Trowbridge-Reitz Sample, near normal, slope_x

Percentage Accurate: 99.0% → 99.0%

Time: 6.0s

Alternatives: 4

Speedup: N/A×

Specification

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

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Julia

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

MATLAB

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

Initial Program: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Julia

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

MATLAB

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

Alternative 1: 99.0% accurate, N/A× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

Alternative 2: 98.7% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{u1 \cdot \left(1 + \left(u1 + u1 \cdot u1\right)\right)}{1 - {u1}^{3}}} \cdot \sin \left(\mathsf{fma}\left(6.28318530718, u2, \frac{\pi}{2}\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sqrt (/ (* u1 (+ 1.0 (+ u1 (* u1 u1)))) (- 1.0 (pow u1 3.0))))
  (sin (fma 6.28318530718 u2 (/ PI 2.0)))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(((u1 * (1.0f + (u1 + (u1 * u1)))) / (1.0f - powf(u1, 3.0f)))) * sinf(fmaf(6.28318530718f, u2, (((float) M_PI) / 2.0f)));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(Float32(u1 * Float32(Float32(1.0) + Float32(u1 + Float32(u1 * u1)))) / Float32(Float32(1.0) - (u1 ^ Float32(3.0))))) * sin(fma(Float32(6.28318530718), u2, Float32(Float32(pi) / Float32(2.0)))))
end

Derivation

1. Initial program 99.1%

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

   1. lift--.f32 N/A

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

   2. flip3-- N/A

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

   3. lower-/.f32 N/A

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

   4. metadata-eval N/A

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

   5. lower--.f32 N/A

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

   6. lower-pow.f32 N/A

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

   7. metadata-eval N/A

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

   8. lower-+.f32 N/A

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

   9. lower-fma.f32 N/A

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

   10. lower-*.f32 99.0

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

4. Applied rewrites 99.0%

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

5. Taylor expanded in u2 around inf

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

   1. lower-*.f32 N/A

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

   2. lower-sqrt.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-+.f32 N/A

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

   6. lower-+.f32 N/A

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

   7. pow2 N/A

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

   8. lift-*.f32 N/A

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

   9. lift-pow.f32 N/A

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

   10. lift--.f32 N/A

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

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

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

   12. lower-sin.f32 N/A

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

7. Applied rewrites 98.9%

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

Alternative 3: 93.3% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\frac{u1}{1 - u1}\right)}^{0.5}\\ \mathsf{fma}\left(\mathsf{fma}\left(\left(-85.45681720672748 \cdot t\_0\right) \cdot \left(u2 \cdot u2\right) - -64.93939402268539 \cdot t\_0, u2 \cdot u2, -19.739208802181317 \cdot t\_0\right), u2 \cdot u2, t\_0\right) \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (pow (/ u1 (- 1.0 u1)) 0.5)))
   (fma
    (fma
     (- (* (* -85.45681720672748 t_0) (* u2 u2)) (* -64.93939402268539 t_0))
     (* u2 u2)
     (* -19.739208802181317 t_0))
    (* u2 u2)
    t_0)))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = powf((u1 / (1.0f - u1)), 0.5f);
	return fmaf(fmaf((((-85.45681720672748f * t_0) * (u2 * u2)) - (-64.93939402268539f * t_0)), (u2 * u2), (-19.739208802181317f * t_0)), (u2 * u2), t_0);
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(u1 / Float32(Float32(1.0) - u1)) ^ Float32(0.5)
	return fma(fma(Float32(Float32(Float32(Float32(-85.45681720672748) * t_0) * Float32(u2 * u2)) - Float32(Float32(-64.93939402268539) * t_0)), Float32(u2 * u2), Float32(Float32(-19.739208802181317) * t_0)), Float32(u2 * u2), t_0)
end

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in u2 around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

5. Applied rewrites 94.0%

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

Alternative 4: 92.8% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\frac{u1}{1 - u1}\right)}^{0.5}\\ t_1 := \left(u2 \cdot u2\right) \cdot \left(\left(u2 \cdot u2\right) \cdot \left(\left(\left(-85.45681720672748 \cdot t\_0\right) \cdot u2\right) \cdot u2 - -64.93939402268539 \cdot t\_0\right) - 19.739208802181317 \cdot t\_0\right)\\ \frac{{t\_0}^{3} + {t\_1}^{3}}{t\_0 \cdot t\_0 + \left(t\_1 \cdot t\_1 - t\_0 \cdot t\_1\right)} \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (pow (/ u1 (- 1.0 u1)) 0.5))
        (t_1
         (*
          (* u2 u2)
          (-
           (*
            (* u2 u2)
            (-
             (* (* (* -85.45681720672748 t_0) u2) u2)
             (* -64.93939402268539 t_0)))
           (* 19.739208802181317 t_0)))))
   (/
    (+ (pow t_0 3.0) (pow t_1 3.0))
    (+ (* t_0 t_0) (- (* t_1 t_1) (* t_0 t_1))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = powf((u1 / (1.0f - u1)), 0.5f);
	float t_1 = (u2 * u2) * (((u2 * u2) * ((((-85.45681720672748f * t_0) * u2) * u2) - (-64.93939402268539f * t_0))) - (19.739208802181317f * t_0));
	return (powf(t_0, 3.0f) + powf(t_1, 3.0f)) / ((t_0 * t_0) + ((t_1 * t_1) - (t_0 * t_1)));
}

