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot \sqrt{\frac{u1}{1 - u1}}\right)\right)\right)} \]

   5. Applied rewrites 95.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-76.70585975309672, \left(u2 \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}, 81.6052492761019 \cdot \sqrt{\frac{u1}{1 - u1}}\right), u2 \cdot u2, -41.341702240407926 \cdot \sqrt{\frac{u1}{1 - u1}}\right), u2 \cdot u2, \sqrt{\frac{u1}{1 - u1}} \cdot 6.28318530718\right) \cdot u2} \]

   6. Taylor expanded in u1 around 0

      \[\leadsto \left(\frac{314159265359}{50000000000} \cdot \sqrt{u1} + \left({u1}^{2} \cdot \left(\frac{314159265359}{100000000000} \cdot \sqrt{\frac{1}{u1}} + \left(u1 \cdot \left(\frac{314159265359}{100000000000} \cdot \sqrt{\frac{1}{u1}} + \left(u1 \cdot \left(\frac{314159265359}{100000000000} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \left(1 - \frac{1}{4} \cdot \frac{1}{u1}\right)\right) + {u2}^{2} \cdot \left(\frac{-31006276680305942139213528068663279}{1500000000000000000000000000000000} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \left(1 - \frac{1}{4} \cdot \frac{1}{u1}\right)\right) + {u2}^{2} \cdot \left(\frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{7875000000000000000000000000000000000000000000000000000000000000000000000000000} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \left({u2}^{2} \cdot \left(1 - \frac{1}{4} \cdot \frac{1}{u1}\right)\right)\right) + \frac{3060196847853821555298148281676017575122444629042460390799}{75000000000000000000000000000000000000000000000000000000} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \left(1 - \frac{1}{4} \cdot \frac{1}{u1}\right)\right)\right)\right)\right) + {u2}^{2} \cdot \left(\frac{-31006276680305942139213528068663279}{1500000000000000000000000000000000} \cdot \sqrt{\frac{1}{u1}} + {u2}^{2} \cdot \left(\frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{7875000000000000000000000000000000000000000000000000000000000000000000000000000} \cdot \left(\sqrt{\frac{1}{u1}} \cdot {u2}^{2}\right) + \frac{3060196847853821555298148281676017575122444629042460390799}{75000000000000000000000000000000000000000000000000000000} \cdot \sqrt{\frac{1}{u1}}\right)\right)\right)\right) + {u2}^{2} \cdot \left(\frac{-31006276680305942139213528068663279}{1500000000000000000000000000000000} \cdot \sqrt{\frac{1}{u1}} + {u2}^{2} \cdot \left(\frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{7875000000000000000000000000000000000000000000000000000000000000000000000000000} \cdot \left(\sqrt{\frac{1}{u1}} \cdot {u2}^{2}\right) + \frac{3060196847853821555298148281676017575122444629042460390799}{75000000000000000000000000000000000000000000000000000000} \cdot \sqrt{\frac{1}{u1}}\right)\right)\right)\right) + {u2}^{2} \cdot \left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot \sqrt{u1} + {u2}^{2} \cdot \left(\frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot \left(\sqrt{u1} \cdot {u2}^{2}\right) + \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot \sqrt{u1}\right)\right)\right)\right) \cdot u2 \]

   8. Applied rewrites 90.4%

      \[\leadsto \mathsf{fma}\left(\sqrt{u1}, 6.28318530718, \mathsf{fma}\left(\mathsf{fma}\left(3.14159265359, \frac{1}{\sqrt{u1}}, \mathsf{fma}\left(\mathsf{fma}\left(3.14159265359, \frac{1}{\sqrt{u1}}, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-38.35292987654836 \cdot \frac{1}{\sqrt{u1}}, \left(u2 \cdot u2\right) \cdot \left(1 - \frac{0.25}{u1}\right), \left(40.80262463805095 \cdot \frac{1}{\sqrt{u1}}\right) \cdot \left(1 - \frac{0.25}{u1}\right)\right), u2 \cdot u2, \left(-20.670851120203963 \cdot \frac{1}{\sqrt{u1}}\right) \cdot \left(1 - \frac{0.25}{u1}\right)\right), u2 \cdot u2, \left(3.14159265359 \cdot \frac{1}{\sqrt{u1}}\right) \cdot \left(1 - \frac{0.25}{u1}\right)\right), u1, \mathsf{fma}\left(\mathsf{fma}\left(-38.35292987654836 \cdot \frac{1}{\sqrt{u1}}, u2 \cdot u2, 40.80262463805095 \cdot \frac{1}{\sqrt{u1}}\right), u2 \cdot u2, -20.670851120203963 \cdot \frac{1}{\sqrt{u1}}\right) \cdot \left(u2 \cdot u2\right)\right)\right), u1, \mathsf{fma}\left(\mathsf{fma}\left(-38.35292987654836 \cdot \frac{1}{\sqrt{u1}}, u2 \cdot u2, 40.80262463805095 \cdot \frac{1}{\sqrt{u1}}\right), u2 \cdot u2, -20.670851120203963 \cdot \frac{1}{\sqrt{u1}}\right) \cdot \left(u2 \cdot u2\right)\right)\right), u1 \cdot u1, \mathsf{fma}\left(\mathsf{fma}\left(-76.70585975309672 \cdot \sqrt{u1}, u2 \cdot u2, \sqrt{u1} \cdot 81.6052492761019\right), u2 \cdot u2, \sqrt{u1} \cdot -41.341702240407926\right) \cdot \left(u2 \cdot u2\right)\right)\right) \cdot u2 \]

