Trowbridge-Reitz Sample, near normal, slope_x

Percentage Accurate: 99.0% → 99.2%

Time: 5.2s

Alternatives: 17

Speedup: 0.9×

Specification

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

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Julia

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

MATLAB

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

Initial Program: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Julia

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

MATLAB

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

Alternative 1: 99.2% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(sin(Float32(Float32(0.5) * Float32(pi))) * cos(Float32(u2 * Float32(-6.28318530718))))
	t_1 = Float32(cos(Float32(Float32(-0.5) * Float32(pi))) * sin(Float32(u2 * Float32(-6.28318530718))))
	return Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * Float32(Float32((t_1 ^ Float32(3.0)) + (t_0 ^ Float32(3.0))) / fma(t_1, t_1, Float32(Float32(t_0 * t_0) - Float32(t_1 * t_0)))))
end

Derivation

1. Initial program 99.0%

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

   1. lift-cos.f32 N/A

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

   2. cos-neg-rev N/A

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

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

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

   4. lower-sin.f32 N/A

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

   5. lift-*.f32 N/A

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

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

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

   7. lower-fma.f32 N/A

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

   8. metadata-eval N/A

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

   9. mult-flip N/A

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

   10. metadata-eval N/A

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

   11. lower-*.f32 N/A

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

   12. lower-PI.f32 99.2

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

3. Applied rewrites 99.2%

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

5. Applied rewrites 99.0%

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

Alternative 2: 99.0% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.0%

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

   1. lift-cos.f32 N/A

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

   2. cos-neg-rev N/A

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

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

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

   4. lower-sin.f32 N/A

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

   5. lift-*.f32 N/A

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

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

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

   7. lower-fma.f32 N/A

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

   8. metadata-eval N/A

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

   9. mult-flip N/A

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

   10. metadata-eval N/A

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

   11. lower-*.f32 N/A

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

   12. lower-PI.f32 99.2

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

3. Applied rewrites 99.2%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\sin \left(\mathsf{fma}\left(-6.28318530718, u2, \pi \cdot 0.5\right)\right)} \]
4. Step-by-step derivation

   1. lift-sin.f32 N/A

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

   2. lift-fma.f32 N/A

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

   3. sin-sum N/A

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

   4. lower-fma.f32 N/A

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

   5. lower-sin.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. cos-neg-rev N/A

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

   9. lower-cos.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. *-commutative N/A

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

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

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

   13. lower-*.f32 N/A

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

   14. metadata-eval N/A

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

   15. *-commutative N/A

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

   16. metadata-eval N/A

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

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

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

   18. lift-*.f32 N/A

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

   19. cos-neg-rev N/A

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

   20. lift-cos.f32 N/A

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

   21. lower-*.f32 N/A

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

5. Applied rewrites 99.0%

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

Alternative 3: 99.0% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.0%

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

   1. lift-cos.f32 N/A

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

   2. cos-neg-rev N/A

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

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

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

   4. lower-sin.f32 N/A

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

   5. lift-*.f32 N/A

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

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

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

   7. lower-fma.f32 N/A

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

   8. metadata-eval N/A

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

   9. mult-flip N/A

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

   10. metadata-eval N/A

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

   11. lower-*.f32 N/A

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

   12. lower-PI.f32 99.2

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

3. Applied rewrites 99.2%

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

Alternative 4: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

Alternative 5: 97.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{u1}{1 - u1}}\\ \mathbf{if}\;u2 \leq 0.03099999949336052:\\ \;\;\;\;\mathsf{fma}\left(t\_0 \cdot \left(\mathsf{fma}\left(u2 \cdot u2, 64.93939402268539, -19.739208802181317\right) \cdot u2\right), u2, t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(u1 - -1\right) \cdot u1} \cdot \cos \left(6.28318530718 \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (/ u1 (- 1.0 u1)))))
   (if (<= u2 0.03099999949336052)
     (fma
      (* t_0 (* (fma (* u2 u2) 64.93939402268539 -19.739208802181317) u2))
      u2
      t_0)
     (* (sqrt (* (- u1 -1.0) u1)) (cos (* 6.28318530718 u2))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sqrtf((u1 / (1.0f - u1)));
	float tmp;
	if (u2 <= 0.03099999949336052f) {
		tmp = fmaf((t_0 * (fmaf((u2 * u2), 64.93939402268539f, -19.739208802181317f) * u2)), u2, t_0);
	} else {
		tmp = sqrtf(((u1 - -1.0f) * u1)) * cosf((6.28318530718f * u2));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sqrt(Float32(u1 / Float32(Float32(1.0) - u1)))
	tmp = Float32(0.0)
	if (u2 <= Float32(0.03099999949336052))
		tmp = fma(Float32(t_0 * Float32(fma(Float32(u2 * u2), Float32(64.93939402268539), Float32(-19.739208802181317)) * u2)), u2, t_0);
	else
		tmp = Float32(sqrt(Float32(Float32(u1 - Float32(-1.0)) * u1)) * cos(Float32(Float32(6.28318530718) * u2)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u2 < 0.0309999995

