Trowbridge-Reitz Sample, near normal, slope_y

Percentage Accurate: 98.4% → 98.3%

Time: 9.6s

Alternatives: 7

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: 98.4% 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

\[\begin{array}{l} \\ {\left({\left(\frac{u1 - 1}{u1}\right)}^{4}\right)}^{-0.125} \cdot \sin \left(6.28318530718 \cdot u2\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (pow (pow (/ (- u1 1.0) u1) 4.0) -0.125) (sin (* 6.28318530718 u2))))

C

float code(float cosTheta_i, float u1, float u2) {
	return powf(powf(((u1 - 1.0f) / u1), 4.0f), -0.125f) * 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) ** 4.0e0) ** (-0.125e0)) * sin((6.28318530718e0 * u2))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(((Float32(Float32(u1 - Float32(1.0)) / u1) ^ Float32(4.0)) ^ Float32(-0.125)) * sin(Float32(Float32(6.28318530718) * u2)))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = ((((u1 - single(1.0)) / u1) ^ single(4.0)) ^ single(-0.125)) * sin((single(6.28318530718) * u2));
end

Derivation

1. Initial program 98.5%

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

4. Applied rewrites 98.5%

   \[\leadsto \color{blue}{{\left(\frac{u1 - 1}{u1} \cdot \frac{u1 - 1}{u1}\right)}^{-0.25}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
5. Step-by-step derivation

   1. lift-pow.f32 N/A

      \[\leadsto \color{blue}{{\left(\frac{u1 - 1}{u1} \cdot \frac{u1 - 1}{u1}\right)}^{\frac{-1}{4}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   2. sqr-pow N/A

      \[\leadsto \color{blue}{\left({\left(\frac{u1 - 1}{u1} \cdot \frac{u1 - 1}{u1}\right)}^{\left(\frac{\frac{-1}{4}}{2}\right)} \cdot {\left(\frac{u1 - 1}{u1} \cdot \frac{u1 - 1}{u1}\right)}^{\left(\frac{\frac{-1}{4}}{2}\right)}\right)} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   3. pow-prod-down N/A

      \[\leadsto \color{blue}{{\left(\left(\frac{u1 - 1}{u1} \cdot \frac{u1 - 1}{u1}\right) \cdot \left(\frac{u1 - 1}{u1} \cdot \frac{u1 - 1}{u1}\right)\right)}^{\left(\frac{\frac{-1}{4}}{2}\right)}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   4. lower-pow.f32 N/A

      \[\leadsto \color{blue}{{\left(\left(\frac{u1 - 1}{u1} \cdot \frac{u1 - 1}{u1}\right) \cdot \left(\frac{u1 - 1}{u1} \cdot \frac{u1 - 1}{u1}\right)\right)}^{\left(\frac{\frac{-1}{4}}{2}\right)}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   5. pow2 N/A

      \[\leadsto {\color{blue}{\left({\left(\frac{u1 - 1}{u1} \cdot \frac{u1 - 1}{u1}\right)}^{2}\right)}}^{\left(\frac{\frac{-1}{4}}{2}\right)} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   6. lift-*.f32 N/A

      \[\leadsto {\left({\color{blue}{\left(\frac{u1 - 1}{u1} \cdot \frac{u1 - 1}{u1}\right)}}^{2}\right)}^{\left(\frac{\frac{-1}{4}}{2}\right)} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   7. pow-prod-down N/A

      \[\leadsto {\color{blue}{\left({\left(\frac{u1 - 1}{u1}\right)}^{2} \cdot {\left(\frac{u1 - 1}{u1}\right)}^{2}\right)}}^{\left(\frac{\frac{-1}{4}}{2}\right)} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   8. pow-prod-up N/A

      \[\leadsto {\color{blue}{\left({\left(\frac{u1 - 1}{u1}\right)}^{\left(2 + 2\right)}\right)}}^{\left(\frac{\frac{-1}{4}}{2}\right)} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   9. lower-pow.f32 N/A

