Trowbridge-Reitz Sample, near normal, slope_x

Percentage Accurate: 99.0% → 99.2%

Time: 4.1s

Alternatives: 12

Speedup: 1.0×

Specification

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

Math

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

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

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

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, 1.0× speedup

Math

\[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\mathsf{fma}\left(-6.28318530718, u2, 1.5707963705062866\right)\right) \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (/ u1 (- 1.0 u1))) (sin (fma -6.28318530718 u2 1.5707963705062866))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - u1))) * sinf(fmaf(-6.28318530718f, u2, 1.5707963705062866f));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * sin(fma(Float32(-6.28318530718), u2, Float32(1.5707963705062866))))
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. *-commutative N/A

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

   12. lower-*.f32 N/A

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

   13. lower-PI.f32 99.2

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

3. Applied rewrites 99.2%

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

4. Evaluated real constant 99.2%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\mathsf{fma}\left(\frac{-314159265359}{50000000000}, u2, \color{blue}{\frac{13176795}{8388608}}\right)\right) \]
5. Add Preprocessing

Alternative 2: 99.0% accurate, 1.0× speedup

Math

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

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 3: 97.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{\frac{u1}{1 - u1}}\\ \mathbf{if}\;t\_0 \cdot \cos \left(6.28318530718 \cdot u2\right) \leq 0.01988000050187111:\\ \;\;\;\;\sqrt{\left(u1 - -1\right) \cdot u1} \cdot \cos \left(u2 \cdot -6.28318530718\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-85.45681720672748, u2 \cdot u2, 64.93939402268539\right), u2 \cdot u2, -19.739208802181317\right), \left(u2 \cdot u2\right) \cdot t\_0, t\_0\right)\\ \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (/ u1 (- 1.0 u1)))))
   (if (<= (* t_0 (cos (* 6.28318530718 u2))) 0.01988000050187111)
     (* (sqrt (* (- u1 -1.0) u1)) (cos (* u2 -6.28318530718)))
     (fma
      (fma
       (fma -85.45681720672748 (* u2 u2) 64.93939402268539)
       (* u2 u2)
       -19.739208802181317)
      (* (* u2 u2) t_0)
      t_0))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sqrtf((u1 / (1.0f - u1)));
	float tmp;
	if ((t_0 * cosf((6.28318530718f * u2))) <= 0.01988000050187111f) {
		tmp = sqrtf(((u1 - -1.0f) * u1)) * cosf((u2 * -6.28318530718f));
	} else {
		tmp = fmaf(fmaf(fmaf(-85.45681720672748f, (u2 * u2), 64.93939402268539f), (u2 * u2), -19.739208802181317f), ((u2 * u2) * t_0), t_0);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sqrt(Float32(u1 / Float32(Float32(1.0) - u1)))
	tmp = Float32(0.0)
	if (Float32(t_0 * cos(Float32(Float32(6.28318530718) * u2))) <= Float32(0.01988000050187111))
		tmp = Float32(sqrt(Float32(Float32(u1 - Float32(-1.0)) * u1)) * cos(Float32(u2 * Float32(-6.28318530718))));
	else
		tmp = fma(fma(fma(Float32(-85.45681720672748), Float32(u2 * u2), Float32(64.93939402268539)), Float32(u2 * u2), Float32(-19.739208802181317)), Float32(Float32(u2 * u2) * t_0), t_0);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (sqrt.f32 (/.f32 u1 (-.f32 #s(literal 1 binary32) u1))) (cos.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2))) < 0.0198800005

   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-/.f32 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. frac-2neg N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f32 N/A

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

      8. lift--.f32 N/A

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

      9. sub-negate-rev N/A

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

      10. lower--.f32 98.9

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

   3. Applied rewrites 98.9%

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

   4. Taylor expanded in u1 around 0

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

      1. lower-+.f32 86.8

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

   6. Applied rewrites 86.8%

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

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. metadata-eval N/A

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

      5. lower--.f32 86.8

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

      6. lower--.f32 N/A

         \[\leadsto \sqrt{\left(u1 - \color{blue}{-1}\right) \cdot u1} \cdot \mathsf{Rewrite=>}\left(lift-cos.f32, \cos \left(\frac{314159265359}{50000000000} \cdot u2\right)\right) \]

      7. lower--.f32 N/A

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

      8. lower--.f32 N/A

         \[\leadsto \sqrt{\left(u1 - \color{blue}{-1}\right) \cdot u1} \cdot \cos \left(\mathsf{neg}\left(\mathsf{Rewrite=>}\left(lift-*.f32, \left(\frac{314159265359}{50000000000} \cdot u2\right)\right)\right)\right) \]