   1. Initial program 99.0%

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

   2. Taylor expanded in u2 around 0

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

      1. lower-+.f32 N/A

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

      2. lower-sqrt.f32 N/A

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

      3. lower-/.f32 N/A

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

      4. lower--.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lower-fma.f32 N/A

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

      8. lower-sqrt.f32 N/A

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

      9. lower-/.f32 N/A

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

      10. lower--.f32 N/A

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

      11. lower-*.f32 N/A

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

      12. lower-*.f32 N/A

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

   4. Applied rewrites 91.4%

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

      1. lift-+.f32 N/A

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

      2. lift-sqrt.f32 N/A

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

      3. lift-/.f32 N/A

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

      4. mult-flip N/A

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

      5. metadata-eval N/A

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

      6. lift--.f32 N/A

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

      7. sub-negate-rev N/A

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

      8. lift--.f32 N/A

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

      9. frac-2neg N/A

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

      10. lift-/.f32 N/A

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

      11. sqrt-unprod N/A

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

      12. lift-sqrt.f32 N/A

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

      13. lift-sqrt.f32 N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f32 91.0

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

   6. Applied rewrites 91.0%

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

      1. lift-fma.f32 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lift-sqrt.f32 N/A

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

      8. lift-sqrt.f32 N/A

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

      9. sqrt-prod N/A

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

      10. lift-/.f32 N/A

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

      11. frac-2neg N/A

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

      12. metadata-eval N/A

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

      13. lift--.f32 N/A

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

      14. sub-negate-rev N/A

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

      15. lift--.f32 N/A

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

      16. mult-flip N/A

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

      17. lift-/.f32 N/A

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

      18. lift-sqrt.f32 N/A

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

   8. Applied rewrites 91.4%

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

   if 0.0309999995 < u2

   1. Initial program 99.0%

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

   2. Taylor expanded in u1 around 0

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

      1. lower-*.f32 N/A

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

      2. lower-+.f32 86.6

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

   4. Applied rewrites 86.6%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f32 86.6

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

      4. lift-+.f32 N/A

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

      5. +-commutative N/A

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

      6. add-flip N/A

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

      7. metadata-eval N/A

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

      8. lower--.f32 86.6

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

   6. Applied rewrites 86.6%

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

Alternative 6: 97.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{u1}{1 - u1}}\\ \mathbf{if}\;u2 \leq 0.03099999949336052:\\ \;\;\;\;\mathsf{fma}\left(t\_0 \cdot \left(\mathsf{fma}\left(u2 \cdot u2, 64.93939402268539, -19.739208802181317\right) \cdot u2\right), u2, t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(u1, u1, u1\right)} \cdot \cos \left(u2 \cdot -6.28318530718\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (/ u1 (- 1.0 u1)))))
   (if (<= u2 0.03099999949336052)
     (fma
      (* t_0 (* (fma (* u2 u2) 64.93939402268539 -19.739208802181317) u2))
      u2
      t_0)
     (* (sqrt (fma u1 u1 u1)) (cos (* u2 -6.28318530718))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sqrtf((u1 / (1.0f - u1)));
	float tmp;
	if (u2 <= 0.03099999949336052f) {
		tmp = fmaf((t_0 * (fmaf((u2 * u2), 64.93939402268539f, -19.739208802181317f) * u2)), u2, t_0);
	} else {
		tmp = sqrtf(fmaf(u1, u1, u1)) * cosf((u2 * -6.28318530718f));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sqrt(Float32(u1 / Float32(Float32(1.0) - u1)))
	tmp = Float32(0.0)
	if (u2 <= Float32(0.03099999949336052))
		tmp = fma(Float32(t_0 * Float32(fma(Float32(u2 * u2), Float32(64.93939402268539), Float32(-19.739208802181317)) * u2)), u2, t_0);
	else
		tmp = Float32(sqrt(fma(u1, u1, u1)) * cos(Float32(u2 * Float32(-6.28318530718))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u2 < 0.0309999995

