Trowbridge-Reitz Sample, near normal, slope_y

Percentage Accurate: 98.3% → 98.2%

Time: 11.1s

Alternatives: 12

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: 98.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ {\left(\frac{1}{u1} + -1\right)}^{-0.5} \cdot \sin \left(6.28318530718 \cdot u2\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (pow (+ (/ 1.0 u1) -1.0) -0.5) (sin (* 6.28318530718 u2))))

C

float code(float cosTheta_i, float u1, float u2) {
	return powf(((1.0f / u1) + -1.0f), -0.5f) * 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 = (((1.0e0 / u1) + (-1.0e0)) ** (-0.5e0)) * sin((6.28318530718e0 * u2))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32((Float32(Float32(Float32(1.0) / u1) + Float32(-1.0)) ^ Float32(-0.5)) * sin(Float32(Float32(6.28318530718) * u2)))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = (((single(1.0) / u1) + single(-1.0)) ^ single(-0.5)) * 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. clear-num 98.4%

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

   2. inv-pow 98.4%

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

4. Applied egg-rr 98.4%

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

   1. sqrt-pow1 98.4%

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

   2. *-un-lft-identity 98.4%

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

   3. div-sub 98.4%

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

   4. *-inverses 98.4%

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

   5. sub-neg 98.4%

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

   6. metadata-eval 98.4%

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

   7. +-commutative 98.4%

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

   8. metadata-eval 98.4%

      \[\leadsto \left(1 \cdot {\left(-1 + \frac{1}{u1}\right)}^{\color{blue}{-0.5}}\right) \cdot \sin \left(6.28318530718 \cdot u2\right) \]

6. Applied egg-rr 98.4%

   \[\leadsto \color{blue}{\left(1 \cdot {\left(-1 + \frac{1}{u1}\right)}^{-0.5}\right)} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
7. Step-by-step derivation

   1. *-lft-identity 98.4%

      \[\leadsto \color{blue}{{\left(-1 + \frac{1}{u1}\right)}^{-0.5}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   2. +-commutative 98.4%

      \[\leadsto {\color{blue}{\left(\frac{1}{u1} + -1\right)}}^{-0.5} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

8. Simplified 98.4%

   \[\leadsto \color{blue}{{\left(\frac{1}{u1} + -1\right)}^{-0.5}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
9. Add Preprocessing

Alternative 2: 93.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.0026000000070780516:\\ \;\;\;\;{\left(\frac{1}{u1} + -1\right)}^{-0.5} \cdot \left(6.28318530718 \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1 \cdot \left(1 + u1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.0026000000070780516)
   (* (pow (+ (/ 1.0 u1) -1.0) -0.5) (* 6.28318530718 u2))
   (* (sin (* 6.28318530718 u2)) (sqrt (* u1 (+ 1.0 u1))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.0026000000070780516f) {
		tmp = powf(((1.0f / u1) + -1.0f), -0.5f) * (6.28318530718f * u2);
	} else {
		tmp = sinf((6.28318530718f * u2)) * sqrtf((u1 * (1.0f + u1)));
	}
	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) <= 0.0026000000070780516e0) then
        tmp = (((1.0e0 / u1) + (-1.0e0)) ** (-0.5e0)) * (6.28318530718e0 * u2)
    else
        tmp = sin((6.28318530718e0 * u2)) * sqrt((u1 * (1.0e0 + u1)))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(6.28318530718) * u2) <= Float32(0.0026000000070780516))
		tmp = Float32((Float32(Float32(Float32(1.0) / u1) + Float32(-1.0)) ^ Float32(-0.5)) * Float32(Float32(6.28318530718) * u2));
	else
		tmp = Float32(sin(Float32(Float32(6.28318530718) * u2)) * sqrt(Float32(u1 * Float32(Float32(1.0) + u1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((single(6.28318530718) * u2) <= single(0.0026000000070780516))
		tmp = (((single(1.0) / u1) + single(-1.0)) ^ single(-0.5)) * (single(6.28318530718) * u2);
	else
		tmp = sin((single(6.28318530718) * u2)) * sqrt((u1 * (single(1.0) + u1)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.00260000001

