Trowbridge-Reitz Sample, near normal, slope_y

Percentage Accurate: 98.3% → 98.3%

Time: 9.9s

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

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.3% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

   1. lift--.f32 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. flip-+ N/A

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

   5. sqr-neg N/A

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

   6. lower-/.f32 N/A

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

   7. metadata-eval N/A

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

   8. lower--.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. lower--.f32 N/A

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

   11. lower-neg.f32 98.5

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

4. Applied rewrites 98.5%

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

6. Applied rewrites 98.5%

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

   1. lift--.f32 N/A

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

   4. +-commutative N/A

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

   5. lift-neg.f32 N/A

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

   6. sub-neg N/A

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

   7. lift--.f32 98.5

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

8. Applied rewrites 98.5%

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

Alternative 2: 98.4% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

4. Applied rewrites 98.5%

   \[\leadsto \sqrt{\color{blue}{\sqrt{{\left(\frac{u1 - 1}{u1}\right)}^{-2}}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

6. Applied rewrites 98.4%

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

   1. lift-/.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-/r/ N/A

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

   4. lift-/.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. associate-/l* N/A

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

   7. lift--.f32 N/A

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

   8. associate-*r* N/A

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

   9. lower-*.f32 N/A

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

   10. lower-*.f32 N/A

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

   11. lower-/.f32 N/A

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

   12. lift--.f32 N/A

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

   13. lower-/.f32 98.5

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

8. Applied rewrites 98.5%

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

Alternative 3: 98.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

   1. lift-/.f32 N/A

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

   2. clear-num N/A

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

   3. frac-2neg N/A

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

   4. associate-/r/ N/A

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

   5. lower-*.f32 N/A

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

   6. metadata-eval N/A

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

   7. frac-2neg N/A

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

   8. lower-/.f32 N/A

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

   9. lower-neg.f32 98.4

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

4. Applied rewrites 98.4%

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

6. Applied rewrites 98.5%

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

Alternative 4: 95.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{u2}{\sqrt{1 - u1}}\\ \mathbf{if}\;\frac{u1}{1 - u1} \leq 0.0003499999875202775:\\ \;\;\;\;\sqrt{\left(-1 - u1\right) \cdot \left(-u1\right)} \cdot \sin \left(6.28318530718 \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \left(t\_0 \cdot \left(\left(u2 \cdot u2\right) \cdot -41.341702240407926\right) + t\_0 \cdot 6.28318530718\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (/ u2 (sqrt (- 1.0 u1)))))
   (if (<= (/ u1 (- 1.0 u1)) 0.0003499999875202775)
     (* (sqrt (* (- -1.0 u1) (- u1))) (sin (* 6.28318530718 u2)))
     (*
      (sqrt u1)
      (+ (* t_0 (* (* u2 u2) -41.341702240407926)) (* t_0 6.28318530718))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = u2 / sqrtf((1.0f - u1));
	float tmp;
	if ((u1 / (1.0f - u1)) <= 0.0003499999875202775f) {
		tmp = sqrtf(((-1.0f - u1) * -u1)) * sinf((6.28318530718f * u2));
	} else {
		tmp = sqrtf(u1) * ((t_0 * ((u2 * u2) * -41.341702240407926f)) + (t_0 * 6.28318530718f));
	}
	return tmp;
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    real(4) :: t_0
    real(4) :: tmp
    t_0 = u2 / sqrt((1.0e0 - u1))
    if ((u1 / (1.0e0 - u1)) <= 0.0003499999875202775e0) then
        tmp = sqrt((((-1.0e0) - u1) * -u1)) * sin((6.28318530718e0 * u2))
    else
        tmp = sqrt(u1) * ((t_0 * ((u2 * u2) * (-41.341702240407926e0))) + (t_0 * 6.28318530718e0))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(u2 / sqrt(Float32(Float32(1.0) - u1)))
	tmp = Float32(0.0)
	if (Float32(u1 / Float32(Float32(1.0) - u1)) <= Float32(0.0003499999875202775))
		tmp = Float32(sqrt(Float32(Float32(Float32(-1.0) - u1) * Float32(-u1))) * sin(Float32(Float32(6.28318530718) * u2)));
	else
		tmp = Float32(sqrt(u1) * Float32(Float32(t_0 * Float32(Float32(u2 * u2) * Float32(-41.341702240407926))) + Float32(t_0 * Float32(6.28318530718))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	t_0 = u2 / sqrt((single(1.0) - u1));
	tmp = single(0.0);
	if ((u1 / (single(1.0) - u1)) <= single(0.0003499999875202775))
		tmp = sqrt(((single(-1.0) - u1) * -u1)) * sin((single(6.28318530718) * u2));
	else
		tmp = sqrt(u1) * ((t_0 * ((u2 * u2) * single(-41.341702240407926))) + (t_0 * single(6.28318530718)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 u1 (-.f32 #s(literal 1 binary32) u1)) < 3.49999988e-4