   1. Initial program 99.0%

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

   2. Taylor expanded in u2 around 0

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

      1. lower-+.f32 N/A

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

      2. lower-sqrt.f32 N/A

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

      3. lower-/.f32 N/A

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

      4. lower--.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lower-fma.f32 N/A

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

      8. lower-sqrt.f32 N/A

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

      9. lower-/.f32 N/A

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

      10. lower--.f32 N/A

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

      11. lower-*.f32 N/A

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

      12. lower-*.f32 N/A

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

   4. Applied rewrites 91.4%

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

      1. lift-+.f32 N/A

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

      2. lift-sqrt.f32 N/A

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

      3. lift-/.f32 N/A

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

      4. mult-flip N/A

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

      5. metadata-eval N/A

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

      6. lift--.f32 N/A

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

      7. sub-negate-rev N/A

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

      8. lift--.f32 N/A

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

      9. frac-2neg N/A

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

      10. lift-/.f32 N/A

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

      11. sqrt-unprod N/A

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

      12. lift-sqrt.f32 N/A

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

      13. lift-sqrt.f32 N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f32 91.0

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

   6. Applied rewrites 91.0%

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

      1. lift-fma.f32 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lift-sqrt.f32 N/A

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

      8. lift-sqrt.f32 N/A

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

      9. sqrt-prod N/A

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

      10. lift-/.f32 N/A

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

      11. frac-2neg N/A

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

      12. metadata-eval N/A

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

      13. lift--.f32 N/A

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

      14. sub-negate-rev N/A

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

      15. lift--.f32 N/A

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

      16. mult-flip N/A

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

      17. lift-/.f32 N/A

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

      18. lift-sqrt.f32 N/A

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

   8. Applied rewrites 91.4%

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

   if 0.0309999995 < u2

   1. Initial program 99.0%

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

   2. Taylor expanded in u1 around 0

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

      1. lower-*.f32 N/A

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

      2. lower-+.f32 86.6

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

   4. Applied rewrites 86.6%

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

      1. lift-*.f32 N/A

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

      2. lift-+.f32 N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lft-identity N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f32 86.6

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

      7. lower-fma.f32 N/A

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

      8. lower-fma.f32 N/A

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

      9. lower-fma.f32 N/A

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

      10. lower-fma.f32 N/A

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

      11. lower-fma.f32 N/A

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

      12. lower-fma.f32 N/A

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

      13. lower-fma.f32 N/A

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

      14. lower-fma.f32 N/A

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

   6. Applied rewrites 86.6%

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

Alternative 7: 95.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{u1}{1 - u1}}\\ \mathbf{if}\;\cos \left(6.28318530718 \cdot u2\right) \leq 0.9599999785423279:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(\mathsf{fma}\left(-6.28318530718, u2, \pi \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0 \cdot \mathsf{fma}\left(64.93939402268539, u2 \cdot u2, -19.739208802181317\right), u2 \cdot u2, t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (/ u1 (- 1.0 u1)))))
   (if (<= (cos (* 6.28318530718 u2)) 0.9599999785423279)
     (* (sqrt u1) (sin (fma -6.28318530718 u2 (* PI 0.5))))
     (fma
      (* t_0 (fma 64.93939402268539 (* u2 u2) -19.739208802181317))
      (* u2 u2)
      t_0))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sqrtf((u1 / (1.0f - u1)));
	float tmp;
	if (cosf((6.28318530718f * u2)) <= 0.9599999785423279f) {
		tmp = sqrtf(u1) * sinf(fmaf(-6.28318530718f, u2, (((float) M_PI) * 0.5f)));
	} else {
		tmp = fmaf((t_0 * fmaf(64.93939402268539f, (u2 * u2), -19.739208802181317f)), (u2 * u2), t_0);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sqrt(Float32(u1 / Float32(Float32(1.0) - u1)))
	tmp = Float32(0.0)
	if (cos(Float32(Float32(6.28318530718) * u2)) <= Float32(0.9599999785423279))
		tmp = Float32(sqrt(u1) * sin(fma(Float32(-6.28318530718), u2, Float32(Float32(pi) * Float32(0.5)))));
	else
		tmp = fma(Float32(t_0 * fma(Float32(64.93939402268539), Float32(u2 * u2), Float32(-19.739208802181317))), Float32(u2 * u2), t_0);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2)) < 0.959999979