   1. Initial program 98.5%

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

      1. clear-num 98.6%

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

      2. inv-pow 98.6%

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

   4. Applied egg-rr 98.6%

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

      1. sqrt-pow1 98.7%

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

      2. *-un-lft-identity 98.7%

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

      3. div-sub 98.7%

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

      4. *-inverses 98.7%

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

      5. sub-neg 98.7%

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

      6. metadata-eval 98.7%

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

      7. +-commutative 98.7%

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

      8. metadata-eval 98.7%

         \[\leadsto \left(1 \cdot {\left(-1 + \frac{1}{u1}\right)}^{\color{blue}{-0.5}}\right) \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   6. Applied egg-rr 98.7%

      \[\leadsto \color{blue}{\left(1 \cdot {\left(-1 + \frac{1}{u1}\right)}^{-0.5}\right)} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
   7. Step-by-step derivation

      1. *-lft-identity 98.7%

         \[\leadsto \color{blue}{{\left(-1 + \frac{1}{u1}\right)}^{-0.5}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

      2. +-commutative 98.7%

         \[\leadsto {\color{blue}{\left(\frac{1}{u1} + -1\right)}}^{-0.5} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   8. Simplified 98.7%

      \[\leadsto \color{blue}{{\left(\frac{1}{u1} + -1\right)}^{-0.5}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   9. Taylor expanded in u2 around 0 98.0%

      \[\leadsto {\left(\frac{1}{u1} + -1\right)}^{-0.5} \cdot \color{blue}{\left(6.28318530718 \cdot u2\right)} \]

   if 0.00260000001 < (*.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 87.1%

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

      1. +-commutative 87.1%

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

   5. Simplified 87.1%

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(u1 + 1\right)}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.0026000000070780516:\\ \;\;\;\;{\left(\frac{1}{u1} + -1\right)}^{-0.5} \cdot \left(6.28318530718 \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1 \cdot \left(1 + u1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.010850000195205212:\\ \;\;\;\;{\left(\frac{1}{u1} + -1\right)}^{-0.5} \cdot \left(6.28318530718 \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.010850000195205212)
   (* (pow (+ (/ 1.0 u1) -1.0) -0.5) (* 6.28318530718 u2))
   (* (sin (* 6.28318530718 u2)) (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.010850000195205212f) {
		tmp = powf(((1.0f / u1) + -1.0f), -0.5f) * (6.28318530718f * u2);
	} else {
		tmp = sinf((6.28318530718f * u2)) * sqrtf(u1);
	}
	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) <= 0.010850000195205212e0) then
        tmp = (((1.0e0 / u1) + (-1.0e0)) ** (-0.5e0)) * (6.28318530718e0 * u2)
    else
        tmp = sin((6.28318530718e0 * u2)) * sqrt(u1)
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(6.28318530718) * u2) <= Float32(0.010850000195205212))
		tmp = Float32((Float32(Float32(Float32(1.0) / u1) + Float32(-1.0)) ^ Float32(-0.5)) * Float32(Float32(6.28318530718) * u2));
	else
		tmp = Float32(sin(Float32(Float32(6.28318530718) * u2)) * sqrt(u1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((single(6.28318530718) * u2) <= single(0.010850000195205212))
		tmp = (((single(1.0) / u1) + single(-1.0)) ^ single(-0.5)) * (single(6.28318530718) * u2);
	else
		tmp = sin((single(6.28318530718) * u2)) * sqrt(u1);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.0108500002

   1. Initial program 98.5%

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

      1. clear-num 98.6%

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

      2. inv-pow 98.6%

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

   4. Applied egg-rr 98.6%

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

      1. sqrt-pow1 98.7%

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

      2. *-un-lft-identity 98.7%

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

      3. div-sub 98.7%

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

      4. *-inverses 98.7%

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

      5. sub-neg 98.7%

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

      6. metadata-eval 98.7%

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

      7. +-commutative 98.7%

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

      8. metadata-eval 98.7%

         \[\leadsto \left(1 \cdot {\left(-1 + \frac{1}{u1}\right)}^{\color{blue}{-0.5}}\right) \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   6. Applied egg-rr 98.7%

      \[\leadsto \color{blue}{\left(1 \cdot {\left(-1 + \frac{1}{u1}\right)}^{-0.5}\right)} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
   7. Step-by-step derivation

