Trowbridge-Reitz Sample, near normal, slope_y

Percentage Accurate: 98.3% → 98.3%

Time: 10.3s

Alternatives: 10

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

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-*r/ N/A

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

   4. *-commutative N/A

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

   5. associate-/l* N/A

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

   6. clear-num N/A

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

   7. div-inv N/A

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

   8. lower-/.f32 N/A

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

   9. lift-sqrt.f32 N/A

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

   10. lift-sqrt.f32 N/A

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

   11. sqrt-undiv N/A

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

   12. lower-sqrt.f32 N/A

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

   13. lower-/.f32 98.6

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

6. Applied rewrites 98.6%

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

7. Final simplification 98.6%

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

Alternative 2: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-*r/ N/A

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

   4. *-commutative N/A

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

   5. associate-/l* N/A

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

   6. clear-num N/A

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

   7. div-inv N/A

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

   8. lower-/.f32 N/A

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

   9. lift-sqrt.f32 N/A

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

   10. lift-sqrt.f32 N/A

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

   11. sqrt-undiv N/A

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

   12. lower-sqrt.f32 N/A

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

   13. lower-/.f32 98.6

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

6. Applied rewrites 98.6%

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

   1. lift-/.f32 N/A

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

   2. lift--.f32 N/A

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

   3. div-sub N/A

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

   4. *-inverses N/A

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

   5. lower--.f32 N/A

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

   6. lower-/.f32 98.5

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

8. Applied rewrites 98.5%

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

9. Final simplification 98.5%

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

Alternative 3: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

3. Final simplification 98.3%

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

Alternative 4: 90.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.007499999832361937:\\ \;\;\;\;\frac{6.28318530718 \cdot u2}{\sqrt{\frac{1 - u1}{u1}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(6.28318530718 \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.007499999832361937)
   (/ (* 6.28318530718 u2) (sqrt (/ (- 1.0 u1) u1)))
   (* (sqrt u1) (sin (* 6.28318530718 u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.007499999832361937f) {
		tmp = (6.28318530718f * u2) / sqrtf(((1.0f - u1) / u1));
	} 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) :: tmp
    if ((6.28318530718e0 * u2) <= 0.007499999832361937e0) then
        tmp = (6.28318530718e0 * u2) / sqrt(((1.0e0 - u1) / u1))
    else
        tmp = sqrt(u1) * sin((6.28318530718e0 * u2))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(6.28318530718) * u2) <= Float32(0.007499999832361937))
		tmp = Float32(Float32(Float32(6.28318530718) * u2) / sqrt(Float32(Float32(Float32(1.0) - u1) / u1)));
	else
		tmp = Float32(sqrt(u1) * sin(Float32(Float32(6.28318530718) * u2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((single(6.28318530718) * u2) <= single(0.007499999832361937))
		tmp = (single(6.28318530718) * u2) / sqrt(((single(1.0) - u1) / u1));
	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) < 0.00749999983

   1. Initial program 98.6%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
   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 98.4

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

   4. Applied rewrites 98.4%

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

      1. lift-*.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. clear-num N/A

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

      7. div-inv N/A

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

      8. lower-/.f32 N/A

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

      9. lift-sqrt.f32 N/A

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

      10. lift-sqrt.f32 N/A

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

      11. sqrt-undiv N/A

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

      12. lower-sqrt.f32 N/A

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

      13. lower-/.f32 98.9

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

   6. Applied rewrites 98.9%

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

   7. Taylor expanded in u2 around 0

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

      1. lower-*.f32 97.2

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

   9. Applied rewrites 97.2%

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

   if 0.00749999983 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 97.4%

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

   3. Taylor expanded in u1 around 0

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

   5. Applied rewrites 79.1%

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

Alternative 5: 81.4% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-*r/ N/A

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

   4. *-commutative N/A

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

   5. associate-/l* N/A

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

   6. clear-num N/A

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

   7. div-inv N/A

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

   8. lower-/.f32 N/A

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

   9. lift-sqrt.f32 N/A

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

   10. lift-sqrt.f32 N/A

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

   11. sqrt-undiv N/A

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

   12. lower-sqrt.f32 N/A

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

   13. lower-/.f32 98.6

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

6. Applied rewrites 98.6%

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

7. Taylor expanded in u2 around 0

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

   1. lower-*.f32 83.6

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

9. Applied rewrites 83.6%

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

Alternative 6: 81.3% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-*r/ N/A

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

   4. *-commutative N/A

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

   5. associate-/l* N/A

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

   6. clear-num N/A

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

   7. div-inv N/A

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

   8. lower-/.f32 N/A

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

   9. lift-sqrt.f32 N/A

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

   10. lift-sqrt.f32 N/A

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

   11. sqrt-undiv N/A

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

   12. lower-sqrt.f32 N/A

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

   13. lower-/.f32 98.6

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

6. Applied rewrites 98.6%

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

   1. lift-/.f32 N/A

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

   2. lift--.f32 N/A

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

   3. div-sub N/A

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

   4. *-inverses N/A

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

   5. lower--.f32 N/A

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

   6. lower-/.f32 98.5

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

8. Applied rewrites 98.5%

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

9. Taylor expanded in u2 around 0

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

    1. lower-*.f32 83.5

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

11. Applied rewrites 83.5%

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

Alternative 7: 81.4% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
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. lower-*.f32 83.5

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

5. Applied rewrites 83.5%

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

6. Final simplification 83.5%

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

Alternative 8: 81.4% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

3. Taylor expanded in u2 around 0

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

5. Applied rewrites 83.3%

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

6. Taylor expanded in u2 around 0

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

8. Applied rewrites 83.3%

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

9. Final simplification 83.3%

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

Alternative 9: 64.3% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
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. lower-*.f32 83.5

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

5. Applied rewrites 83.5%

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

6. Taylor expanded in u1 around 0

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

8. Applied rewrites 65.2%

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

9. Final simplification 65.2%

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

Alternative 10: 19.6% accurate, 22.5× speedup

Math

\[\begin{array}{l} \\ 6.28318530718 \cdot u2 \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

4. Applied rewrites 98.3%

   \[\leadsto \color{blue}{{\left(\frac{u1 - 1}{u1} \cdot \frac{u1 - 1}{u1}\right)}^{-0.25}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

5. Taylor expanded in u1 around inf

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

   1. lower-sin.f32 N/A

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

   2. lower-*.f32 20.1

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

7. Applied rewrites 20.1%

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

8. Taylor expanded in u2 around 0

   \[\leadsto \frac{314159265359}{50000000000} \cdot \color{blue}{u2} \]
9. Step-by-step derivation

10. Applied rewrites 19.7%

    \[\leadsto 6.28318530718 \cdot \color{blue}{u2} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024272 
(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))))