   1. Initial program 99.0%

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

      1. lift-cos.f32 N/A

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

      2. cos-neg-rev N/A

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

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

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

      4. lower-sin.f32 N/A

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

      5. lift-*.f32 N/A

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

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

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

      7. lower-fma.f32 N/A

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

      8. metadata-eval N/A

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

      9. mult-flip N/A

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

      10. metadata-eval N/A

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

      11. lower-*.f32 N/A

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

      12. lower-PI.f32 99.2

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

   3. Applied rewrites 99.2%

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

   4. Taylor expanded in u1 around 0

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

   6. Applied rewrites 74.4%

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

   if 0.959999979 < (cos.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2))

   1. Initial program 99.0%

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

   2. Taylor expanded in u2 around 0

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

      1. lower-+.f32 N/A

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

      2. lower-sqrt.f32 N/A

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

      3. lower-/.f32 N/A

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

      4. lower--.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lower-fma.f32 N/A

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

      8. lower-sqrt.f32 N/A

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

      9. lower-/.f32 N/A

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

      10. lower--.f32 N/A

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

      11. lower-*.f32 N/A

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

      12. lower-*.f32 N/A

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

   4. Applied rewrites 91.4%

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

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 91.4

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

   6. Applied rewrites 91.4%

      \[\leadsto \mathsf{fma}\left(\sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(64.93939402268539, u2 \cdot u2, -19.739208802181317\right), \color{blue}{u2 \cdot u2}, \sqrt{\frac{u1}{1 - u1}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 95.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(6.28318530718 \cdot u2\right)\\ t_1 := \sqrt{\frac{u1}{1 - u1}}\\ \mathbf{if}\;t\_0 \leq 0.9599999785423279:\\ \;\;\;\;\sqrt{u1} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_1 \cdot \mathsf{fma}\left(64.93939402268539, u2 \cdot u2, -19.739208802181317\right), u2 \cdot u2, t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* 6.28318530718 u2))) (t_1 (sqrt (/ u1 (- 1.0 u1)))))
   (if (<= t_0 0.9599999785423279)
     (* (sqrt u1) t_0)
     (fma
      (* t_1 (fma 64.93939402268539 (* u2 u2) -19.739208802181317))
      (* u2 u2)
      t_1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = cosf((6.28318530718f * u2));
	float t_1 = sqrtf((u1 / (1.0f - u1)));
	float tmp;
	if (t_0 <= 0.9599999785423279f) {
		tmp = sqrtf(u1) * t_0;
	} else {
		tmp = fmaf((t_1 * fmaf(64.93939402268539f, (u2 * u2), -19.739208802181317f)), (u2 * u2), t_1);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = cos(Float32(Float32(6.28318530718) * u2))
	t_1 = sqrt(Float32(u1 / Float32(Float32(1.0) - u1)))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.9599999785423279))
		tmp = Float32(sqrt(u1) * t_0);
	else
		tmp = fma(Float32(t_1 * fma(Float32(64.93939402268539), Float32(u2 * u2), Float32(-19.739208802181317))), Float32(u2 * u2), t_1);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2)) < 0.959999979

   1. Initial program 99.0%

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

   2. Taylor expanded in u1 around 0

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

   4. Applied rewrites 74.3%

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

   if 0.959999979 < (cos.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2))

   1. Initial program 99.0%

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

   2. Taylor expanded in u2 around 0

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

      1. lower-+.f32 N/A

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

      2. lower-sqrt.f32 N/A

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

      3. lower-/.f32 N/A

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

      4. lower--.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lower-fma.f32 N/A

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

      8. lower-sqrt.f32 N/A

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

      9. lower-/.f32 N/A

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

      10. lower--.f32 N/A

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

      11. lower-*.f32 N/A

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

      12. lower-*.f32 N/A

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

   4. Applied rewrites 91.4%

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

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 91.4

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

   6. Applied rewrites 91.4%

      \[\leadsto \mathsf{fma}\left(\sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(64.93939402268539, u2 \cdot u2, -19.739208802181317\right), \color{blue}{u2 \cdot u2}, \sqrt{\frac{u1}{1 - u1}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 91.4% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.0%