   if 2.49999994e-6 < (*.f32 (sqrt.f32 (/.f32 u1 (-.f32 #s(literal 1 binary32) u1))) (sin.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2)))

   1. Initial program 98.6%

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

   3. Taylor expanded in u2 around 0

      \[\leadsto \color{blue}{u2 \cdot \left(\frac{314159265359}{50000000000} \cdot \sqrt{\frac{u1}{1 - u1}} + {u2}^{2} \cdot \left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot \sqrt{\frac{u1}{1 - u1}} + {u2}^{2} \cdot \left(\frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot {u2}^{2}\right) + \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot \sqrt{\frac{u1}{1 - u1}}\right)\right)\right)} \]

   5. Applied rewrites 94.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-76.70585975309672, \left(u2 \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}, 81.6052492761019 \cdot \sqrt{\frac{u1}{1 - u1}}\right), u2 \cdot u2, -41.341702240407926 \cdot \sqrt{\frac{u1}{1 - u1}}\right), u2 \cdot u2, \sqrt{\frac{u1}{1 - u1}} \cdot 6.28318530718\right) \cdot u2} \]

   6. Taylor expanded in u2 around -inf

      \[\leadsto -1 \cdot \color{blue}{\left({u2}^{7} \cdot \left(\frac{-3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot \frac{1}{{u2}^{2}}\right) + \left(\frac{-314159265359}{50000000000} \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot \frac{1}{{u2}^{6}}\right) + \left(\frac{31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot \frac{1}{{u2}^{4}}\right) + \frac{302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot \sqrt{\frac{u1}{1 - u1}}\right)\right)\right)\right)} \]