   1. Initial program 98.5%

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

      1. lift-/.f32 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f32 N/A

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

      6. metadata-eval N/A

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

      7. frac-2neg N/A

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

      8. lower-/.f32 N/A

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

      9. lower-neg.f32 98.4

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

   4. Applied rewrites 98.4%

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

   5. Taylor expanded in u1 around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

      6. lower--.f32 98.5

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

   7. Applied rewrites 98.5%

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

   if 3.49999988e-4 < (/.f32 u1 (-.f32 #s(literal 1 binary32) u1))

   1. Initial program 98.5%

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

      1. lift-*.f32 N/A

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

      2. lift-sqrt.f32 N/A

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

      3. lift-/.f32 N/A

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

      4. sqrt-div N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f32 N/A

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

      8. lower-sqrt.f32 N/A

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

      9. lower-/.f32 N/A

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

      10. lift-*.f32 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f32 N/A

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

      13. lower-sqrt.f32 98.1

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

   4. Applied rewrites 98.1%

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

   5. Taylor expanded in u2 around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-out N/A

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

      8. *-commutative N/A

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

      9. distribute-lft-in N/A

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

      10. lower-*.f32 N/A

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

      11. lower-sqrt.f32 N/A

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

      12. lower-/.f32 N/A

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

      13. lower--.f32 N/A

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

   7. Applied rewrites 80.6%

      \[\leadsto \sqrt{u1} \cdot \color{blue}{\left(\sqrt{\frac{1}{1 - u1}} \cdot \left(\mathsf{fma}\left(u2 \cdot u2, -41.341702240407926, 6.28318530718\right) \cdot u2\right)\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 80.7%

      \[\leadsto \color{blue}{\sqrt{u1} \cdot \frac{\mathsf{fma}\left(-41.341702240407926, u2 \cdot u2, 6.28318530718\right) \cdot u2}{\sqrt{1 - u1}}} \]
   10. Step-by-step derivation