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

2. Taylor expanded in u2 around 0

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

   1. lower-+.f32 N/A

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

   2. lower-sqrt.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. lower--.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-pow.f32 N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-sqrt.f32 N/A

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

   9. lower-/.f32 N/A

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

   10. lower--.f32 N/A

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

   11. lower-*.f32 N/A

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

   12. lower-*.f32 N/A

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

4. Applied rewrites 91.4%

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

   1. lift-+.f32 N/A

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

   2. lift-sqrt.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. mult-flip N/A

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

   5. metadata-eval N/A

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

   6. lift--.f32 N/A

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

   7. sub-negate-rev N/A

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

   8. lift--.f32 N/A

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

   9. frac-2neg N/A

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

   10. lift-/.f32 N/A

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

   11. sqrt-unprod N/A

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

   12. lift-sqrt.f32 N/A

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

   13. lift-sqrt.f32 N/A

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

   14. *-commutative N/A

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

   15. lower-fma.f32 91.0

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

6. Applied rewrites 91.0%

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

   1. lift-fma.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. lift-sqrt.f32 N/A

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

   8. lift-sqrt.f32 N/A

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

   9. sqrt-prod N/A

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

   10. lift-/.f32 N/A

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

   11. frac-2neg N/A

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

   12. metadata-eval N/A

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

   13. lift--.f32 N/A

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

   14. sub-negate-rev N/A

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

   15. lift--.f32 N/A

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

   16. mult-flip N/A

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

   17. lift-/.f32 N/A

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

   18. lift-sqrt.f32 N/A

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

8. Applied rewrites 91.4%

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

Alternative 10: 89.2% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.0%

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

2. Taylor expanded in u2 around 0

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

   1. lower-+.f32 N/A

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

   2. lower-sqrt.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. lower--.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-pow.f32 N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-sqrt.f32 N/A

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

   9. lower-/.f32 N/A

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

   10. lower--.f32 N/A

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

   11. lower-*.f32 N/A

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

   12. lower-*.f32 N/A

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

4. Applied rewrites 91.4%

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

   1. lift-+.f32 N/A

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

   2. lift-sqrt.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. mult-flip N/A

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

   5. metadata-eval N/A

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

   6. lift--.f32 N/A

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

   7. sub-negate-rev N/A

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

   8. lift--.f32 N/A

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

   9. frac-2neg N/A

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

   10. lift-/.f32 N/A

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

   11. sqrt-unprod N/A

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

   12. lift-sqrt.f32 N/A

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

   13. lift-sqrt.f32 N/A

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

   14. *-commutative N/A

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

   15. lower-fma.f32 91.0

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

6. Applied rewrites 91.0%

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

7. Taylor expanded in u1 around 0

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

   1. lower-sqrt.f32 89.2

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

9. Applied rewrites 89.2%

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

Alternative 11: 88.4% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (/ u1 (- 1.0 u1))) (+ 1.0 (* -19.739208802181317 (pow u2 2.0)))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - u1))) * (1.0f + (-19.739208802181317f * powf(u2, 2.0f)));
}

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 + ((-19.739208802181317e0) * (u2 ** 2.0e0)))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * Float32(Float32(1.0) + Float32(Float32(-19.739208802181317) * (u2 ^ Float32(2.0)))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((u1 / (single(1.0) - u1))) * (single(1.0) + (single(-19.739208802181317) * (u2 ^ single(2.0))));
end

Derivation

1. Initial program 99.0%

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

2. Taylor expanded in u2 around 0

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

   1. lower-+.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-pow.f32 88.4

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

4. Applied rewrites 88.4%

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

Alternative 12: 87.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.0%

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

2. Taylor expanded in u2 around 0

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

   1. lower-+.f32 N/A

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

   2. lower-sqrt.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. lower--.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-pow.f32 N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-sqrt.f32 N/A

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

   9. lower-/.f32 N/A

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

   10. lower--.f32 N/A

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

   11. lower-*.f32 N/A

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

   12. lower-*.f32 N/A

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

4. Applied rewrites 91.4%

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

   1. lift-+.f32 N/A

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

   2. lift-sqrt.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. mult-flip N/A