   8. Applied rewrites 93.3%

      \[\leadsto -\mathsf{fma}\left(-81.6052492761019 \cdot \sqrt{\frac{u1}{1 - u1}}, {u2}^{-2}, \mathsf{fma}\left(-6.28318530718 \cdot \sqrt{\frac{u1}{1 - u1}}, {u2}^{-6}, \mathsf{fma}\left({u2}^{-4} \cdot \sqrt{\frac{u1}{1 - u1}}, 41.341702240407926, 76.70585975309672 \cdot \sqrt{\frac{u1}{1 - u1}}\right)\right)\right) \cdot {u2}^{7} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \leq 2.499999936844688 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{u1}, 6.28318530718, \mathsf{fma}\left(\mathsf{fma}\left(3.14159265359, \frac{1}{\sqrt{u1}}, \mathsf{fma}\left(\mathsf{fma}\left(3.14159265359, \frac{1}{\sqrt{u1}}, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-38.35292987654836 \cdot \frac{1}{\sqrt{u1}}, \left(u2 \cdot u2\right) \cdot \left(1 - \frac{0.25}{u1}\right), \left(40.80262463805095 \cdot \frac{1}{\sqrt{u1}}\right) \cdot \left(1 - \frac{0.25}{u1}\right)\right), u2 \cdot u2, \left(-20.670851120203963 \cdot \frac{1}{\sqrt{u1}}\right) \cdot \left(1 - \frac{0.25}{u1}\right)\right), u2 \cdot u2, \left(3.14159265359 \cdot \frac{1}{\sqrt{u1}}\right) \cdot \left(1 - \frac{0.25}{u1}\right)\right), u1, \mathsf{fma}\left(\mathsf{fma}\left(-38.35292987654836 \cdot \frac{1}{\sqrt{u1}}, u2 \cdot u2, 40.80262463805095 \cdot \frac{1}{\sqrt{u1}}\right), u2 \cdot u2, -20.670851120203963 \cdot \frac{1}{\sqrt{u1}}\right) \cdot \left(u2 \cdot u2\right)\right)\right), u1, \mathsf{fma}\left(\mathsf{fma}\left(-38.35292987654836 \cdot \frac{1}{\sqrt{u1}}, u2 \cdot u2, 40.80262463805095 \cdot \frac{1}{\sqrt{u1}}\right), u2 \cdot u2, -20.670851120203963 \cdot \frac{1}{\sqrt{u1}}\right) \cdot \left(u2 \cdot u2\right)\right)\right), u1 \cdot u1, \mathsf{fma}\left(\mathsf{fma}\left(-76.70585975309672 \cdot \sqrt{u1}, u2 \cdot u2, \sqrt{u1} \cdot 81.6052492761019\right), u2 \cdot u2, \sqrt{u1} \cdot -41.341702240407926\right) \cdot \left(u2 \cdot u2\right)\right)\right) \cdot u2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-81.6052492761019 \cdot \sqrt{\frac{u1}{1 - u1}}, {u2}^{-2}, \mathsf{fma}\left(-6.28318530718 \cdot \sqrt{\frac{u1}{1 - u1}}, {u2}^{-6}, \mathsf{fma}\left({u2}^{-4} \cdot \sqrt{\frac{u1}{1 - u1}}, 41.341702240407926, 76.70585975309672 \cdot \sqrt{\frac{u1}{1 - u1}}\right)\right)\right) \cdot \left(-{u2}^{7}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 91.0% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sqrt{u1}}\\ t_1 := u2 \cdot \mathsf{fma}\left(3.14159265359, t\_0, \left(u2 \cdot u2\right) \cdot \mathsf{fma}\left(-20.670851120203963, t\_0, \left(u2 \cdot u2\right) \cdot \mathsf{fma}\left(-38.35292987654836, t\_0 \cdot \left(u2 \cdot u2\right), 40.80262463805095 \cdot t\_0\right)\right)\right)\\ t_2 := \sqrt{\frac{u1}{1 - u1}}\\ t_3 := 1 - 0.25 \cdot \frac{1}{u1}\\ t_4 := t\_0 \cdot t\_3\\ \mathbf{if}\;t\_2 \cdot \sin \left(6.28318530718 \cdot u2\right) \leq 2.499999936844688 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(u2, \mathsf{fma}\left(6.28318530718, \sqrt{u1}, \left(u2 \cdot u2\right) \cdot \mathsf{fma}\left(-41.341702240407926, \sqrt{u1}, \left(u2 \cdot u2\right) \cdot \mathsf{fma}\left(-76.70585975309672, \sqrt{u1} \cdot \left(u2 \cdot u2\right), 81.6052492761019 \cdot \sqrt{u1}\right)\right)\right), \left(u1 \cdot u1\right) \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u2 \cdot \mathsf{fma}\left(3.14159265359, t\_4, \left(u2 \cdot u2\right) \cdot \mathsf{fma}\left(-20.670851120203963, t\_4, \left(u2 \cdot u2\right) \cdot \mathsf{fma}\left(-38.35292987654836, t\_0 \cdot \left(\left(u2 \cdot u2\right) \cdot t\_3\right), 40.80262463805095 \cdot t\_4\right)\right)\right), t\_1\right), t\_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-81.6052492761019 \cdot t\_2, {u2}^{-2}, \mathsf{fma}\left(-6.28318530718 \cdot t\_2, {u2}^{-6}, \mathsf{fma}\left({u2}^{-4} \cdot t\_2, 41.341702240407926, 76.70585975309672 \cdot t\_2\right)\right)\right) \cdot \left(-{u2}^{7}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (/ 1.0 (sqrt u1)))
        (t_1
         (*
          u2
          (fma
           3.14159265359
           t_0
           (*
            (* u2 u2)
            (fma
             -20.670851120203963
             t_0
