Trowbridge-Reitz Sample, near normal, slope_y

Percentage Accurate: 98.3% → 98.3%

Time: 11.0s

Alternatives: 9

Speedup: 1.0×

Specification

\[\left(\left(cosTheta\_i > 0.9999 \land cosTheta\_i \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u2 \land u2 \leq 1\right)\]

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(6.28318530718 \cdot u2\right)\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (/ u1 (- 1.0 u1))) (log1p (expm1 (sin (* 6.28318530718 u2))))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - u1))) * log1pf(expm1f(sinf((6.28318530718f * u2))));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * log1p(expm1(sin(Float32(Float32(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. Step-by-step derivation

   1. log1p-expm1-u 98.5%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(6.28318530718 \cdot u2\right)\right)\right)} \]

4. Applied egg-rr 98.5%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(6.28318530718 \cdot u2\right)\right)\right)} \]
5. Add Preprocessing

Alternative 2: 96.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.11999999731779099:\\ \;\;\;\;u2 \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot \left(6.28318530718 + u2 \cdot \left(u2 \cdot -41.341702240407926\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1 \cdot \left(u1 + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.11999999731779099)
   (*
    u2
    (*
     (sqrt (/ u1 (- 1.0 u1)))
     (+ 6.28318530718 (* u2 (* u2 -41.341702240407926)))))
   (* (sin (* 6.28318530718 u2)) (sqrt (* u1 (+ u1 1.0))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.11999999731779099f) {
		tmp = u2 * (sqrtf((u1 / (1.0f - u1))) * (6.28318530718f + (u2 * (u2 * -41.341702240407926f))));
	} else {
		tmp = sinf((6.28318530718f * u2)) * sqrtf((u1 * (u1 + 1.0f)));
	}
	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) <= 0.11999999731779099e0) then
        tmp = u2 * (sqrt((u1 / (1.0e0 - u1))) * (6.28318530718e0 + (u2 * (u2 * (-41.341702240407926e0)))))
    else
        tmp = sin((6.28318530718e0 * u2)) * sqrt((u1 * (u1 + 1.0e0)))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(6.28318530718) * u2) <= Float32(0.11999999731779099))
		tmp = Float32(u2 * Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * Float32(Float32(6.28318530718) + Float32(u2 * Float32(u2 * Float32(-41.341702240407926))))));
	else
		tmp = Float32(sin(Float32(Float32(6.28318530718) * u2)) * sqrt(Float32(u1 * Float32(u1 + Float32(1.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((single(6.28318530718) * u2) <= single(0.11999999731779099))
		tmp = u2 * (sqrt((u1 / (single(1.0) - u1))) * (single(6.28318530718) + (u2 * (u2 * single(-41.341702240407926)))));
	else
		tmp = sin((single(6.28318530718) * u2)) * sqrt((u1 * (u1 + single(1.0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.119999997

   1. Initial program 98.9%

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

   3. Taylor expanded in u2 around 0 98.3%

      \[\leadsto \color{blue}{u2 \cdot \left(-41.341702240407926 \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot {u2}^{2}\right) + 6.28318530718 \cdot \sqrt{\frac{u1}{1 - u1}}\right)} \]
   4. Step-by-step derivation

      1. *-commutative 98.3%

         \[\leadsto u2 \cdot \left(\color{blue}{\left(\sqrt{\frac{u1}{1 - u1}} \cdot {u2}^{2}\right) \cdot -41.341702240407926} + 6.28318530718 \cdot \sqrt{\frac{u1}{1 - u1}}\right) \]

      2. associate-*l* 98.3%

         \[\leadsto u2 \cdot \left(\color{blue}{\sqrt{\frac{u1}{1 - u1}} \cdot \left({u2}^{2} \cdot -41.341702240407926\right)} + 6.28318530718 \cdot \sqrt{\frac{u1}{1 - u1}}\right) \]

      3. *-commutative 98.3%

         \[\leadsto u2 \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\left(-41.341702240407926 \cdot {u2}^{2}\right)} + 6.28318530718 \cdot \sqrt{\frac{u1}{1 - u1}}\right) \]

