Trowbridge-Reitz Sample, near normal, slope_y

Percentage Accurate: 98.3% → 98.3%

Time: 8.3s

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: 98.3% 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, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

   1. lift-/.f32 N/A

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

   2. frac-2neg N/A

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

   3. neg-sub0 N/A

      \[\leadsto \sqrt{\frac{\color{blue}{0 - u1}}{\mathsf{neg}\left(\left(1 - u1\right)\right)}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   4. div-sub N/A

      \[\leadsto \sqrt{\color{blue}{\frac{0}{\mathsf{neg}\left(\left(1 - u1\right)\right)} - \frac{u1}{\mathsf{neg}\left(\left(1 - u1\right)\right)}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   5. distribute-frac-neg2 N/A

      \[\leadsto \sqrt{\frac{0}{\mathsf{neg}\left(\left(1 - u1\right)\right)} - \color{blue}{\left(\mathsf{neg}\left(\frac{u1}{1 - u1}\right)\right)}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   6. distribute-frac-neg N/A

      \[\leadsto \sqrt{\frac{0}{\mathsf{neg}\left(\left(1 - u1\right)\right)} - \color{blue}{\frac{\mathsf{neg}\left(u1\right)}{1 - u1}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   7. frac-sub N/A

      \[\leadsto \sqrt{\color{blue}{\frac{0 \cdot \left(1 - u1\right) - \left(\mathsf{neg}\left(\left(1 - u1\right)\right)\right) \cdot \left(\mathsf{neg}\left(u1\right)\right)}{\left(\mathsf{neg}\left(\left(1 - u1\right)\right)\right) \cdot \left(1 - u1\right)}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   8. lower-/.f32 N/A

      \[\leadsto \sqrt{\color{blue}{\frac{0 \cdot \left(1 - u1\right) - \left(\mathsf{neg}\left(\left(1 - u1\right)\right)\right) \cdot \left(\mathsf{neg}\left(u1\right)\right)}{\left(\mathsf{neg}\left(\left(1 - u1\right)\right)\right) \cdot \left(1 - u1\right)}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   9. lower--.f32 N/A

      \[\leadsto \sqrt{\frac{\color{blue}{0 \cdot \left(1 - u1\right) - \left(\mathsf{neg}\left(\left(1 - u1\right)\right)\right) \cdot \left(\mathsf{neg}\left(u1\right)\right)}}{\left(\mathsf{neg}\left(\left(1 - u1\right)\right)\right) \cdot \left(1 - u1\right)}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   10. lower-*.f32 N/A

       \[\leadsto \sqrt{\frac{\color{blue}{0 \cdot \left(1 - u1\right)} - \left(\mathsf{neg}\left(\left(1 - u1\right)\right)\right) \cdot \left(\mathsf{neg}\left(u1\right)\right)}{\left(\mathsf{neg}\left(\left(1 - u1\right)\right)\right) \cdot \left(1 - u1\right)}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   11. lower-*.f32 N/A

       \[\leadsto \sqrt{\frac{0 \cdot \left(1 - u1\right) - \color{blue}{\left(\mathsf{neg}\left(\left(1 - u1\right)\right)\right) \cdot \left(\mathsf{neg}\left(u1\right)\right)}}{\left(\mathsf{neg}\left(\left(1 - u1\right)\right)\right) \cdot \left(1 - u1\right)}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   12. neg-sub0 N/A

       \[\leadsto \sqrt{\frac{0 \cdot \left(1 - u1\right) - \color{blue}{\left(0 - \left(1 - u1\right)\right)} \cdot \left(\mathsf{neg}\left(u1\right)\right)}{\left(\mathsf{neg}\left(\left(1 - u1\right)\right)\right) \cdot \left(1 - u1\right)}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   13. lift--.f32 N/A

       \[\leadsto \sqrt{\frac{0 \cdot \left(1 - u1\right) - \left(0 - \color{blue}{\left(1 - u1\right)}\right) \cdot \left(\mathsf{neg}\left(u1\right)\right)}{\left(\mathsf{neg}\left(\left(1 - u1\right)\right)\right) \cdot \left(1 - u1\right)}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   14. sub-neg N/A

       \[\leadsto \sqrt{\frac{0 \cdot \left(1 - u1\right) - \left(0 - \color{blue}{\left(1 + \left(\mathsf{neg}\left(u1\right)\right)\right)}\right) \cdot \left(\mathsf{neg}\left(u1\right)\right)}{\left(\mathsf{neg}\left(\left(1 - u1\right)\right)\right) \cdot \left(1 - u1\right)}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   15. +-commutative N/A

       \[\leadsto \sqrt{\frac{0 \cdot \left(1 - u1\right) - \left(0 - \color{blue}{\left(\left(\mathsf{neg}\left(u1\right)\right) + 1\right)}\right) \cdot \left(\mathsf{neg}\left(u1\right)\right)}{\left(\mathsf{neg}\left(\left(1 - u1\right)\right)\right) \cdot \left(1 - u1\right)}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   16. associate--r+ N/A

