Trowbridge-Reitz Sample, near normal, slope_y

Percentage Accurate: 98.3% → 98.2%

Time: 5.5s

Alternatives: 15

Speedup: 1.0×

Specification

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

Math

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

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: 98.3% 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: 98.2% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.3%

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

2. Taylor expanded in u1 around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. lower-/.f32 98.2

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

4. Applied rewrites 98.2%

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

Alternative 2: 97.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{u1}{1 - u1}}\\ \mathbf{if}\;u1 \leq 0.007000000216066837:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(1 + u1, u1, 1\right) \cdot u1} \cdot \sin \left(6.28318530718 \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0 \cdot 6.28318530718, u2, \left(\mathsf{fma}\left(t\_0 \cdot \mathsf{fma}\left(u2 \cdot u2, -76.70585975309672, 81.6052492761019\right), u2 \cdot u2, -41.341702240407926 \cdot t\_0\right) \cdot \left(u2 \cdot u2\right)\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (/ u1 (- 1.0 u1)))))
   (if (<= u1 0.007000000216066837)
     (* (sqrt (* (fma (+ 1.0 u1) u1 1.0) u1)) (sin (* 6.28318530718 u2)))
     (fma
      (* t_0 6.28318530718)
      u2
      (*
       (*
        (fma
         (* t_0 (fma (* u2 u2) -76.70585975309672 81.6052492761019))
         (* u2 u2)
         (* -41.341702240407926 t_0))
        (* u2 u2))
       u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sqrtf((u1 / (1.0f - u1)));
	float tmp;
	if (u1 <= 0.007000000216066837f) {
		tmp = sqrtf((fmaf((1.0f + u1), u1, 1.0f) * u1)) * sinf((6.28318530718f * u2));
	} else {
		tmp = fmaf((t_0 * 6.28318530718f), u2, ((fmaf((t_0 * fmaf((u2 * u2), -76.70585975309672f, 81.6052492761019f)), (u2 * u2), (-41.341702240407926f * t_0)) * (u2 * u2)) * u2));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sqrt(Float32(u1 / Float32(Float32(1.0) - u1)))
	tmp = Float32(0.0)
	if (u1 <= Float32(0.007000000216066837))
		tmp = Float32(sqrt(Float32(fma(Float32(Float32(1.0) + u1), u1, Float32(1.0)) * u1)) * sin(Float32(Float32(6.28318530718) * u2)));
	else
		tmp = fma(Float32(t_0 * Float32(6.28318530718)), u2, Float32(Float32(fma(Float32(t_0 * fma(Float32(u2 * u2), Float32(-76.70585975309672), Float32(81.6052492761019))), Float32(u2 * u2), Float32(Float32(-41.341702240407926) * t_0)) * Float32(u2 * u2)) * u2));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u1 < 0.00700000022

   1. Initial program 98.3%

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

   2. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 N/A

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

      6. lower-+.f32 98.3

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

   4. Applied rewrites 98.3%

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

   if 0.00700000022 < u1

   1. Initial program 98.3%

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

   2. Taylor expanded in u1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 98.0

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

   4. Applied rewrites 98.0%

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

      1. lift--.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. flip3-- N/A

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

      4. lower-/.f32 N/A

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

      5. metadata-eval N/A

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

      6. lower--.f32 N/A

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

      7. lower-pow.f32 N/A

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

      8. lift-/.f32 N/A

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

      9. lower-fma.f32 N/A

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

      10. lift-/.f32 N/A

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

      11. lift-/.f32 N/A

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

      12. metadata-eval N/A

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

      13. lower-+.f32 N/A

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

      14. lower-*.f32 N/A

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

      15. lift-/.f32 97.9

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

   6. Applied rewrites 97.9%

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

   7. Taylor expanded in u2 around 0

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

   9. Applied rewrites 94.4%

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

   11. Applied rewrites 94.5%

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

Alternative 3: 98.3% 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

Derivation

1. Initial program 98.3%

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

Alternative 4: 94.2% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.3%

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

2. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

4. Applied rewrites 94.1%

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

6. Applied rewrites 94.2%

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

Alternative 5: 94.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * Float32(Float32(Float32(Float32(Float32(Float32(fma(Float32(u2 * u2), Float32(-76.70585975309672), Float32(81.6052492761019)) * u2) * u2) - Float32(41.341702240407926)) * Float32(u2 * u2)) + 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. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