   11. Applied rewrites 90.2%

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

Alternative 5: 98.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

4. Applied rewrites 98.5%

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

Alternative 6: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

Alternative 7: 93.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{u2}{\sqrt{1 - u1}}\\ \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.30000001192092896:\\ \;\;\;\;\sqrt{u1} \cdot \left(t\_0 \cdot \left(\left(u2 \cdot u2\right) \cdot -41.341702240407926\right) + t\_0 \cdot 6.28318530718\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(6.28318530718 \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (/ u2 (sqrt (- 1.0 u1)))))
   (if (<= (* 6.28318530718 u2) 0.30000001192092896)
     (*
      (sqrt u1)
      (+ (* t_0 (* (* u2 u2) -41.341702240407926)) (* t_0 6.28318530718)))
     (* (sqrt u1) (sin (* 6.28318530718 u2))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = u2 / sqrtf((1.0f - u1));
	float tmp;
	if ((6.28318530718f * u2) <= 0.30000001192092896f) {
		tmp = sqrtf(u1) * ((t_0 * ((u2 * u2) * -41.341702240407926f)) + (t_0 * 6.28318530718f));
	} else {
		tmp = sqrtf(u1) * sinf((6.28318530718f * u2));
	}
	return tmp;
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    real(4) :: t_0
    real(4) :: tmp
    t_0 = u2 / sqrt((1.0e0 - u1))
    if ((6.28318530718e0 * u2) <= 0.30000001192092896e0) then
        tmp = sqrt(u1) * ((t_0 * ((u2 * u2) * (-41.341702240407926e0))) + (t_0 * 6.28318530718e0))
    else
        tmp = sqrt(u1) * sin((6.28318530718e0 * u2))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(u2 / sqrt(Float32(Float32(1.0) - u1)))
	tmp = Float32(0.0)
	if (Float32(Float32(6.28318530718) * u2) <= Float32(0.30000001192092896))
		tmp = Float32(sqrt(u1) * Float32(Float32(t_0 * Float32(Float32(u2 * u2) * Float32(-41.341702240407926))) + Float32(t_0 * Float32(6.28318530718))));
	else
		tmp = Float32(sqrt(u1) * sin(Float32(Float32(6.28318530718) * u2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	t_0 = u2 / sqrt((single(1.0) - u1));
	tmp = single(0.0);
	if ((single(6.28318530718) * u2) <= single(0.30000001192092896))
		tmp = sqrt(u1) * ((t_0 * ((u2 * u2) * single(-41.341702240407926))) + (t_0 * single(6.28318530718)));
	else
		tmp = sqrt(u1) * sin((single(6.28318530718) * u2));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.300000012

   1. Initial program 98.7%

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

      1. lift-*.f32 N/A

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

      2. lift-sqrt.f32 N/A

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

      3. lift-/.f32 N/A

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

      4. sqrt-div N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f32 N/A

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

      8. lower-sqrt.f32 N/A

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

      9. lower-/.f32 N/A

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

      10. lift-*.f32 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f32 N/A

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

      13. lower-sqrt.f32 98.3

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

   4. Applied rewrites 98.3%

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

   5. Taylor expanded in u2 around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-out N/A

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

      8. *-commutative N/A

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

      9. distribute-lft-in N/A

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

      10. lower-*.f32 N/A

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

      11. lower-sqrt.f32 N/A

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

      12. lower-/.f32 N/A

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

      13. lower--.f32 N/A

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

   7. Applied rewrites 88.9%

      \[\leadsto \sqrt{u1} \cdot \color{blue}{\left(\sqrt{\frac{1}{1 - u1}} \cdot \left(\mathsf{fma}\left(u2 \cdot u2, -41.341702240407926, 6.28318530718\right) \cdot u2\right)\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 88.9%

      \[\leadsto \color{blue}{\sqrt{u1} \cdot \frac{\mathsf{fma}\left(-41.341702240407926, u2 \cdot u2, 6.28318530718\right) \cdot u2}{\sqrt{1 - u1}}} \]
   10. Step-by-step derivation

   11. Applied rewrites 96.8%

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

   if 0.300000012 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 96.9%

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

   3. Taylor expanded in u1 around 0

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

      1. lower-sqrt.f32 78.5

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

   5. Applied rewrites 78.5%

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

Alternative 8: 88.9% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    real(4) :: t_0
    t_0 = u2 / sqrt((1.0e0 - u1))
    code = sqrt(u1) * ((t_0 * ((u2 * u2) * (-41.341702240407926e0))) + (t_0 * 6.28318530718e0))
end function

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	t_0 = u2 / sqrt((single(1.0) - u1));
	tmp = sqrt(u1) * ((t_0 * ((u2 * u2) * single(-41.341702240407926))) + (t_0 * single(6.28318530718)));
end