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
    real(4) :: t_0
    real(4) :: t_1
    t_0 = (u1 / (1.0e0 - u1)) ** 0.5e0
    t_1 = (u2 * u2) * (((u2 * u2) * (((((-85.45681720672748e0) * t_0) * u2) * u2) - ((-64.93939402268539e0) * t_0))) - (19.739208802181317e0 * t_0))
    code = ((t_0 ** 3.0e0) + (t_1 ** 3.0e0)) / ((t_0 * t_0) + ((t_1 * t_1) - (t_0 * t_1)))
end function

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(u1 / Float32(Float32(1.0) - u1)) ^ Float32(0.5)
	t_1 = Float32(Float32(u2 * u2) * Float32(Float32(Float32(u2 * u2) * Float32(Float32(Float32(Float32(Float32(-85.45681720672748) * t_0) * u2) * u2) - Float32(Float32(-64.93939402268539) * t_0))) - Float32(Float32(19.739208802181317) * t_0)))
	return Float32(Float32((t_0 ^ Float32(3.0)) + (t_1 ^ Float32(3.0))) / Float32(Float32(t_0 * t_0) + Float32(Float32(t_1 * t_1) - Float32(t_0 * t_1))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	t_0 = (u1 / (single(1.0) - u1)) ^ single(0.5);
	t_1 = (u2 * u2) * (((u2 * u2) * ((((single(-85.45681720672748) * t_0) * u2) * u2) - (single(-64.93939402268539) * t_0))) - (single(19.739208802181317) * t_0));
	tmp = ((t_0 ^ single(3.0)) + (t_1 ^ single(3.0))) / ((t_0 * t_0) + ((t_1 * t_1) - (t_0 * t_1)));
end

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in u2 around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

5. Applied rewrites 94.0%

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

7. Applied rewrites 93.6%

   \[\leadsto \frac{{\left({\left(\frac{u1}{1 - u1}\right)}^{0.5}\right)}^{3} + {\left(\left(u2 \cdot u2\right) \cdot \left(\left(u2 \cdot u2\right) \cdot \left(\left(\left(-85.45681720672748 \cdot {\left(\frac{u1}{1 - u1}\right)}^{0.5}\right) \cdot u2\right) \cdot u2 - -64.93939402268539 \cdot {\left(\frac{u1}{1 - u1}\right)}^{0.5}\right) - 19.739208802181317 \cdot {\left(\frac{u1}{1 - u1}\right)}^{0.5}\right)\right)}^{3}}{\color{blue}{{\left(\frac{u1}{1 - u1}\right)}^{0.5} \cdot {\left(\frac{u1}{1 - u1}\right)}^{0.5} + \left(\left(\left(u2 \cdot u2\right) \cdot \left(\left(u2 \cdot u2\right) \cdot \left(\left(\left(-85.45681720672748 \cdot {\left(\frac{u1}{1 - u1}\right)}^{0.5}\right) \cdot u2\right) \cdot u2 - -64.93939402268539 \cdot {\left(\frac{u1}{1 - u1}\right)}^{0.5}\right) - 19.739208802181317 \cdot {\left(\frac{u1}{1 - u1}\right)}^{0.5}\right)\right) \cdot \left(\left(u2 \cdot u2\right) \cdot \left(\left(u2 \cdot u2\right) \cdot \left(\left(\left(-85.45681720672748 \cdot {\left(\frac{u1}{1 - u1}\right)}^{0.5}\right) \cdot u2\right) \cdot u2 - -64.93939402268539 \cdot {\left(\frac{u1}{1 - u1}\right)}^{0.5}\right) - 19.739208802181317 \cdot {\left(\frac{u1}{1 - u1}\right)}^{0.5}\right)\right) - {\left(\frac{u1}{1 - u1}\right)}^{0.5} \cdot \left(\left(u2 \cdot u2\right) \cdot \left(\left(u2 \cdot u2\right) \cdot \left(\left(\left(-85.45681720672748 \cdot {\left(\frac{u1}{1 - u1}\right)}^{0.5}\right) \cdot u2\right) \cdot u2 - -64.93939402268539 \cdot {\left(\frac{u1}{1 - u1}\right)}^{0.5}\right) - 19.739208802181317 \cdot {\left(\frac{u1}{1 - u1}\right)}^{0.5}\right)\right)\right)}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025066 
(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))))