Trowbridge-Reitz Sample, near normal, slope_y

Percentage Accurate: 98.3% → 98.3%

Time: 9.8s

Alternatives: 9

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, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
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. *-commutative N/A

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

   3. lift-sqrt.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. sqrt-div N/A

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

   6. associate-*r/ N/A

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

   7. lower-/.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. lower-sqrt.f32 N/A

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

   11. lift-*.f32 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f32 N/A

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

   14. lower-sqrt.f32 98.2

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

4. Applied rewrites 98.2%

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

   1. lift-/.f32 N/A

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

   2. clear-num N/A

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

   3. lower-/.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f32 N/A

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

   8. associate-/r* N/A

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

   9. lower-/.f32 N/A

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

   10. lift-sqrt.f32 N/A

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

   11. lift-sqrt.f32 N/A

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

   12. sqrt-div N/A

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

   13. lower-sqrt.f32 N/A

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

   14. lower-/.f32 98.3

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

   15. lift-*.f32 N/A

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

   16. *-commutative N/A

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

   17. lift-*.f32 98.3

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

6. Applied rewrites 98.3%

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

   1. lift-/.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. clear-num N/A

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

   4. lower-/.f32 98.5

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 98.5

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

8. Applied rewrites 98.5%

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

9. Final simplification 98.5%

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

Alternative 2: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 = sin((u2 * 6.28318530718e0)) / sqrt(((1.0e0 / u1) - 1.0e0))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
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. *-commutative N/A

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

   3. lift-sqrt.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. sqrt-div N/A

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

   6. associate-*r/ N/A

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

   7. lower-/.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. lower-sqrt.f32 N/A

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

   11. lift-*.f32 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f32 N/A

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

   14. lower-sqrt.f32 98.2

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

4. Applied rewrites 98.2%

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

   1. lift-/.f32 N/A

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

   2. clear-num N/A

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

   3. lower-/.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f32 N/A

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

   8. associate-/r* N/A

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

   9. lower-/.f32 N/A

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

   10. lift-sqrt.f32 N/A

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

   11. lift-sqrt.f32 N/A

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

   12. sqrt-div N/A

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

   13. lower-sqrt.f32 N/A

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

   14. lower-/.f32 98.3

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

   15. lift-*.f32 N/A

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

   16. *-commutative N/A

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

   17. lift-*.f32 98.3

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

6. Applied rewrites 98.3%

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

   1. lift-/.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. clear-num N/A

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

   4. lower-/.f32 98.5

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 98.5

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

8. Applied rewrites 98.5%

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

   1. lift-/.f32 N/A

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

   2. lift--.f32 N/A

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

   3. div-sub N/A

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

   4. lift-/.f32 N/A

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

   5. *-inverses N/A

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

   6. lift--.f32 98.4

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

10. Applied rewrites 98.4%

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

11. Final simplification 98.4%

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

Alternative 3: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

3. Final simplification 98.4%

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

Alternative 4: 94.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.35499998927116394:\\ \;\;\;\;\left(\left(\left(\mathsf{fma}\left(u2 \cdot u2, 81.6052492761019, -41.341702240407926\right) \cdot u2\right) \cdot u2 + 6.28318530718\right) \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(u2 \cdot 6.28318530718\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 6.28318530718) 0.35499998927116394)
   (*
    (*
     (+
      (* (* (fma (* u2 u2) 81.6052492761019 -41.341702240407926) u2) u2)
      6.28318530718)
     u2)
    (sqrt (/ u1 (- 1.0 u1))))
   (* (sqrt u1) (sin (* u2 6.28318530718)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * 6.28318530718f) <= 0.35499998927116394f) {
		tmp = ((((fmaf((u2 * u2), 81.6052492761019f, -41.341702240407926f) * u2) * u2) + 6.28318530718f) * u2) * sqrtf((u1 / (1.0f - u1)));
	} else {
		tmp = sqrtf(u1) * sinf((u2 * 6.28318530718f));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(u2 * Float32(6.28318530718)) <= Float32(0.35499998927116394))
		tmp = Float32(Float32(Float32(Float32(Float32(fma(Float32(u2 * u2), Float32(81.6052492761019), Float32(-41.341702240407926)) * u2) * u2) + Float32(6.28318530718)) * u2) * sqrt(Float32(u1 / Float32(Float32(1.0) - u1))));
	else
		tmp = Float32(sqrt(u1) * sin(Float32(u2 * Float32(6.28318530718))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.354999989

