Trowbridge-Reitz Sample, near normal, slope_y

Percentage Accurate: 98.3% → 98.3%

Time: 8.7s

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, 0.4× speedup

Math

\[\begin{array}{l} \\ {\left({\left(\frac{u1}{1 - u1}\right)}^{2}\right)}^{0.25} \cdot \sin \left(6.28318530718 \cdot u2\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (pow (pow (/ u1 (- 1.0 u1)) 2.0) 0.25) (sin (* 6.28318530718 u2))))

C

float code(float cosTheta_i, float u1, float u2) {
	return powf(powf((u1 / (1.0f - u1)), 2.0f), 0.25f) * 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 = (((u1 / (1.0e0 - u1)) ** 2.0e0) ** 0.25e0) * sin((6.28318530718e0 * u2))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(((Float32(u1 / Float32(Float32(1.0) - u1)) ^ Float32(2.0)) ^ Float32(0.25)) * sin(Float32(Float32(6.28318530718) * u2)))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = (((u1 / (single(1.0) - u1)) ^ single(2.0)) ^ single(0.25)) * sin((single(6.28318530718) * u2));
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-sqrt.f32 N/A

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

   2. pow1/2 N/A

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

   3. sqr-pow N/A

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

   4. pow-prod-down N/A

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

   5. lower-pow.f32 N/A

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

   6. pow2 N/A

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

   7. lower-pow.f32 N/A

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

   8. metadata-eval 98.5

      \[\leadsto {\left({\left(\frac{u1}{1 - u1}\right)}^{2}\right)}^{\color{blue}{0.25}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

4. Applied rewrites 98.5%

   \[\leadsto \color{blue}{{\left({\left(\frac{u1}{1 - u1}\right)}^{2}\right)}^{0.25}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
5. Add Preprocessing

Alternative 2: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

Alternative 3: 92.1% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = sqrt(((((-1.0e0) - u1) * u1) / ((-1.0e0) + (u1 * u1)))) * ((6.28318530718e0 + ((u2 * u2) * (((u2 * u2) * 81.6052492761019e0) - 41.341702240407926e0))) * u2)
end function

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((((single(-1.0) - u1) * u1) / (single(-1.0) + (u1 * u1)))) * ((single(6.28318530718) + ((u2 * u2) * (((u2 * u2) * single(81.6052492761019)) - single(41.341702240407926)))) * u2);
end

Derivation

1. Initial program 98.4%

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

4. Applied rewrites 98.4%

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

5. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. lower--.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f32 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f32 81.1

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

7. Applied rewrites 80.7%

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

9. Applied rewrites 92.9%

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

10. Final simplification 92.9%

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

Alternative 4: 92.1% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = sqrt(((((-1.0e0) - u1) * u1) / ((-1.0e0) + (u1 * u1)))) * (((((((u2 * u2) * 81.6052492761019e0) - 41.341702240407926e0) * u2) * u2) + 6.28318530718e0) * u2)
end function

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((((single(-1.0) - u1) * u1) / (single(-1.0) + (u1 * u1)))) * (((((((u2 * u2) * single(81.6052492761019)) - single(41.341702240407926)) * u2) * u2) + single(6.28318530718)) * u2);
end

Derivation

1. Initial program 98.4%

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

4. Applied rewrites 98.4%

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

5. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. lower--.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f32 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f32 81.1

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

7. Applied rewrites 80.7%

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

9. Applied rewrites 92.9%

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

Alternative 5: 82.3% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

3. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-sqrt.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. lower--.f32 N/A

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

   8. lower-*.f32 81.1

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

5. Applied rewrites 81.1%

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

Alternative 6: 65.1% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

3. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-sqrt.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. lower--.f32 N/A

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

   8. lower-*.f32 81.1

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

5. Applied rewrites 81.1%

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

7. Applied rewrites 62.9%

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

8. Taylor expanded in u1 around 0

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

10. Applied rewrites 64.6%

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

Alternative 7: 65.1% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

3. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-sqrt.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. lower--.f32 N/A

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

   8. lower-*.f32 81.1

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

5. Applied rewrites 81.1%

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

6. Taylor expanded in u1 around 0

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

8. Applied rewrites 64.5%

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

10. Applied rewrites 64.5%

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

Alternative 8: 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 98.4%

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

3. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-sqrt.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. lower--.f32 N/A

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

   8. lower-*.f32 81.1

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

5. Applied rewrites 81.1%

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

7. Applied rewrites 62.9%

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

8. Taylor expanded in u1 around inf

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

10. Applied rewrites 19.9%

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

Alternative 9: 4.3% 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 98.4%

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

3. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-sqrt.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. lower--.f32 N/A

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

   8. lower-*.f32 81.1

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

5. Applied rewrites 81.1%

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

7. Applied rewrites 62.9%

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

8. Taylor expanded in u1 around -inf

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

10. Applied rewrites 4.1%

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

Reproduce

herbie shell --seed 2024339 
(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))))