             (*
              (* u2 u2)
              (fma
               -38.35292987654836
               (* t_0 (* u2 u2))
               (* 40.80262463805095 t_0))))))))
        (t_2 (sqrt (/ u1 (- 1.0 u1))))
        (t_3 (- 1.0 (* 0.25 (/ 1.0 u1))))
        (t_4 (* t_0 t_3)))
   (if (<= (* t_2 (sin (* 6.28318530718 u2))) 2.499999936844688e-6)
     (fma
      u2
      (fma
       6.28318530718
       (sqrt u1)
       (*
        (* u2 u2)
        (fma
         -41.341702240407926
         (sqrt u1)
         (*
          (* u2 u2)
          (fma
           -76.70585975309672
           (* (sqrt u1) (* u2 u2))
           (* 81.6052492761019 (sqrt u1)))))))
      (*
       (* u1 u1)
       (fma
        u1
        (fma
         u1
         (*
          u2
          (fma
           3.14159265359
           t_4
           (*
            (* u2 u2)
            (fma
             -20.670851120203963
             t_4
             (*
              (* u2 u2)
              (fma
               -38.35292987654836
               (* t_0 (* (* u2 u2) t_3))
               (* 40.80262463805095 t_4)))))))
         t_1)
        t_1)))
     (*
      (fma
       (* -81.6052492761019 t_2)
       (pow u2 -2.0)
       (fma
        (* -6.28318530718 t_2)
        (pow u2 -6.0)
        (fma
         (* (pow u2 -4.0) t_2)
         41.341702240407926
         (* 76.70585975309672 t_2))))
      (- (pow u2 7.0))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = 1.0f / sqrtf(u1);
	float t_1 = u2 * fmaf(3.14159265359f, t_0, ((u2 * u2) * fmaf(-20.670851120203963f, t_0, ((u2 * u2) * fmaf(-38.35292987654836f, (t_0 * (u2 * u2)), (40.80262463805095f * t_0))))));
	float t_2 = sqrtf((u1 / (1.0f - u1)));
	float t_3 = 1.0f - (0.25f * (1.0f / u1));
	float t_4 = t_0 * t_3;
	float tmp;
	if ((t_2 * sinf((6.28318530718f * u2))) <= 2.499999936844688e-6f) {
		tmp = fmaf(u2, fmaf(6.28318530718f, sqrtf(u1), ((u2 * u2) * fmaf(-41.341702240407926f, sqrtf(u1), ((u2 * u2) * fmaf(-76.70585975309672f, (sqrtf(u1) * (u2 * u2)), (81.6052492761019f * sqrtf(u1))))))), ((u1 * u1) * fmaf(u1, fmaf(u1, (u2 * fmaf(3.14159265359f, t_4, ((u2 * u2) * fmaf(-20.670851120203963f, t_4, ((u2 * u2) * fmaf(-38.35292987654836f, (t_0 * ((u2 * u2) * t_3)), (40.80262463805095f * t_4))))))), t_1), t_1)));
	} else {
		tmp = fmaf((-81.6052492761019f * t_2), powf(u2, -2.0f), fmaf((-6.28318530718f * t_2), powf(u2, -6.0f), fmaf((powf(u2, -4.0f) * t_2), 41.341702240407926f, (76.70585975309672f * t_2)))) * -powf(u2, 7.0f);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(Float32(1.0) / sqrt(u1))
	t_1 = Float32(u2 * fma(Float32(3.14159265359), t_0, Float32(Float32(u2 * u2) * fma(Float32(-20.670851120203963), t_0, Float32(Float32(u2 * u2) * fma(Float32(-38.35292987654836), Float32(t_0 * Float32(u2 * u2)), Float32(Float32(40.80262463805095) * t_0)))))))
	t_2 = sqrt(Float32(u1 / Float32(Float32(1.0) - u1)))
	t_3 = Float32(Float32(1.0) - Float32(Float32(0.25) * Float32(Float32(1.0) / u1)))
	t_4 = Float32(t_0 * t_3)
	tmp = Float32(0.0)
	if (Float32(t_2 * sin(Float32(Float32(6.28318530718) * u2))) <= Float32(2.499999936844688e-6))
		tmp = fma(u2, fma(Float32(6.28318530718), sqrt(u1), Float32(Float32(u2 * u2) * fma(Float32(-41.341702240407926), sqrt(u1), Float32(Float32(u2 * u2) * fma(Float32(-76.70585975309672), Float32(sqrt(u1) * Float32(u2 * u2)), Float32(Float32(81.6052492761019) * sqrt(u1))))))), Float32(Float32(u1 * u1) * fma(u1, fma(u1, Float32(u2 * fma(Float32(3.14159265359), t_4, Float32(Float32(u2 * u2) * fma(Float32(-20.670851120203963), t_4, Float32(Float32(u2 * u2) * fma(Float32(-38.35292987654836), Float32(t_0 * Float32(Float32(u2 * u2) * t_3)), Float32(Float32(40.80262463805095) * t_4))))))), t_1), t_1)));
	else
		tmp = Float32(fma(Float32(Float32(-81.6052492761019) * t_2), (u2 ^ Float32(-2.0)), fma(Float32(Float32(-6.28318530718) * t_2), (u2 ^ Float32(-6.0)), fma(Float32((u2 ^ Float32(-4.0)) * t_2), Float32(41.341702240407926), Float32(Float32(76.70585975309672) * t_2)))) * Float32(-(u2 ^ Float32(7.0))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (sqrt.f32 (/.f32 u1 (-.f32 #s(literal 1 binary32) u1))) (sin.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2))) < 2.49999994e-6