      9. lower--.f32 N/A

         \[\leadsto \sqrt{\left(u1 - \color{blue}{-1}\right) \cdot u1} \cdot \cos \mathsf{Rewrite=>}\left(distribute-lft-neg-in, \left(\left(\mathsf{neg}\left(\frac{314159265359}{50000000000}\right)\right) \cdot u2\right)\right) \]

      10. lower--.f32 N/A

          \[\leadsto \sqrt{\left(u1 - \color{blue}{-1}\right) \cdot u1} \cdot \cos \left(\mathsf{Rewrite=>}\left(metadata-eval, \frac{-314159265359}{50000000000}\right) \cdot u2\right) \]

      11. lower--.f32 N/A

          \[\leadsto \sqrt{\left(u1 - \color{blue}{-1}\right) \cdot u1} \cdot \mathsf{Rewrite=>}\left(lower-cos.f32, \cos \left(\frac{-314159265359}{50000000000} \cdot u2\right)\right) \]

      12. lower--.f32 N/A

          \[\leadsto \sqrt{\left(u1 - \color{blue}{-1}\right) \cdot u1} \cdot \cos \mathsf{Rewrite=>}\left(*-commutative, \left(u2 \cdot \frac{-314159265359}{50000000000}\right)\right) \]

      13. lower--.f32 N/A

          \[\leadsto \sqrt{\left(u1 - \color{blue}{-1}\right) \cdot u1} \cdot \cos \mathsf{Rewrite=>}\left(lower-*.f32, \left(u2 \cdot \frac{-314159265359}{50000000000}\right)\right) \]

   8. Applied rewrites 86.8%

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

   if 0.0198800005 < (*.f32 (sqrt.f32 (/.f32 u1 (-.f32 #s(literal 1 binary32) u1))) (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

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

      1. lower-+.f32 N/A

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

      2. lower-sqrt.f32 N/A

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

      3. lower-/.f32 N/A

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

      4. lower--.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lower-fma.f32 N/A

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

   4. Applied rewrites 93.9%

      \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}} + {u2}^{2} \cdot \mathsf{fma}\left(-19.739208802181317, \sqrt{\frac{u1}{1 - u1}}, {u2}^{2} \cdot \mathsf{fma}\left(-85.45681720672748, {u2}^{2} \cdot \sqrt{\frac{u1}{1 - u1}}, 64.93939402268539 \cdot \sqrt{\frac{u1}{1 - u1}}\right)\right)} \]
   5. Step-by-step derivation

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 93.9

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-19.739208802181317, \sqrt{\frac{u1}{1 - u1}}, {u2}^{2} \cdot \mathsf{fma}\left(-85.45681720672748, {u2}^{2} \cdot \sqrt{\frac{u1}{1 - u1}}, 64.93939402268539 \cdot \sqrt{\frac{u1}{1 - u1}}\right)\right), \color{blue}{{u2}^{2}}, \sqrt{\frac{u1}{1 - u1}}\right) \]

   6. Applied rewrites 93.9%

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

      1. lift-fma.f32 N/A

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

      2. lift-fma.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f32 N/A

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

      6. *-commutative N/A

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

      7. distribute-lft-out N/A

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

      8. associate-*l* N/A

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

      9. lower-fma.f32 N/A

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

   8. Applied rewrites 93.9%

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

      1. lift-fma.f32 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f32 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f32 N/A

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

      6. lift-fma.f32 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f32 N/A

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

      9. lower-*.f32 93.9

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

   10. Applied rewrites 93.9%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-85.45681720672748, u2 \cdot u2, 64.93939402268539\right), u2 \cdot u2, -19.739208802181317\right), \color{blue}{\left(u2 \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}}, \sqrt{\frac{u1}{1 - u1}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 93.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

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

   1. lower-+.f32 N/A

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

   2. lower-sqrt.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. lower--.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-pow.f32 N/A

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

   7. lower-fma.f32 N/A

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

4. Applied rewrites 93.9%

   \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}} + {u2}^{2} \cdot \mathsf{fma}\left(-19.739208802181317, \sqrt{\frac{u1}{1 - u1}}, {u2}^{2} \cdot \mathsf{fma}\left(-85.45681720672748, {u2}^{2} \cdot \sqrt{\frac{u1}{1 - u1}}, 64.93939402268539 \cdot \sqrt{\frac{u1}{1 - u1}}\right)\right)} \]
5. Step-by-step derivation