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

   5. metadata-eval N/A

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

   6. lift--.f32 N/A

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

   7. sub-negate-rev N/A

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

   8. lift--.f32 N/A

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

   9. frac-2neg N/A

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

   10. lift-/.f32 N/A

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

   11. sqrt-unprod N/A

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

   12. lift-sqrt.f32 N/A

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

   13. lift-sqrt.f32 N/A

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

   14. *-commutative N/A

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

   15. lower-fma.f32 91.0

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

6. Applied rewrites 91.0%

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

7. Taylor expanded in u2 around 0

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

9. Applied rewrites 87.9%

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

Alternative 13: 85.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(6.28318530718 \cdot u2\right) \leq 0.9998999834060669:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\frac{-1}{-1}}, \sqrt{u1}, \left(\sqrt{u1} \cdot \mathsf{fma}\left(64.93939402268539, u2 \cdot u2, -19.739208802181317\right)\right) \cdot \left(u2 \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (cos (* 6.28318530718 u2)) 0.9998999834060669)
   (fma
    (sqrt (/ -1.0 -1.0))
    (sqrt u1)
    (*
     (* (sqrt u1) (fma 64.93939402268539 (* u2 u2) -19.739208802181317))
     (* u2 u2)))
   (sqrt (/ u1 (- 1.0 u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (cosf((6.28318530718f * u2)) <= 0.9998999834060669f) {
		tmp = fmaf(sqrtf((-1.0f / -1.0f)), sqrtf(u1), ((sqrtf(u1) * fmaf(64.93939402268539f, (u2 * u2), -19.739208802181317f)) * (u2 * u2)));
	} else {
		tmp = sqrtf((u1 / (1.0f - u1)));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (cos(Float32(Float32(6.28318530718) * u2)) <= Float32(0.9998999834060669))
		tmp = fma(sqrt(Float32(Float32(-1.0) / Float32(-1.0))), sqrt(u1), Float32(Float32(sqrt(u1) * fma(Float32(64.93939402268539), Float32(u2 * u2), Float32(-19.739208802181317))) * Float32(u2 * u2)));
	else
		tmp = sqrt(Float32(u1 / Float32(Float32(1.0) - u1)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2)) < 0.999899983

   1. Initial program 99.0%

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

   2. Taylor expanded in u2 around 0

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

      1. lower-+.f32 N/A

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

      2. lower-sqrt.f32 N/A

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

      3. lower-/.f32 N/A

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

      4. lower--.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lower-fma.f32 N/A

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

      8. lower-sqrt.f32 N/A

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

      9. lower-/.f32 N/A

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

      10. lower--.f32 N/A

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

      11. lower-*.f32 N/A

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

      12. lower-*.f32 N/A

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

   4. Applied rewrites 91.4%

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

      1. lift-+.f32 N/A

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

      2. lift-sqrt.f32 N/A

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

      3. lift-/.f32 N/A

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

      4. mult-flip N/A

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

      5. metadata-eval N/A

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

      6. lift--.f32 N/A

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

      7. sub-negate-rev N/A

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

      8. lift--.f32 N/A

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

      9. frac-2neg N/A

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

      10. lift-/.f32 N/A

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

      11. sqrt-unprod N/A

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

      12. lift-sqrt.f32 N/A

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

      13. lift-sqrt.f32 N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f32 91.0

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

   6. Applied rewrites 91.0%

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

   7. Taylor expanded in u1 around 0

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

   9. Applied rewrites 69.7%

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

   10. Taylor expanded in u1 around 0

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

   12. Applied rewrites 69.8%

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

   if 0.999899983 < (cos.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2))