   1. Initial program 98.2%

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

   3. Taylor expanded in u2 around 0

      \[\leadsto \color{blue}{u2 \cdot \left(\frac{314159265359}{50000000000} \cdot \sqrt{\frac{u1}{1 - u1}} + {u2}^{2} \cdot \left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot \sqrt{\frac{u1}{1 - u1}} + {u2}^{2} \cdot \left(\frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot {u2}^{2}\right) + \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot \sqrt{\frac{u1}{1 - u1}}\right)\right)\right)} \]

   5. Applied rewrites 95.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-76.70585975309672, \left(u2 \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}, 81.6052492761019 \cdot \sqrt{\frac{u1}{1 - u1}}\right), u2 \cdot u2, -41.341702240407926 \cdot \sqrt{\frac{u1}{1 - u1}}\right), u2 \cdot u2, \sqrt{\frac{u1}{1 - u1}} \cdot 6.28318530718\right) \cdot u2} \]

   6. Taylor expanded in u2 around -inf

      \[\leadsto -1 \cdot \color{blue}{\left({u2}^{7} \cdot \left(\frac{-3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot \frac{1}{{u2}^{2}}\right) + \left(\frac{-314159265359}{50000000000} \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot \frac{1}{{u2}^{6}}\right) + \left(\frac{31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot \frac{1}{{u2}^{4}}\right) + \frac{302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot \sqrt{\frac{u1}{1 - u1}}\right)\right)\right)\right)} \]

   8. Applied rewrites 35.7%

      \[\leadsto -\mathsf{fma}\left(-81.6052492761019 \cdot \sqrt{\frac{u1}{1 - u1}}, {u2}^{-2}, \mathsf{fma}\left(-6.28318530718 \cdot \sqrt{\frac{u1}{1 - u1}}, {u2}^{-6}, \mathsf{fma}\left({u2}^{-4} \cdot \sqrt{\frac{u1}{1 - u1}}, 41.341702240407926, 76.70585975309672 \cdot \sqrt{\frac{u1}{1 - u1}}\right)\right)\right) \cdot {u2}^{7} \]

   9. Taylor expanded in u1 around 0

      \[\leadsto u2 \cdot \left(\frac{314159265359}{50000000000} \cdot \sqrt{u1} + {u2}^{2} \cdot \left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot \sqrt{u1} + {u2}^{2} \cdot \left(\frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot \left(\sqrt{u1} \cdot {u2}^{2}\right) + \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot \sqrt{u1}\right)\right)\right) + \color{blue}{{u1}^{2} \cdot \left(u1 \cdot \left(u1 \cdot \left(u2 \cdot \left(\frac{314159265359}{100000000000} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \left(1 - \frac{1}{4} \cdot \frac{1}{u1}\right)\right) + {u2}^{2} \cdot \left(\frac{-31006276680305942139213528068663279}{1500000000000000000000000000000000} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \left(1 - \frac{1}{4} \cdot \frac{1}{u1}\right)\right) + {u2}^{2} \cdot \left(\frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{7875000000000000000000000000000000000000000000000000000000000000000000000000000} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \left({u2}^{2} \cdot \left(1 - \frac{1}{4} \cdot \frac{1}{u1}\right)\right)\right) + \frac{3060196847853821555298148281676017575122444629042460390799}{75000000000000000000000000000000000000000000000000000000} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \left(1 - \frac{1}{4} \cdot \frac{1}{u1}\right)\right)\right)\right)\right)\right) + u2 \cdot \left(\frac{314159265359}{100000000000} \cdot \sqrt{\frac{1}{u1}} + {u2}^{2} \cdot \left(\frac{-31006276680305942139213528068663279}{1500000000000000000000000000000000} \cdot \sqrt{\frac{1}{u1}} + {u2}^{2} \cdot \left(\frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{7875000000000000000000000000000000000000000000000000000000000000000000000000000} \cdot \left(\sqrt{\frac{1}{u1}} \cdot {u2}^{2}\right) + \frac{3060196847853821555298148281676017575122444629042460390799}{75000000000000000000000000000000000000000000000000000000} \cdot \sqrt{\frac{1}{u1}}\right)\right)\right)\right) + u2 \cdot \left(\frac{314159265359}{100000000000} \cdot \sqrt{\frac{1}{u1}} + {u2}^{2} \cdot \left(\frac{-31006276680305942139213528068663279}{1500000000000000000000000000000000} \cdot \sqrt{\frac{1}{u1}} + {u2}^{2} \cdot \left(\frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{7875000000000000000000000000000000000000000000000000000000000000000000000000000} \cdot \left(\sqrt{\frac{1}{u1}} \cdot {u2}^{2}\right) + \frac{3060196847853821555298148281676017575122444629042460390799}{75000000000000000000000000000000000000000000000000000000} \cdot \sqrt{\frac{1}{u1}}\right)\right)\right)\right)} \]