   1. Initial program 98.4%

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

   3. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 87.9

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

   5. Applied rewrites 87.9%

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

   6. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f32 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f32 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f32 87.9

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

   8. Applied rewrites 87.9%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(81.6052492761019, u2 \cdot u2, -41.341702240407926\right), u2 \cdot u2, 6.28318530718\right) \cdot u2\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 96.5%

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

   if 0.354999989 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 97.8%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
   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 75.5

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

   5. Applied rewrites 75.5%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.35499998927116394:\\ \;\;\;\;\left(\left(\left(\mathsf{fma}\left(u2 \cdot u2, 81.6052492761019, -41.341702240407926\right) \cdot u2\right) \cdot u2 + 6.28318530718\right) \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(u2 \cdot 6.28318530718\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.5% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.4%

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

3. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 82.1

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

5. Applied rewrites 82.1%

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

6. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. sub-neg N/A

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

   7. metadata-eval N/A

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

   8. lower-fma.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f32 82.1

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

8. Applied rewrites 82.1%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(81.6052492761019, u2 \cdot u2, -41.341702240407926\right), u2 \cdot u2, 6.28318530718\right) \cdot u2\right)} \]
9. Step-by-step derivation

10. Applied rewrites 90.6%

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

11. Final simplification 90.9%

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

Alternative 6: 81.8% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
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. *-commutative N/A

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

   3. lift-sqrt.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. sqrt-div N/A

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

   6. associate-*r/ N/A

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

   7. lower-/.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. lower-sqrt.f32 N/A

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

   11. lift-*.f32 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f32 N/A

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

   14. lower-sqrt.f32 98.2

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

4. Applied rewrites 98.2%

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

   1. lift-/.f32 N/A

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

   2. clear-num N/A

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

   3. lower-/.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f32 N/A

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

   8. associate-/r* N/A

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

   9. lower-/.f32 N/A

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

   10. lift-sqrt.f32 N/A

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

   11. lift-sqrt.f32 N/A

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

   12. sqrt-div N/A

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

   13. lower-sqrt.f32 N/A

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

   14. lower-/.f32 98.3

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

   15. lift-*.f32 N/A

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

   16. *-commutative N/A

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

   17. lift-*.f32 98.3

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

6. Applied rewrites 98.3%

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

   1. lift-/.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. clear-num N/A

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

   4. lower-/.f32 98.5

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 98.5

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

8. Applied rewrites 98.5%

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

9. Taylor expanded in u2 around 0

   \[\leadsto \frac{\color{blue}{\frac{314159265359}{50000000000} \cdot u2}}{\sqrt{\frac{1 - u1}{u1}}} \]
10. Step-by-step derivation

    1. *-commutative N/A

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

    2. lower-*.f32 82.3

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

11. Applied rewrites 82.3%

    \[\leadsto \frac{\color{blue}{u2 \cdot 6.28318530718}}{\sqrt{\frac{1 - u1}{u1}}} \]
12. Add Preprocessing

Alternative 7: 81.8% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

3. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 82.1

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

5. Applied rewrites 82.1%

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

6. Final simplification 82.1%

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

Alternative 8: 68.9% accurate, 4.0× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return (((-41.341702240407926f * (u2 * u2)) + 6.28318530718f) * u2) * sqrtf(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 = ((((-41.341702240407926e0) * (u2 * u2)) + 6.28318530718e0) * u2) * sqrt(u1)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

3. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 82.1

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

5. Applied rewrites 82.1%

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

6. Taylor expanded in u1 around 0

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

   1. lower-sqrt.f32 63.7

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

8. Applied rewrites 63.7%

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

9. Taylor expanded in u2 around 0

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

    1. *-commutative N/A

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

    2. lower-*.f32 N/A

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

    3. +-commutative N/A

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

    4. *-commutative N/A

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

    5. lower-fma.f32 N/A

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

    6. unpow2 N/A

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

    7. lower-*.f32 63.7

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

11. Applied rewrites 63.4%

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

13. Applied rewrites 67.7%

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

14. Final simplification 67.7%

    \[\leadsto \left(\left(-41.341702240407926 \cdot \left(u2 \cdot u2\right) + 6.28318530718\right) \cdot u2\right) \cdot \sqrt{u1} \]
15. Add Preprocessing

Alternative 9: 64.9% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

3. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 82.1

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

5. Applied rewrites 82.1%

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

6. Taylor expanded in u1 around 0

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

   1. lower-sqrt.f32 63.7

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

8. Applied rewrites 63.7%

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

9. Final simplification 63.7%

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

Reproduce

herbie shell --seed 2024268 
(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))))