       \[\leadsto \sqrt{\frac{0 \cdot \left(1 - u1\right) - \color{blue}{\left(\left(0 - \left(\mathsf{neg}\left(u1\right)\right)\right) - 1\right)} \cdot \left(\mathsf{neg}\left(u1\right)\right)}{\left(\mathsf{neg}\left(\left(1 - u1\right)\right)\right) \cdot \left(1 - u1\right)}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   17. neg-sub0 N/A

       \[\leadsto \sqrt{\frac{0 \cdot \left(1 - u1\right) - \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(u1\right)\right)\right)\right)} - 1\right) \cdot \left(\mathsf{neg}\left(u1\right)\right)}{\left(\mathsf{neg}\left(\left(1 - u1\right)\right)\right) \cdot \left(1 - u1\right)}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   18. remove-double-neg N/A

       \[\leadsto \sqrt{\frac{0 \cdot \left(1 - u1\right) - \left(\color{blue}{u1} - 1\right) \cdot \left(\mathsf{neg}\left(u1\right)\right)}{\left(\mathsf{neg}\left(\left(1 - u1\right)\right)\right) \cdot \left(1 - u1\right)}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   19. lower--.f32 N/A

       \[\leadsto \sqrt{\frac{0 \cdot \left(1 - u1\right) - \color{blue}{\left(u1 - 1\right)} \cdot \left(\mathsf{neg}\left(u1\right)\right)}{\left(\mathsf{neg}\left(\left(1 - u1\right)\right)\right) \cdot \left(1 - u1\right)}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   20. lower-neg.f32 N/A

       \[\leadsto \sqrt{\frac{0 \cdot \left(1 - u1\right) - \left(u1 - 1\right) \cdot \color{blue}{\left(-u1\right)}}{\left(\mathsf{neg}\left(\left(1 - u1\right)\right)\right) \cdot \left(1 - u1\right)}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   21. lower-*.f32 N/A

       \[\leadsto \sqrt{\frac{0 \cdot \left(1 - u1\right) - \left(u1 - 1\right) \cdot \left(-u1\right)}{\color{blue}{\left(\mathsf{neg}\left(\left(1 - u1\right)\right)\right) \cdot \left(1 - u1\right)}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

4. Applied rewrites 98.5%

   \[\leadsto \sqrt{\color{blue}{\frac{0 \cdot \left(1 - u1\right) - \left(u1 - 1\right) \cdot \left(-u1\right)}{\left(u1 - 1\right) \cdot \left(1 - u1\right)}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

5. Final simplification 98.5%

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

Alternative 2: 98.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

4. Applied rewrites 98.4%

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

5. Final simplification 98.4%

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

Alternative 3: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

3. Final simplification 98.4%

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

Alternative 4: 89.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.019999999552965164:\\ \;\;\;\;\left(u2 \cdot 6.28318530718\right) \cdot \sqrt{\frac{-1}{\frac{u1 - 1}{u1}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(u2 \cdot 6.28318530718\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 6.28318530718) 0.019999999552965164)
   (* (* u2 6.28318530718) (sqrt (/ -1.0 (/ (- u1 1.0) u1))))
   (* (sqrt u1) (sin (* u2 6.28318530718)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * 6.28318530718f) <= 0.019999999552965164f) {
		tmp = (u2 * 6.28318530718f) * sqrtf((-1.0f / ((u1 - 1.0f) / 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) <= 0.019999999552965164e0) then
        tmp = (u2 * 6.28318530718e0) * sqrt(((-1.0e0) / ((u1 - 1.0e0) / 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(0.019999999552965164))
		tmp = Float32(Float32(u2 * Float32(6.28318530718)) * sqrt(Float32(Float32(-1.0) / Float32(Float32(u1 - Float32(1.0)) / 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(0.019999999552965164))
		tmp = (u2 * single(6.28318530718)) * sqrt((single(-1.0) / ((u1 - single(1.0)) / 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) < 0.0199999996

   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

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\left(\frac{314159265359}{50000000000} \cdot u2\right)} \]
   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 95.4

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

   5. Applied rewrites 95.4%

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

      1. lift-/.f32 N/A

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

      2. clear-num N/A

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

      3. lower-/.f32 N/A

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

      4. lower-/.f32 95.5

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

   7. Applied rewrites 95.5%

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

   if 0.0199999996 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 98.1%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
   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 76.4

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

   5. Applied rewrites 76.4%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.019999999552965164:\\ \;\;\;\;\left(u2 \cdot 6.28318530718\right) \cdot \sqrt{\frac{-1}{\frac{u1 - 1}{u1}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(u2 \cdot 6.28318530718\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.5% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

3. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 81.5

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

5. Applied rewrites 81.5%

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

   1. lift-/.f32 N/A

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

   2. clear-num N/A

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

   3. lower-/.f32 N/A

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

   4. lower-/.f32 81.6

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

7. Applied rewrites 81.6%

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

8. Final simplification 81.6%

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

Alternative 6: 81.5% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

3. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 81.5

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

5. Applied rewrites 81.5%

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

6. Final simplification 81.5%

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

Alternative 7: 64.4% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

3. 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 81.5

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

5. Applied rewrites 81.5%

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

6. Taylor expanded in u1 around 0

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

   1. lower-sqrt.f32 63.4

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

8. Applied rewrites 63.4%

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

Reproduce

herbie shell --seed 2024271 
(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))))