   11. Applied rewrites 90.4%

       \[\leadsto \mathsf{fma}\left(u2, \color{blue}{\mathsf{fma}\left(6.28318530718, \sqrt{u1}, \left(u2 \cdot u2\right) \cdot \mathsf{fma}\left(-41.341702240407926, \sqrt{u1}, \left(u2 \cdot u2\right) \cdot \mathsf{fma}\left(-76.70585975309672, \sqrt{u1} \cdot \left(u2 \cdot u2\right), 81.6052492761019 \cdot \sqrt{u1}\right)\right)\right)}, \left(u1 \cdot u1\right) \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u2 \cdot \mathsf{fma}\left(3.14159265359, \frac{1}{\sqrt{u1}} \cdot \left(1 - 0.25 \cdot \frac{1}{u1}\right), \left(u2 \cdot u2\right) \cdot \mathsf{fma}\left(-20.670851120203963, \frac{1}{\sqrt{u1}} \cdot \left(1 - 0.25 \cdot \frac{1}{u1}\right), \left(u2 \cdot u2\right) \cdot \mathsf{fma}\left(-38.35292987654836, \frac{1}{\sqrt{u1}} \cdot \left(\left(u2 \cdot u2\right) \cdot \left(1 - 0.25 \cdot \frac{1}{u1}\right)\right), 40.80262463805095 \cdot \left(\frac{1}{\sqrt{u1}} \cdot \left(1 - 0.25 \cdot \frac{1}{u1}\right)\right)\right)\right)\right), u2 \cdot \mathsf{fma}\left(3.14159265359, \frac{1}{\sqrt{u1}}, \left(u2 \cdot u2\right) \cdot \mathsf{fma}\left(-20.670851120203963, \frac{1}{\sqrt{u1}}, \left(u2 \cdot u2\right) \cdot \mathsf{fma}\left(-38.35292987654836, \frac{1}{\sqrt{u1}} \cdot \left(u2 \cdot u2\right), 40.80262463805095 \cdot \frac{1}{\sqrt{u1}}\right)\right)\right)\right), u2 \cdot \mathsf{fma}\left(3.14159265359, \frac{1}{\sqrt{u1}}, \left(u2 \cdot u2\right) \cdot \mathsf{fma}\left(-20.670851120203963, \frac{1}{\sqrt{u1}}, \left(u2 \cdot u2\right) \cdot \mathsf{fma}\left(-38.35292987654836, \frac{1}{\sqrt{u1}} \cdot \left(u2 \cdot u2\right), 40.80262463805095 \cdot \frac{1}{\sqrt{u1}}\right)\right)\right)\right)\right) \]

   if 2.49999994e-6 < (*.f32 (sqrt.f32 (/.f32 u1 (-.f32 #s(literal 1 binary32) u1))) (sin.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2)))

   1. Initial program 98.6%

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

   3. Taylor expanded in u2 around 0

      \[\leadsto \color{blue}{u2 \cdot \left(\frac{314159265359}{50000000000} \cdot \sqrt{\frac{u1}{1 - u1}} + {u2}^{2} \cdot \left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot \sqrt{\frac{u1}{1 - u1}} + {u2}^{2} \cdot \left(\frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot {u2}^{2}\right) + \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot \sqrt{\frac{u1}{1 - u1}}\right)\right)\right)} \]