Derivation

1. Initial program 98.5%

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

   1. lift-*.f32 N/A

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

   2. lift-sqrt.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. sqrt-div N/A

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

   5. associate-*l/ N/A

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

   6. associate-/l* N/A

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

   7. lower-*.f32 N/A

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

   8. lower-sqrt.f32 N/A

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

   9. lower-/.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 N/A

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

   13. lower-sqrt.f32 98.2

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

4. Applied rewrites 98.2%

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

5. Taylor expanded in u2 around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. associate-*r* N/A

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

   7. distribute-rgt-out N/A

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

   8. *-commutative N/A

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

   9. distribute-lft-in N/A

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

   10. lower-*.f32 N/A

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

   11. lower-sqrt.f32 N/A

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

   12. lower-/.f32 N/A

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

   13. lower--.f32 N/A

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

7. Applied rewrites 80.9%

   \[\leadsto \sqrt{u1} \cdot \color{blue}{\left(\sqrt{\frac{1}{1 - u1}} \cdot \left(\mathsf{fma}\left(u2 \cdot u2, -41.341702240407926, 6.28318530718\right) \cdot u2\right)\right)} \]
8. Step-by-step derivation

9. Applied rewrites 80.9%

   \[\leadsto \color{blue}{\sqrt{u1} \cdot \frac{\mathsf{fma}\left(-41.341702240407926, u2 \cdot u2, 6.28318530718\right) \cdot u2}{\sqrt{1 - u1}}} \]
10. Step-by-step derivation

11. Applied rewrites 89.3%

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

Alternative 9: 88.9% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = sqrt(u1) * (((((u2 * u2) * (-41.341702240407926e0)) + 6.28318530718e0) * u2) / sqrt((1.0e0 - u1)))
end function

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt(u1) * (((((u2 * u2) * single(-41.341702240407926)) + single(6.28318530718)) * u2) / sqrt((single(1.0) - u1)));
end

Derivation

1. Initial program 98.5%

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

   1. lift-*.f32 N/A

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

   2. lift-sqrt.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. sqrt-div N/A

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

   5. associate-*l/ N/A

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

   6. associate-/l* N/A

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

   7. lower-*.f32 N/A

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

   8. lower-sqrt.f32 N/A

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

   9. lower-/.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 N/A

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

   13. lower-sqrt.f32 98.2

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

4. Applied rewrites 98.2%

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

5. Taylor expanded in u2 around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. associate-*r* N/A

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

   7. distribute-rgt-out N/A

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

   8. *-commutative N/A

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

   9. distribute-lft-in N/A

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

   10. lower-*.f32 N/A

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

   11. lower-sqrt.f32 N/A

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

   12. lower-/.f32 N/A

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

   13. lower--.f32 N/A

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

7. Applied rewrites 80.9%

   \[\leadsto \sqrt{u1} \cdot \color{blue}{\left(\sqrt{\frac{1}{1 - u1}} \cdot \left(\mathsf{fma}\left(u2 \cdot u2, -41.341702240407926, 6.28318530718\right) \cdot u2\right)\right)} \]
8. Step-by-step derivation

9. Applied rewrites 80.9%

   \[\leadsto \color{blue}{\sqrt{u1} \cdot \frac{\mathsf{fma}\left(-41.341702240407926, u2 \cdot u2, 6.28318530718\right) \cdot u2}{\sqrt{1 - u1}}} \]
10. Step-by-step derivation

11. Applied rewrites 89.2%

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

Alternative 10: 81.5% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

5. Applied rewrites 81.5%

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

7. Applied rewrites 81.5%

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

Alternative 11: 81.5% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

5. Applied rewrites 81.5%

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

Alternative 12: 64.4% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = (sqrt(u1) * 6.28318530718e0) * u2
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

5. Applied rewrites 81.5%

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

6. Taylor expanded in u1 around 0

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

8. Applied rewrites 66.6%

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

10. Applied rewrites 66.6%

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

Reproduce

herbie shell --seed 2024302 
(FPCore (cosTheta_i u1 u2)
  :name "Trowbridge-Reitz Sample, near normal, slope_y"
  :precision binary32
  :pre (and (and (and (> cosTheta_i 0.9999) (<= cosTheta_i 1.0)) (and (<= 2.328306437e-10 u1) (<= u1 1.0))) (and (<= 2.328306437e-10 u2) (<= u2 1.0)))
  (* (sqrt (/ u1 (- 1.0 u1))) (sin (* 6.28318530718 u2))))