      \[\leadsto {\color{blue}{\left({\left(\frac{u1 - 1}{u1}\right)}^{\left(2 + 2\right)}\right)}}^{\left(\frac{\frac{-1}{4}}{2}\right)} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   10. metadata-eval N/A

       \[\leadsto {\left({\left(\frac{u1 - 1}{u1}\right)}^{\color{blue}{4}}\right)}^{\left(\frac{\frac{-1}{4}}{2}\right)} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   11. metadata-eval 98.5

       \[\leadsto {\left({\left(\frac{u1 - 1}{u1}\right)}^{4}\right)}^{\color{blue}{-0.125}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

6. Applied rewrites 98.5%

   \[\leadsto \color{blue}{{\left({\left(\frac{u1 - 1}{u1}\right)}^{4}\right)}^{-0.125}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
7. Add Preprocessing

Alternative 2: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

Alternative 3: 94.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.3199999928474426:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \left(\left(-41.341702240407926 \cdot \left(u2 \cdot u2\right)\right) \cdot u2 + u2 \cdot 6.28318530718\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(6.28318530718 \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.3199999928474426)
   (*
    (sqrt (/ u1 (- 1.0 u1)))
    (+ (* (* -41.341702240407926 (* u2 u2)) u2) (* u2 6.28318530718)))
   (* (sqrt u1) (sin (* 6.28318530718 u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.3199999928474426f) {
		tmp = sqrtf((u1 / (1.0f - u1))) * (((-41.341702240407926f * (u2 * u2)) * u2) + (u2 * 6.28318530718f));
	} else {
		tmp = sqrtf(u1) * sinf((6.28318530718f * u2));
	}
	return tmp;
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    real(4) :: tmp
    if ((6.28318530718e0 * u2) <= 0.3199999928474426e0) then
        tmp = sqrt((u1 / (1.0e0 - u1))) * ((((-41.341702240407926e0) * (u2 * u2)) * u2) + (u2 * 6.28318530718e0))
    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(0.3199999928474426))
		tmp = Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * Float32(Float32(Float32(Float32(-41.341702240407926) * Float32(u2 * u2)) * u2) + Float32(u2 * Float32(6.28318530718))));
	else
		tmp = Float32(sqrt(u1) * sin(Float32(Float32(6.28318530718) * u2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((single(6.28318530718) * u2) <= single(0.3199999928474426))
		tmp = sqrt((u1 / (single(1.0) - u1))) * (((single(-41.341702240407926) * (u2 * u2)) * u2) + (u2 * single(6.28318530718)));
	else
		tmp = sqrt(u1) * sin((single(6.28318530718) * u2));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.319999993

   1. Initial program 98.7%

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

   3. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f32 89.6

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

   5. Applied rewrites 89.2%

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

   7. Applied rewrites 97.1%

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

   if 0.319999993 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 97.6%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
   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 77.4

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

   5. Applied rewrites 77.4%

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

Alternative 4: 89.0% accurate, 2.5× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - u1))) * (((-41.341702240407926f * (u2 * u2)) * u2) + (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))) * ((((-41.341702240407926e0) * (u2 * u2)) * u2) + (u2 * 6.28318530718e0))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f32 80.1

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

5. Applied rewrites 80.1%

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

7. Applied rewrites 88.4%

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

Alternative 5: 89.1% accurate, 2.8× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - u1))) * (((-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((u1 / (1.0e0 - u1))) * ((((-41.341702240407926e0) * (u2 * u2)) + 6.28318530718e0) * u2)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f32 80.1

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

5. Applied rewrites 80.0%

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

7. Applied rewrites 88.4%

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

Alternative 6: 81.5% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

5. Applied rewrites 80.1%

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

Alternative 7: 64.6% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

5. Applied rewrites 80.1%

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

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 64.6%

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

10. Applied rewrites 64.7%

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

Reproduce

herbie shell --seed 2024303 
(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))))