   5. Applied rewrites 94.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-76.70585975309672, \left(u2 \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}, 81.6052492761019 \cdot \sqrt{\frac{u1}{1 - u1}}\right), u2 \cdot u2, -41.341702240407926 \cdot \sqrt{\frac{u1}{1 - u1}}\right), u2 \cdot u2, \sqrt{\frac{u1}{1 - u1}} \cdot 6.28318530718\right) \cdot u2} \]

   6. Taylor expanded in u2 around -inf

      \[\leadsto -1 \cdot \color{blue}{\left({u2}^{7} \cdot \left(\frac{-3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot \frac{1}{{u2}^{2}}\right) + \left(\frac{-314159265359}{50000000000} \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot \frac{1}{{u2}^{6}}\right) + \left(\frac{31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot \frac{1}{{u2}^{4}}\right) + \frac{302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot \sqrt{\frac{u1}{1 - u1}}\right)\right)\right)\right)} \]

   8. Applied rewrites 93.3%

      \[\leadsto -\mathsf{fma}\left(-81.6052492761019 \cdot \sqrt{\frac{u1}{1 - u1}}, {u2}^{-2}, \mathsf{fma}\left(-6.28318530718 \cdot \sqrt{\frac{u1}{1 - u1}}, {u2}^{-6}, \mathsf{fma}\left({u2}^{-4} \cdot \sqrt{\frac{u1}{1 - u1}}, 41.341702240407926, 76.70585975309672 \cdot \sqrt{\frac{u1}{1 - u1}}\right)\right)\right) \cdot {u2}^{7} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \leq 2.499999936844688 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(u2, \mathsf{fma}\left(6.28318530718, \sqrt{u1}, \left(u2 \cdot u2\right) \cdot \mathsf{fma}\left(-41.341702240407926, \sqrt{u1}, \left(u2 \cdot u2\right) \cdot \mathsf{fma}\left(-76.70585975309672, \sqrt{u1} \cdot \left(u2 \cdot u2\right), 81.6052492761019 \cdot \sqrt{u1}\right)\right)\right), \left(u1 \cdot u1\right) \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u2 \cdot \mathsf{fma}\left(3.14159265359, \frac{1}{\sqrt{u1}} \cdot \left(1 - 0.25 \cdot \frac{1}{u1}\right), \left(u2 \cdot u2\right) \cdot \mathsf{fma}\left(-20.670851120203963, \frac{1}{\sqrt{u1}} \cdot \left(1 - 0.25 \cdot \frac{1}{u1}\right), \left(u2 \cdot u2\right) \cdot \mathsf{fma}\left(-38.35292987654836, \frac{1}{\sqrt{u1}} \cdot \left(\left(u2 \cdot u2\right) \cdot \left(1 - 0.25 \cdot \frac{1}{u1}\right)\right), 40.80262463805095 \cdot \left(\frac{1}{\sqrt{u1}} \cdot \left(1 - 0.25 \cdot \frac{1}{u1}\right)\right)\right)\right)\right), u2 \cdot \mathsf{fma}\left(3.14159265359, \frac{1}{\sqrt{u1}}, \left(u2 \cdot u2\right) \cdot \mathsf{fma}\left(-20.670851120203963, \frac{1}{\sqrt{u1}}, \left(u2 \cdot u2\right) \cdot \mathsf{fma}\left(-38.35292987654836, \frac{1}{\sqrt{u1}} \cdot \left(u2 \cdot u2\right), 40.80262463805095 \cdot \frac{1}{\sqrt{u1}}\right)\right)\right)\right), u2 \cdot \mathsf{fma}\left(3.14159265359, \frac{1}{\sqrt{u1}}, \left(u2 \cdot u2\right) \cdot \mathsf{fma}\left(-20.670851120203963, \frac{1}{\sqrt{u1}}, \left(u2 \cdot u2\right) \cdot \mathsf{fma}\left(-38.35292987654836, \frac{1}{\sqrt{u1}} \cdot \left(u2 \cdot u2\right), 40.80262463805095 \cdot \frac{1}{\sqrt{u1}}\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-81.6052492761019 \cdot \sqrt{\frac{u1}{1 - u1}}, {u2}^{-2}, \mathsf{fma}\left(-6.28318530718 \cdot \sqrt{\frac{u1}{1 - u1}}, {u2}^{-6}, \mathsf{fma}\left({u2}^{-4} \cdot \sqrt{\frac{u1}{1 - u1}}, 41.341702240407926, 76.70585975309672 \cdot \sqrt{\frac{u1}{1 - u1}}\right)\right)\right) \cdot \left(-{u2}^{7}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 58.5% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{u1}{1 - u1}}\\ \mathsf{fma}\left(-81.6052492761019 \cdot t\_0, {u2}^{-2}, \mathsf{fma}\left(-6.28318530718 \cdot t\_0, {u2}^{-6}, \mathsf{fma}\left({u2}^{-4} \cdot t\_0, 41.341702240407926, 76.70585975309672 \cdot t\_0\right)\right)\right) \cdot \left(-{u2}^{7}\right) \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (/ u1 (- 1.0 u1)))))
   (*
    (fma
     (* -81.6052492761019 t_0)
     (pow u2 -2.0)
     (fma
      (* -6.28318530718 t_0)
      (pow u2 -6.0)
      (fma
       (* (pow u2 -4.0) t_0)
       41.341702240407926
       (* 76.70585975309672 t_0))))
    (- (pow u2 7.0)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sqrtf((u1 / (1.0f - u1)));
	return fmaf((-81.6052492761019f * t_0), powf(u2, -2.0f), fmaf((-6.28318530718f * t_0), powf(u2, -6.0f), fmaf((powf(u2, -4.0f) * t_0), 41.341702240407926f, (76.70585975309672f * t_0)))) * -powf(u2, 7.0f);
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sqrt(Float32(u1 / Float32(Float32(1.0) - u1)))
	return Float32(fma(Float32(Float32(-81.6052492761019) * t_0), (u2 ^ Float32(-2.0)), fma(Float32(Float32(-6.28318530718) * t_0), (u2 ^ Float32(-6.0)), fma(Float32((u2 ^ Float32(-4.0)) * t_0), Float32(41.341702240407926), Float32(Float32(76.70585975309672) * t_0)))) * Float32(-(u2 ^ Float32(7.0))))
end