4. Applied rewrites 94.1%

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

6. Applied rewrites 94.1%

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

Alternative 6: 94.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * Float32(fma(Float32(Float32(Float32(fma(Float32(u2 * u2), Float32(-76.70585975309672), Float32(81.6052492761019)) * u2) * u2) - Float32(41.341702240407926)), Float32(u2 * u2), 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. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

4. Applied rewrites 94.1%

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

6. Applied rewrites 94.1%

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

Alternative 7: 94.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * Float32(fma(Float32(Float32(Float32(Float32(fma(Float32(u2 * u2), Float32(-76.70585975309672), Float32(81.6052492761019)) * u2) * u2) - Float32(41.341702240407926)) * u2), u2, 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. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

4. Applied rewrites 94.1%

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

6. Applied rewrites 94.1%

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

Alternative 8: 92.0% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * Float32(fma(Float32(Float32(Float32(81.6052492761019) * Float32(u2 * u2)) - Float32(41.341702240407926)), Float32(u2 * u2), 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. Taylor expanded in u2 around 0

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\left(u2 \cdot \left(\frac{314159265359}{50000000000} + {u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)\right)} \]
3. Step-by-step derivation

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(\mathsf{fma}\left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}, {u2}^{2}, \frac{314159265359}{50000000000}\right) \cdot u2\right) \]

   6. lower--.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(\mathsf{fma}\left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}, {u2}^{2}, \frac{314159265359}{50000000000}\right) \cdot u2\right) \]

   7. lower-*.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(\mathsf{fma}\left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}, {u2}^{2}, \frac{314159265359}{50000000000}\right) \cdot u2\right) \]

   8. unpow2 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(\mathsf{fma}\left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot \left(u2 \cdot u2\right) - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}, {u2}^{2}, \frac{314159265359}{50000000000}\right) \cdot u2\right) \]

   9. lower-*.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(\mathsf{fma}\left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot \left(u2 \cdot u2\right) - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}, {u2}^{2}, \frac{314159265359}{50000000000}\right) \cdot u2\right) \]

   10. unpow2 N/A

       \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(\mathsf{fma}\left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot \left(u2 \cdot u2\right) - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}, u2 \cdot u2, \frac{314159265359}{50000000000}\right) \cdot u2\right) \]

   11. lower-*.f32 92.0

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

4. Applied rewrites 92.0%

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

Alternative 9: 87.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u1 \leq 0.001560000004246831:\\ \;\;\;\;\sqrt{\left(1 + u1\right) \cdot u1} \cdot \left(\mathsf{fma}\left(u2 \cdot u2, -41.341702240407926, 6.28318530718\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot 6.28318530718\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u1 0.001560000004246831)
   (*
    (sqrt (* (+ 1.0 u1) u1))
    (* (fma (* u2 u2) -41.341702240407926 6.28318530718) u2))
   (* (sqrt (/ u1 (- 1.0 u1))) (* u2 6.28318530718))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (u1 <= 0.001560000004246831f) {
		tmp = sqrtf(((1.0f + u1) * u1)) * (fmaf((u2 * u2), -41.341702240407926f, 6.28318530718f) * u2);
	} else {
		tmp = sqrtf((u1 / (1.0f - u1))) * (u2 * 6.28318530718f);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (u1 <= Float32(0.001560000004246831))
		tmp = Float32(sqrt(Float32(Float32(Float32(1.0) + u1) * u1)) * Float32(fma(Float32(u2 * u2), Float32(-41.341702240407926), Float32(6.28318530718)) * u2));
	else
		tmp = Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * Float32(u2 * Float32(6.28318530718)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u1 < 0.00156