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 93.9

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-19.739208802181317, \sqrt{\frac{u1}{1 - u1}}, {u2}^{2} \cdot \mathsf{fma}\left(-85.45681720672748, {u2}^{2} \cdot \sqrt{\frac{u1}{1 - u1}}, 64.93939402268539 \cdot \sqrt{\frac{u1}{1 - u1}}\right)\right), \color{blue}{{u2}^{2}}, \sqrt{\frac{u1}{1 - u1}}\right) \]

6. Applied rewrites 93.9%

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

   1. lift-fma.f32 N/A

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

   2. lift-fma.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*l* N/A

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. distribute-lft-out N/A

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

   8. associate-*l* N/A

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

   9. lower-fma.f32 N/A

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

8. Applied rewrites 93.9%

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

   1. lift-fma.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 93.9

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. associate-*r* N/A

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

   7. lower-*.f32 N/A

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

   8. lower-*.f32 93.9

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

   9. lift-fma.f32 N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f32 93.9

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

10. Applied rewrites 93.9%

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

Alternative 5: 93.9% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (/ u1 (- 1.0 u1)))))
   (fma
    t_0
    (*
     (fma
      (* (fma -85.45681720672748 (* u2 u2) 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)));
	return fmaf(t_0, (fmaf((fmaf(-85.45681720672748f, (u2 * u2), 64.93939402268539f) * u2), u2, -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(t_0, Float32(fma(Float32(fma(Float32(-85.45681720672748), Float32(u2 * u2), Float32(64.93939402268539)) * u2), u2, Float32(-19.739208802181317)) * Float32(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

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

   1. lower-+.f32 N/A

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

   2. lower-sqrt.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. lower--.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-pow.f32 N/A

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

   7. lower-fma.f32 N/A

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

4. Applied rewrites 93.9%

   \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}} + {u2}^{2} \cdot \mathsf{fma}\left(-19.739208802181317, \sqrt{\frac{u1}{1 - u1}}, {u2}^{2} \cdot \mathsf{fma}\left(-85.45681720672748, {u2}^{2} \cdot \sqrt{\frac{u1}{1 - u1}}, 64.93939402268539 \cdot \sqrt{\frac{u1}{1 - u1}}\right)\right)} \]
5. Step-by-step derivation

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 93.9

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-19.739208802181317, \sqrt{\frac{u1}{1 - u1}}, {u2}^{2} \cdot \mathsf{fma}\left(-85.45681720672748, {u2}^{2} \cdot \sqrt{\frac{u1}{1 - u1}}, 64.93939402268539 \cdot \sqrt{\frac{u1}{1 - u1}}\right)\right), \color{blue}{{u2}^{2}}, \sqrt{\frac{u1}{1 - u1}}\right) \]

6. Applied rewrites 93.9%

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

   1. lift-fma.f32 N/A

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

   2. lift-fma.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*l* N/A

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. distribute-lft-out N/A

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

   8. associate-*l* N/A

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

   9. lower-fma.f32 N/A

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

8. Applied rewrites 93.9%

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

   1. lift-fma.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*r* N/A

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

   4. lower-fma.f32 N/A

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

   5. lower-*.f32 93.9

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

   6. lift-fma.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f32 93.9

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

10. Applied rewrites 93.9%

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

Alternative 6: 93.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

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

   1. lower-+.f32 N/A

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

   2. lower-sqrt.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. lower--.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-pow.f32 N/A

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

   7. lower-fma.f32 N/A

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

4. Applied rewrites 93.9%

   \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}} + {u2}^{2} \cdot \mathsf{fma}\left(-19.739208802181317, \sqrt{\frac{u1}{1 - u1}}, {u2}^{2} \cdot \mathsf{fma}\left(-85.45681720672748, {u2}^{2} \cdot \sqrt{\frac{u1}{1 - u1}}, 64.93939402268539 \cdot \sqrt{\frac{u1}{1 - u1}}\right)\right)} \]
5. Step-by-step derivation

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 93.9

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-19.739208802181317, \sqrt{\frac{u1}{1 - u1}}, {u2}^{2} \cdot \mathsf{fma}\left(-85.45681720672748, {u2}^{2} \cdot \sqrt{\frac{u1}{1 - u1}}, 64.93939402268539 \cdot \sqrt{\frac{u1}{1 - u1}}\right)\right), \color{blue}{{u2}^{2}}, \sqrt{\frac{u1}{1 - u1}}\right) \]