Derivation

1. Initial program 98.3%

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

3. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{u2 \cdot \left(\frac{314159265359}{50000000000} \cdot \sqrt{\frac{u1}{1 - u1}} + {u2}^{2} \cdot \left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot \sqrt{\frac{u1}{1 - u1}} + {u2}^{2} \cdot \left(\frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot {u2}^{2}\right) + \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot \sqrt{\frac{u1}{1 - u1}}\right)\right)\right)} \]

5. Applied rewrites 94.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-76.70585975309672, \left(u2 \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}, 81.6052492761019 \cdot \sqrt{\frac{u1}{1 - u1}}\right), u2 \cdot u2, -41.341702240407926 \cdot \sqrt{\frac{u1}{1 - u1}}\right), u2 \cdot u2, \sqrt{\frac{u1}{1 - u1}} \cdot 6.28318530718\right) \cdot u2} \]

6. Taylor expanded in u2 around -inf

   \[\leadsto -1 \cdot \color{blue}{\left({u2}^{7} \cdot \left(\frac{-3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot \frac{1}{{u2}^{2}}\right) + \left(\frac{-314159265359}{50000000000} \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot \frac{1}{{u2}^{6}}\right) + \left(\frac{31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot \frac{1}{{u2}^{4}}\right) + \frac{302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot \sqrt{\frac{u1}{1 - u1}}\right)\right)\right)\right)} \]

8. Applied rewrites 59.1%

   \[\leadsto -\mathsf{fma}\left(-81.6052492761019 \cdot \sqrt{\frac{u1}{1 - u1}}, {u2}^{-2}, \mathsf{fma}\left(-6.28318530718 \cdot \sqrt{\frac{u1}{1 - u1}}, {u2}^{-6}, \mathsf{fma}\left({u2}^{-4} \cdot \sqrt{\frac{u1}{1 - u1}}, 41.341702240407926, 76.70585975309672 \cdot \sqrt{\frac{u1}{1 - u1}}\right)\right)\right) \cdot {u2}^{7} \]

9. Final simplification 59.1%

   \[\leadsto \mathsf{fma}\left(-81.6052492761019 \cdot \sqrt{\frac{u1}{1 - u1}}, {u2}^{-2}, \mathsf{fma}\left(-6.28318530718 \cdot \sqrt{\frac{u1}{1 - u1}}, {u2}^{-6}, \mathsf{fma}\left({u2}^{-4} \cdot \sqrt{\frac{u1}{1 - u1}}, 41.341702240407926, 76.70585975309672 \cdot \sqrt{\frac{u1}{1 - u1}}\right)\right)\right) \cdot \left(-{u2}^{7}\right) \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025057 
(FPCore (cosTheta_i u1 u2)
  :name "Trowbridge-Reitz Sample, near normal, slope_y"
  :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))) (sin (* 6.28318530718 u2))))