      1. *-lft-identity 98.7%

         \[\leadsto \color{blue}{{\left(-1 + \frac{1}{u1}\right)}^{-0.5}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

      2. +-commutative 98.7%

         \[\leadsto {\color{blue}{\left(\frac{1}{u1} + -1\right)}}^{-0.5} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   8. Simplified 98.7%

      \[\leadsto \color{blue}{{\left(\frac{1}{u1} + -1\right)}^{-0.5}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   9. Taylor expanded in u2 around 0 96.8%

      \[\leadsto {\left(\frac{1}{u1} + -1\right)}^{-0.5} \cdot \color{blue}{\left(6.28318530718 \cdot u2\right)} \]

   if 0.0108500002 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 98.0%

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

   3. Taylor expanded in u1 around 0 74.6%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.010850000195205212:\\ \;\;\;\;{\left(\frac{1}{u1} + -1\right)}^{-0.5} \cdot \left(6.28318530718 \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

3. Final simplification 98.4%

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

Alternative 5: 81.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ {\left(\frac{1}{u1} + -1\right)}^{-0.5} \cdot \left(6.28318530718 \cdot u2\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (pow (+ (/ 1.0 u1) -1.0) -0.5) (* 6.28318530718 u2)))

C

float code(float cosTheta_i, float u1, float u2) {
	return powf(((1.0f / u1) + -1.0f), -0.5f) * (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 = (((1.0e0 / u1) + (-1.0e0)) ** (-0.5e0)) * (6.28318530718e0 * u2)
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32((Float32(Float32(Float32(1.0) / u1) + Float32(-1.0)) ^ Float32(-0.5)) * Float32(Float32(6.28318530718) * u2))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = (((single(1.0) / u1) + single(-1.0)) ^ single(-0.5)) * (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. clear-num 98.4%

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

   2. inv-pow 98.4%

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

4. Applied egg-rr 98.4%

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

   1. sqrt-pow1 98.4%

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

   2. *-un-lft-identity 98.4%

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

   3. div-sub 98.4%

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

   4. *-inverses 98.4%

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

   5. sub-neg 98.4%

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

   6. metadata-eval 98.4%

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

   7. +-commutative 98.4%

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

   8. metadata-eval 98.4%

      \[\leadsto \left(1 \cdot {\left(-1 + \frac{1}{u1}\right)}^{\color{blue}{-0.5}}\right) \cdot \sin \left(6.28318530718 \cdot u2\right) \]

6. Applied egg-rr 98.4%

   \[\leadsto \color{blue}{\left(1 \cdot {\left(-1 + \frac{1}{u1}\right)}^{-0.5}\right)} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
7. Step-by-step derivation

   1. *-lft-identity 98.4%

      \[\leadsto \color{blue}{{\left(-1 + \frac{1}{u1}\right)}^{-0.5}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   2. +-commutative 98.4%

      \[\leadsto {\color{blue}{\left(\frac{1}{u1} + -1\right)}}^{-0.5} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

8. Simplified 98.4%

   \[\leadsto \color{blue}{{\left(\frac{1}{u1} + -1\right)}^{-0.5}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

9. Taylor expanded in u2 around 0 79.0%

   \[\leadsto {\left(\frac{1}{u1} + -1\right)}^{-0.5} \cdot \color{blue}{\left(6.28318530718 \cdot u2\right)} \]
10. Add Preprocessing

Alternative 6: 81.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

3. Taylor expanded in u2 around 0 78.9%

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

   1. *-commutative 78.9%

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

   2. associate-*r* 78.9%

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

5. Simplified 78.9%

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

Alternative 7: 81.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

3. Taylor expanded in u2 around 0 78.9%

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

4. Final simplification 78.9%

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

Alternative 8: 73.3% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

3. Taylor expanded in u2 around 0 78.9%

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

4. Taylor expanded in u1 around 0 71.0%

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

   1. +-commutative 86.4%

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

6. Simplified 71.0%

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

7. Final simplification 71.0%

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

Alternative 9: 65.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \left(6.28318530718 \cdot u2\right) \cdot {u1}^{0.5} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (* 6.28318530718 u2) (pow u1 0.5)))