   1. Initial program 99.0%

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

   2. Taylor expanded in u2 around 0

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

      1. lower-sqrt.f32 N/A

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

      2. lower-/.f32 N/A

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

      3. lower--.f32 80.1

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

   4. Applied rewrites 80.1%

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

Alternative 14: 83.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{u1}{1 - u1}}\\ \mathbf{if}\;\cos \left(6.28318530718 \cdot u2\right) \leq 0.9998999834060669:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\frac{-1}{-1}}, \sqrt{u1}, \left(t\_0 \cdot -19.739208802181317\right) \cdot \left(u2 \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (/ u1 (- 1.0 u1)))))
   (if (<= (cos (* 6.28318530718 u2)) 0.9998999834060669)
     (fma
      (sqrt (/ -1.0 -1.0))
      (sqrt u1)
      (* (* t_0 -19.739208802181317) (* u2 u2)))
     t_0)))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sqrtf((u1 / (1.0f - u1)));
	float tmp;
	if (cosf((6.28318530718f * u2)) <= 0.9998999834060669f) {
		tmp = fmaf(sqrtf((-1.0f / -1.0f)), sqrtf(u1), ((t_0 * -19.739208802181317f) * (u2 * u2)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sqrt(Float32(u1 / Float32(Float32(1.0) - u1)))
	tmp = Float32(0.0)
	if (cos(Float32(Float32(6.28318530718) * u2)) <= Float32(0.9998999834060669))
		tmp = fma(sqrt(Float32(Float32(-1.0) / Float32(-1.0))), sqrt(u1), Float32(Float32(t_0 * Float32(-19.739208802181317)) * Float32(u2 * u2)));
	else
		tmp = t_0;
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2)) < 0.999899983

   1. Initial program 99.0%

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

   2. Taylor expanded in u2 around 0

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

      1. lower-+.f32 N/A

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

      2. lower-sqrt.f32 N/A

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

      3. lower-/.f32 N/A

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

      4. lower--.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lower-fma.f32 N/A

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

      8. lower-sqrt.f32 N/A

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

      9. lower-/.f32 N/A

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

      10. lower--.f32 N/A

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

      11. lower-*.f32 N/A

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

      12. lower-*.f32 N/A

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

   4. Applied rewrites 91.4%

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

      1. lift-+.f32 N/A

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

      2. lift-sqrt.f32 N/A

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

      3. lift-/.f32 N/A

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

      4. mult-flip N/A

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

      5. metadata-eval N/A

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

      6. lift--.f32 N/A

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

      7. sub-negate-rev N/A

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

      8. lift--.f32 N/A

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

      9. frac-2neg N/A

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

      10. lift-/.f32 N/A

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

      11. sqrt-unprod N/A

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

      12. lift-sqrt.f32 N/A

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

      13. lift-sqrt.f32 N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f32 91.0

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

   6. Applied rewrites 91.0%

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

   7. Taylor expanded in u1 around 0

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

   9. Applied rewrites 69.7%

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

   10. Taylor expanded in u2 around 0

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

   12. Applied rewrites 67.9%

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

   if 0.999899983 < (cos.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2))

   1. Initial program 99.0%

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

   2. Taylor expanded in u2 around 0

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

      1. lower-sqrt.f32 N/A

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

      2. lower-/.f32 N/A

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

      3. lower--.f32 80.1

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

   4. Applied rewrites 80.1%

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

Alternative 15: 80.1% accurate, 5.3× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

2. Taylor expanded in u2 around 0

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

   1. lower-sqrt.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower--.f32 80.1

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

4. Applied rewrites 80.1%

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

Alternative 16: 71.7% accurate, 6.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.0%

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

2. Taylor expanded in u2 around 0

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

   1. lower-sqrt.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower--.f32 80.1

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

4. Applied rewrites 80.1%

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

5. Taylor expanded in u1 around 0

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

   1. lower-*.f32 N/A

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

   2. lower-+.f32 71.7

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

7. Applied rewrites 71.7%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-+.f32 N/A

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

   4. +-commutative N/A

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

   5. distribute-lft1-in N/A

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

   6. lift-fma.f32 71.7

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

9. Applied rewrites 71.7%

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

Alternative 17: 63.4% accurate, 16.2× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

2. Taylor expanded in u2 around 0

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

   1. lower-sqrt.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower--.f32 80.1

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

4. Applied rewrites 80.1%

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

5. Taylor expanded in u1 around 0

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

7. Applied rewrites 63.4%

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

Reproduce

herbie shell --seed 2025156 
(FPCore (cosTheta_i u1 u2)
  :name "Trowbridge-Reitz Sample, near normal, slope_x"
  :precision binary32
  :pre (and (and (and (> cosTheta_i 0.9999) (<= cosTheta_i 1.0)) (and (<= 2.328306437e-10 u1) (<= u1 1.0))) (and (<= 2.328306437e-10 u2) (<= u2 1.0)))
  (* (sqrt (/ u1 (- 1.0 u1))) (cos (* 6.28318530718 u2))))