   1. Initial program 98.3%

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

   2. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower-+.f32 97.9

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

   4. Applied rewrites 97.9%

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

   5. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{\left(1 + u1\right) \cdot u1} \cdot \color{blue}{\left(u2 \cdot \left(\frac{314159265359}{50000000000} + \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2}\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 N/A

         \[\leadsto \sqrt{\left(1 + u1\right) \cdot u1} \cdot \left(\mathsf{fma}\left({u2}^{2}, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right) \cdot u2\right) \]

      6. pow2 N/A

         \[\leadsto \sqrt{\left(1 + u1\right) \cdot u1} \cdot \left(\mathsf{fma}\left(u2 \cdot u2, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right) \cdot u2\right) \]

      7. lift-*.f32 89.1

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

   7. Applied rewrites 89.1%

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

   if 0.00156 < u1

   1. Initial program 98.3%

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

   2. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 82.3

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

   4. Applied rewrites 82.3%

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

Alternative 10: 89.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{u1}{1 - u1}} \cdot \left(\mathsf{fma}\left(u2 \cdot u2, -41.341702240407926, 6.28318530718\right) \cdot u2\right) \end{array} \]

FPCore

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

C

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

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * Float32(fma(Float32(u2 * u2), Float32(-41.341702240407926), 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. Taylor expanded in u2 around 0

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\left(u2 \cdot \left(\frac{314159265359}{50000000000} + \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2}\right)\right)} \]
3. Step-by-step derivation

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(\mathsf{fma}\left({u2}^{2}, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right) \cdot u2\right) \]

   6. unpow2 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(\mathsf{fma}\left(u2 \cdot u2, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right) \cdot u2\right) \]

   7. lower-*.f32 89.6

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

4. Applied rewrites 89.6%

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

Alternative 11: 82.1% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((u1 / (single(1.0) - u1))) * (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. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 82.1

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

4. Applied rewrites 82.1%

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

Alternative 12: 82.1% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = (sqrt((u1 / (single(1.0) - u1))) * 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. Taylor expanded in u2 around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. lift--.f32 N/A

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

   7. lift-sqrt.f32 82.1

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

4. Applied rewrites 82.1%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-sqrt.f32 N/A

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

   4. lift--.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. *-commutative N/A

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. lower-*.f32 N/A

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

   11. lift-/.f32 N/A

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

   12. lift--.f32 N/A

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

   13. lift-sqrt.f32 82.1

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

6. Applied rewrites 82.1%

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

Alternative 13: 82.1% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

2. Taylor expanded in u2 around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. lift--.f32 N/A

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

   7. lift-sqrt.f32 82.1

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

4. Applied rewrites 82.1%

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

Alternative 14: 73.4% accurate, 4.7× speedup

Math

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

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (* (+ 1.0 u1) u1)) (* u2 6.28318530718)))

C

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

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(Float32(Float32(1.0) + u1) * u1)) * Float32(u2 * Float32(6.28318530718)))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt(((single(1.0) + u1) * u1)) * (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. Taylor expanded in u1 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower-+.f32 86.2

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

4. Applied rewrites 86.2%

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

5. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 73.4

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

7. Applied rewrites 73.4%

   \[\leadsto \sqrt{\left(1 + u1\right) \cdot u1} \cdot \color{blue}{\left(u2 \cdot 6.28318530718\right)} \]
8. Add Preprocessing

Alternative 15: 65.1% accurate, 6.4× speedup

Math

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

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt u1) (* u2 6.28318530718)))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(u1) * (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) * (u2 * 6.28318530718e0)
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(u1) * Float32(u2 * Float32(6.28318530718)))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt(u1) * (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. Taylor expanded in u2 around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. lift--.f32 N/A

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

   7. lift-sqrt.f32 82.1

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

4. Applied rewrites 82.1%

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

5. Taylor expanded in u1 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-sqrt.f32 65.1

      \[\leadsto \left(\sqrt{u1} \cdot u2\right) \cdot 6.28318530718 \]

7. Applied rewrites 65.1%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-sqrt.f32 N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. lift-sqrt.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 65.1

      \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot 6.28318530718\right) \]

9. Applied rewrites 65.1%

   \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \color{blue}{6.28318530718}\right) \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025100 
(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))))