      4. *-commutative 98.3%

         \[\leadsto u2 \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot \left(-41.341702240407926 \cdot {u2}^{2}\right) + \color{blue}{\sqrt{\frac{u1}{1 - u1}} \cdot 6.28318530718}\right) \]

      5. distribute-lft-out 98.3%

         \[\leadsto u2 \cdot \color{blue}{\left(\sqrt{\frac{u1}{1 - u1}} \cdot \left(-41.341702240407926 \cdot {u2}^{2} + 6.28318530718\right)\right)} \]

      6. fma-define 98.3%

         \[\leadsto u2 \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\mathsf{fma}\left(-41.341702240407926, {u2}^{2}, 6.28318530718\right)}\right) \]

   5. Simplified 98.3%

      \[\leadsto \color{blue}{u2 \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(-41.341702240407926, {u2}^{2}, 6.28318530718\right)\right)} \]
   6. Step-by-step derivation

      1. fma-undefine 98.3%

         \[\leadsto u2 \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\left(-41.341702240407926 \cdot {u2}^{2} + 6.28318530718\right)}\right) \]

   7. Applied egg-rr 98.3%

      \[\leadsto u2 \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\left(-41.341702240407926 \cdot {u2}^{2} + 6.28318530718\right)}\right) \]
   8. Step-by-step derivation

      1. unpow2 98.3%

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

      2. associate-*r* 98.3%

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

   9. Applied egg-rr 98.3%

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

   if 0.119999997 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 96.9%

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

   3. Taylor expanded in u1 around 0 83.6%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.11999999731779099:\\ \;\;\;\;u2 \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot \left(6.28318530718 + u2 \cdot \left(u2 \cdot -41.341702240407926\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1 \cdot \left(u1 + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.25:\\ \;\;\;\;u2 \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot \left(6.28318530718 + u2 \cdot \left(u2 \cdot -41.341702240407926\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.25)
   (*
    u2
    (*
     (sqrt (/ u1 (- 1.0 u1)))
     (+ 6.28318530718 (* u2 (* u2 -41.341702240407926)))))
   (* (sin (* 6.28318530718 u2)) (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.25f) {
		tmp = u2 * (sqrtf((u1 / (1.0f - u1))) * (6.28318530718f + (u2 * (u2 * -41.341702240407926f))));
	} 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) <= 0.25e0) then
        tmp = u2 * (sqrt((u1 / (1.0e0 - u1))) * (6.28318530718e0 + (u2 * (u2 * (-41.341702240407926e0)))))
    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(0.25))
		tmp = Float32(u2 * Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * Float32(Float32(6.28318530718) + Float32(u2 * Float32(u2 * Float32(-41.341702240407926))))));
	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(0.25))
		tmp = u2 * (sqrt((u1 / (single(1.0) - u1))) * (single(6.28318530718) + (u2 * (u2 * single(-41.341702240407926)))));
	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) < 0.25

   1. Initial program 98.8%

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

   3. Taylor expanded in u2 around 0 97.7%

      \[\leadsto \color{blue}{u2 \cdot \left(-41.341702240407926 \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot {u2}^{2}\right) + 6.28318530718 \cdot \sqrt{\frac{u1}{1 - u1}}\right)} \]
   4. Step-by-step derivation

      1. *-commutative 97.7%

         \[\leadsto u2 \cdot \left(\color{blue}{\left(\sqrt{\frac{u1}{1 - u1}} \cdot {u2}^{2}\right) \cdot -41.341702240407926} + 6.28318530718 \cdot \sqrt{\frac{u1}{1 - u1}}\right) \]

      2. associate-*l* 97.7%

         \[\leadsto u2 \cdot \left(\color{blue}{\sqrt{\frac{u1}{1 - u1}} \cdot \left({u2}^{2} \cdot -41.341702240407926\right)} + 6.28318530718 \cdot \sqrt{\frac{u1}{1 - u1}}\right) \]

      3. *-commutative 97.7%

         \[\leadsto u2 \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\left(-41.341702240407926 \cdot {u2}^{2}\right)} + 6.28318530718 \cdot \sqrt{\frac{u1}{1 - u1}}\right) \]