C

float code(float cosTheta_i, float u1, float u2) {
	return (6.28318530718f * u2) * powf(u1, 0.5f);
}

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) * (u1 ** 0.5e0)
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(Float32(Float32(6.28318530718) * u2) * (u1 ^ Float32(0.5)))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = (single(6.28318530718) * u2) * (u1 ^ single(0.5));
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 78.9%

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

4. Taylor expanded in u1 around 0 62.8%

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

5. Taylor expanded in u1 around -inf -0.0%

   \[\leadsto \color{blue}{-6.28318530718 \cdot \left(\sqrt{u1} \cdot \left(u2 \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)} \]
6. Step-by-step derivation

   1. *-commutative -0.0%

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

   2. *-commutative -0.0%

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

   3. *-commutative -0.0%

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

   4. unpow2 -0.0%

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

   5. rem-square-sqrt 62.8%

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

   6. associate-*l* 62.8%

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

   7. *-commutative 62.8%

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

   8. associate-*l* 62.8%

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

   9. metadata-eval 62.8%

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

   10. *-commutative 62.8%

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

   11. associate-*r* 62.8%

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

   12. *-commutative 62.8%

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

7. Simplified 62.8%

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

   1. pow1/2 62.8%

      \[\leadsto \color{blue}{{u1}^{0.5}} \cdot \left(6.28318530718 \cdot u2\right) \]

9. Applied egg-rr 62.8%

   \[\leadsto \color{blue}{{u1}^{0.5}} \cdot \left(6.28318530718 \cdot u2\right) \]

10. Final simplification 62.8%

    \[\leadsto \left(6.28318530718 \cdot u2\right) \cdot {u1}^{0.5} \]
11. Add Preprocessing

Alternative 10: 65.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

3. Taylor expanded in u2 around 0 78.9%

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

4. Taylor expanded in u1 around 0 62.8%

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

5. Taylor expanded in u1 around -inf -0.0%

   \[\leadsto \color{blue}{-6.28318530718 \cdot \left(\sqrt{u1} \cdot \left(u2 \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)} \]
6. Step-by-step derivation

   1. *-commutative -0.0%

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

   2. *-commutative -0.0%

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

   3. *-commutative -0.0%

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

   4. unpow2 -0.0%

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

   5. rem-square-sqrt 62.8%

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

   6. associate-*l* 62.8%

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

   7. *-commutative 62.8%

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

   8. associate-*l* 62.8%

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

   9. metadata-eval 62.8%

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

   10. *-commutative 62.8%

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

   11. associate-*r* 62.8%

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

   12. *-commutative 62.8%

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

7. Simplified 62.8%

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

8. Final simplification 62.8%

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

Alternative 11: 65.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

3. Taylor expanded in u2 around 0 78.9%

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

4. Taylor expanded in u1 around 0 62.8%

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

5. Final simplification 62.8%

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

Alternative 12: 4.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = (u2 * sqrt(u1)) * 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 78.9%

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

4. Taylor expanded in u1 around 0 62.8%

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

   1. add-sqr-sqrt 62.5%

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

   2. sqrt-unprod 62.8%

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

   3. *-commutative 62.8%

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

   4. *-commutative 62.8%

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

   5. swap-sqr 62.8%

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

   6. swap-sqr 62.8%

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

   7. add-sqr-sqrt 62.8%

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

   8. pow2 62.8%

      \[\leadsto \sqrt{\left(u1 \cdot \color{blue}{{u2}^{2}}\right) \cdot \left(6.28318530718 \cdot 6.28318530718\right)} \]

   9. metadata-eval 62.7%

      \[\leadsto \sqrt{\left(u1 \cdot {u2}^{2}\right) \cdot \color{blue}{39.47841760436263}} \]

6. Applied egg-rr 62.7%

   \[\leadsto \color{blue}{\sqrt{\left(u1 \cdot {u2}^{2}\right) \cdot 39.47841760436263}} \]

7. Taylor expanded in u2 around -inf 4.8%

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

8. Final simplification 4.8%

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

Reproduce

herbie shell --seed 2024160 
(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))))