Trowbridge-Reitz Sample, near normal, slope_y

Percentage Accurate: 98.3% → 98.3%

Time: 8.5s

Alternatives: 6

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

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sin((u2 * single(6.28318530718))) * 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%

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

Alternative 2: 93.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.4000000059604645:\\ \;\;\;\;\frac{\left(\left(u2 \cdot u2\right) \cdot -41.341702240407926 + 6.28318530718\right) \cdot u2}{\sqrt{1 - u1}} \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(u2 \cdot 6.28318530718\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 6.28318530718) 0.4000000059604645)
   (*
    (/
     (* (+ (* (* u2 u2) -41.341702240407926) 6.28318530718) u2)
     (sqrt (- 1.0 u1)))
    (sqrt u1))
   (* (sqrt u1) (sin (* u2 6.28318530718)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * 6.28318530718f) <= 0.4000000059604645f) {
		tmp = (((((u2 * u2) * -41.341702240407926f) + 6.28318530718f) * u2) / sqrtf((1.0f - u1))) * sqrtf(u1);
	} else {
		tmp = sqrtf(u1) * sinf((u2 * 6.28318530718f));
	}
	return tmp;
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    real(4) :: tmp
    if ((u2 * 6.28318530718e0) <= 0.4000000059604645e0) then
        tmp = (((((u2 * u2) * (-41.341702240407926e0)) + 6.28318530718e0) * u2) / sqrt((1.0e0 - u1))) * sqrt(u1)
    else
        tmp = sqrt(u1) * sin((u2 * 6.28318530718e0))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(u2 * Float32(6.28318530718)) <= Float32(0.4000000059604645))
		tmp = Float32(Float32(Float32(Float32(Float32(Float32(u2 * u2) * Float32(-41.341702240407926)) + Float32(6.28318530718)) * u2) / sqrt(Float32(Float32(1.0) - u1))) * sqrt(u1));
	else
		tmp = Float32(sqrt(u1) * sin(Float32(u2 * Float32(6.28318530718))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((u2 * single(6.28318530718)) <= single(0.4000000059604645))
		tmp = (((((u2 * u2) * single(-41.341702240407926)) + single(6.28318530718)) * u2) / sqrt((single(1.0) - u1))) * sqrt(u1);
	else
		tmp = sqrt(u1) * sin((u2 * single(6.28318530718)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.400000006

   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 98.1

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

   4. Applied rewrites 98.1%

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

   5. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f32 89.3

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

   7. Applied rewrites 88.8%

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

   9. Applied rewrites 96.7%

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

   if 0.400000006 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 96.8%

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

   3. Taylor expanded in u1 around 0

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

      1. lower-sqrt.f32 71.1

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

   5. Applied rewrites 71.1%

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

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.4000000059604645:\\ \;\;\;\;\frac{\left(\left(u2 \cdot u2\right) \cdot -41.341702240407926 + 6.28318530718\right) \cdot u2}{\sqrt{1 - u1}} \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(u2 \cdot 6.28318530718\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.7% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = (((((u2 * u2) * single(-41.341702240407926)) + single(6.28318530718)) * u2) / sqrt((single(1.0) - u1))) * 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. 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 97.9

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

4. Applied rewrites 97.9%

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

5. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f32 79.1

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

7. Applied rewrites 78.7%

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

9. Applied rewrites 86.7%

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

10. Final simplification 86.7%

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

Alternative 4: 81.3% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

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

   2. lower-*.f32 79.3

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

5. Applied rewrites 79.3%

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

6. Final simplification 79.3%

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

Alternative 5: 73.4% accurate, 4.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

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

   2. lower-*.f32 79.3

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

5. Applied rewrites 79.3%

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

6. Taylor expanded in u1 around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-rgt-identity N/A

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

   4. lower-fma.f32 62.4

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

8. Applied rewrites 62.1%

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

10. Applied rewrites 71.1%

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

Alternative 6: 65.1% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

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

   2. lower-*.f32 79.3

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

5. Applied rewrites 79.3%

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

6. Taylor expanded in u1 around 0

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

   1. lower-sqrt.f32 62.4

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

8. Applied rewrites 62.4%

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

9. Final simplification 62.4%

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

Reproduce

herbie shell --seed 2024270 
(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))))