6. Applied rewrites 93.9%

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

   1. lift-fma.f32 N/A

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

   2. lift-fma.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*l* N/A

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. distribute-lft-out N/A

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

   8. associate-*l* N/A

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

   9. lower-fma.f32 N/A

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

8. Applied rewrites 93.9%

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

   1. lift-fma.f32 N/A

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

   2. *-commutative N/A

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

   3. distribute-lft1-in N/A

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

   4. lower-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-*r* N/A

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

   8. lower-fma.f32 N/A

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

   9. lower-*.f32 93.8

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

   10. lift-fma.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f32 93.8

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

10. Applied rewrites 93.8%

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

Alternative 7: 91.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{\frac{u1}{1 - u1}}\\ \mathbf{if}\;\cos \left(6.28318530718 \cdot u2\right) \leq 0.9800000190734863:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{u1}, \mathsf{fma}\left(\mathsf{fma}\left(u2 \cdot u2, -85.45681720672748, 64.93939402268539\right), u2 \cdot u2, -19.739208802181317\right) \cdot \left(u2 \cdot u2\right), \sqrt{u1}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-19.739208802181317 \cdot t\_0, u2 \cdot u2, t\_0\right)\\ \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (/ u1 (- 1.0 u1)))))
   (if (<= (cos (* 6.28318530718 u2)) 0.9800000190734863)
     (fma
      (sqrt u1)
      (*
       (fma
        (fma (* u2 u2) -85.45681720672748 64.93939402268539)
        (* u2 u2)
        -19.739208802181317)
       (* u2 u2))
      (sqrt u1))
     (fma (* -19.739208802181317 t_0) (* 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)) <= 0.9800000190734863f) {
		tmp = fmaf(sqrtf(u1), (fmaf(fmaf((u2 * u2), -85.45681720672748f, 64.93939402268539f), (u2 * u2), -19.739208802181317f) * (u2 * u2)), sqrtf(u1));
	} else {
		tmp = fmaf((-19.739208802181317f * t_0), (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(0.9800000190734863))
		tmp = fma(sqrt(u1), Float32(fma(fma(Float32(u2 * u2), Float32(-85.45681720672748), Float32(64.93939402268539)), Float32(u2 * u2), Float32(-19.739208802181317)) * Float32(u2 * u2)), sqrt(u1));
	else
		tmp = fma(Float32(Float32(-19.739208802181317) * t_0), 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)) < 0.980000019

   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

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

      1. lower-+.f32 N/A

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

      2. lower-sqrt.f32 N/A

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

      3. lower-/.f32 N/A

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

      4. lower--.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lower-fma.f32 N/A

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

   4. Applied rewrites 93.9%

      \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}} + {u2}^{2} \cdot \mathsf{fma}\left(-19.739208802181317, \sqrt{\frac{u1}{1 - u1}}, {u2}^{2} \cdot \mathsf{fma}\left(-85.45681720672748, {u2}^{2} \cdot \sqrt{\frac{u1}{1 - u1}}, 64.93939402268539 \cdot \sqrt{\frac{u1}{1 - u1}}\right)\right)} \]
   5. Step-by-step derivation

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 93.9

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-19.739208802181317, \sqrt{\frac{u1}{1 - u1}}, {u2}^{2} \cdot \mathsf{fma}\left(-85.45681720672748, {u2}^{2} \cdot \sqrt{\frac{u1}{1 - u1}}, 64.93939402268539 \cdot \sqrt{\frac{u1}{1 - u1}}\right)\right), \color{blue}{{u2}^{2}}, \sqrt{\frac{u1}{1 - u1}}\right) \]

   6. Applied rewrites 93.9%

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

      1. lift-fma.f32 N/A

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

      2. lift-fma.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f32 N/A

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

      6. *-commutative N/A

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

      7. distribute-lft-out N/A

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

      8. associate-*l* N/A

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

      9. lower-fma.f32 N/A

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

   8. Applied rewrites 93.9%

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

   9. Taylor expanded in u1 around 0

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

   11. Applied rewrites 91.5%

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

   12. Taylor expanded in u1 around 0

       \[\leadsto \mathsf{fma}\left(\sqrt{u1}, \mathsf{fma}\left(\mathsf{fma}\left(u2 \cdot u2, \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000}, \frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}\right), u2 \cdot u2, \frac{-98696044010906577398881}{5000000000000000000000}\right) \cdot \left(u2 \cdot u2\right), \sqrt{u1}\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 71.4%

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

   if 0.980000019 < (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

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

      1. lower-+.f32 N/A

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

      2. lower-sqrt.f32 N/A

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

      3. lower-/.f32 N/A

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

      4. lower--.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lower-fma.f32 N/A

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

   4. Applied rewrites 93.9%

      \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}} + {u2}^{2} \cdot \mathsf{fma}\left(-19.739208802181317, \sqrt{\frac{u1}{1 - u1}}, {u2}^{2} \cdot \mathsf{fma}\left(-85.45681720672748, {u2}^{2} \cdot \sqrt{\frac{u1}{1 - u1}}, 64.93939402268539 \cdot \sqrt{\frac{u1}{1 - u1}}\right)\right)} \]
   5. Step-by-step derivation

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 93.9

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-19.739208802181317, \sqrt{\frac{u1}{1 - u1}}, {u2}^{2} \cdot \mathsf{fma}\left(-85.45681720672748, {u2}^{2} \cdot \sqrt{\frac{u1}{1 - u1}}, 64.93939402268539 \cdot \sqrt{\frac{u1}{1 - u1}}\right)\right), \color{blue}{{u2}^{2}}, \sqrt{\frac{u1}{1 - u1}}\right) \]

   6. Applied rewrites 93.9%

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

   7. Taylor expanded in u2 around 0

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

      1. lower-*.f32 N/A

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

      2. lower-sqrt.f32 N/A

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

      3. lower-/.f32 N/A

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

      4. lower--.f32 88.7

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

   9. Applied rewrites 88.7%

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

Alternative 8: 88.7% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

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

   1. lower-+.f32 N/A

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

   2. lower-sqrt.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. lower--.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-pow.f32 N/A

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

   7. lower-fma.f32 N/A

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

4. Applied rewrites 93.9%

   \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}} + {u2}^{2} \cdot \mathsf{fma}\left(-19.739208802181317, \sqrt{\frac{u1}{1 - u1}}, {u2}^{2} \cdot \mathsf{fma}\left(-85.45681720672748, {u2}^{2} \cdot \sqrt{\frac{u1}{1 - u1}}, 64.93939402268539 \cdot \sqrt{\frac{u1}{1 - u1}}\right)\right)} \]
5. Step-by-step derivation

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 93.9

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-19.739208802181317, \sqrt{\frac{u1}{1 - u1}}, {u2}^{2} \cdot \mathsf{fma}\left(-85.45681720672748, {u2}^{2} \cdot \sqrt{\frac{u1}{1 - u1}}, 64.93939402268539 \cdot \sqrt{\frac{u1}{1 - u1}}\right)\right), \color{blue}{{u2}^{2}}, \sqrt{\frac{u1}{1 - u1}}\right) \]

6. Applied rewrites 93.9%

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

7. Taylor expanded in u2 around 0

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

   1. lower-*.f32 N/A

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

   2. lower-sqrt.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. lower--.f32 88.7

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

9. Applied rewrites 88.7%

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

Alternative 9: 80.3% accurate, 5.3× speedup

Math

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

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 80.3

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

4. Applied rewrites 80.3%

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

Alternative 10: 72.0% accurate, 5.7× speedup

Math

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

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(Float32(u1 - 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 80.3

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

4. Applied rewrites 80.3%

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

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 71.9

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

7. Applied rewrites 71.9%

   \[\leadsto \sqrt{u1 \cdot \left(1 + u1\right)} \]
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. lower-*.f32 71.9

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

   4. lift-+.f32 N/A

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

   5. +-commutative N/A

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

   6. add-flip N/A

      \[\leadsto \sqrt{\left(u1 - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot u1} \]

   7. metadata-eval N/A

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

   8. lower--.f32 71.9

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

9. Applied rewrites 71.9%

   \[\leadsto \sqrt{\left(u1 - -1\right) \cdot u1} \]
10. Add Preprocessing

Alternative 11: 71.9% accurate, 6.0× speedup

Math

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

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 80.3

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

4. Applied rewrites 80.3%

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

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 71.9

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

7. Applied rewrites 71.9%

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

   1. lift-*.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. +-commutative N/A

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

   4. distribute-rgt-in N/A

      \[\leadsto \sqrt{u1 \cdot u1 + 1 \cdot u1} \]

   5. *-lft-identity N/A

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

   6. lower-fma.f32 72.0

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

9. Applied rewrites 72.0%

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

Alternative 12: 63.6% accurate, 16.2× speedup

Math

\[\sqrt{u1} \]

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 80.3

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

4. Applied rewrites 80.3%

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

5. Taylor expanded in u1 around 0

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

7. Applied rewrites 63.6%

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

Reproduce

herbie shell --seed 2025171 
(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))))