      4. *-commutative 97.7%

         \[\leadsto u2 \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot \left(-41.341702240407926 \cdot {u2}^{2}\right) + \color{blue}{\sqrt{\frac{u1}{1 - u1}} \cdot 6.28318530718}\right) \]

      5. distribute-lft-out 97.7%

         \[\leadsto u2 \cdot \color{blue}{\left(\sqrt{\frac{u1}{1 - u1}} \cdot \left(-41.341702240407926 \cdot {u2}^{2} + 6.28318530718\right)\right)} \]

      6. fma-define 97.7%

         \[\leadsto u2 \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\mathsf{fma}\left(-41.341702240407926, {u2}^{2}, 6.28318530718\right)}\right) \]

   5. Simplified 97.7%

      \[\leadsto \color{blue}{u2 \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(-41.341702240407926, {u2}^{2}, 6.28318530718\right)\right)} \]
   6. Step-by-step derivation

      1. fma-undefine 97.7%

         \[\leadsto u2 \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\left(-41.341702240407926 \cdot {u2}^{2} + 6.28318530718\right)}\right) \]

   7. Applied egg-rr 97.7%

      \[\leadsto u2 \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\left(-41.341702240407926 \cdot {u2}^{2} + 6.28318530718\right)}\right) \]
   8. Step-by-step derivation

      1. unpow2 97.7%

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

      2. associate-*r* 97.7%

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

   9. Applied egg-rr 97.7%

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

   if 0.25 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 96.6%

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

   3. Taylor expanded in u1 around 0 74.4%

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

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.25:\\ \;\;\;\;u2 \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot \left(6.28318530718 + u2 \cdot \left(u2 \cdot -41.341702240407926\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.3% accurate, 1.0× speedup

Math

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

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 5: 89.5% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

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 / (1.0e0 - u1))) * (6.28318530718e0 + (u2 * (u2 * (-41.341702240407926e0)))))
end function

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = u2 * (sqrt((u1 / (single(1.0) - u1))) * (single(6.28318530718) + (u2 * (u2 * single(-41.341702240407926)))));
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 88.4%

   \[\leadsto \color{blue}{u2 \cdot \left(-41.341702240407926 \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot {u2}^{2}\right) + 6.28318530718 \cdot \sqrt{\frac{u1}{1 - u1}}\right)} \]
4. Step-by-step derivation

   1. *-commutative 88.4%

      \[\leadsto u2 \cdot \left(\color{blue}{\left(\sqrt{\frac{u1}{1 - u1}} \cdot {u2}^{2}\right) \cdot -41.341702240407926} + 6.28318530718 \cdot \sqrt{\frac{u1}{1 - u1}}\right) \]

   2. associate-*l* 88.4%

      \[\leadsto u2 \cdot \left(\color{blue}{\sqrt{\frac{u1}{1 - u1}} \cdot \left({u2}^{2} \cdot -41.341702240407926\right)} + 6.28318530718 \cdot \sqrt{\frac{u1}{1 - u1}}\right) \]

   3. *-commutative 88.4%

      \[\leadsto u2 \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\left(-41.341702240407926 \cdot {u2}^{2}\right)} + 6.28318530718 \cdot \sqrt{\frac{u1}{1 - u1}}\right) \]

   4. *-commutative 88.4%

      \[\leadsto u2 \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot \left(-41.341702240407926 \cdot {u2}^{2}\right) + \color{blue}{\sqrt{\frac{u1}{1 - u1}} \cdot 6.28318530718}\right) \]

   5. distribute-lft-out 88.4%

      \[\leadsto u2 \cdot \color{blue}{\left(\sqrt{\frac{u1}{1 - u1}} \cdot \left(-41.341702240407926 \cdot {u2}^{2} + 6.28318530718\right)\right)} \]

   6. fma-define 88.4%

      \[\leadsto u2 \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\mathsf{fma}\left(-41.341702240407926, {u2}^{2}, 6.28318530718\right)}\right) \]

5. Simplified 88.4%

   \[\leadsto \color{blue}{u2 \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(-41.341702240407926, {u2}^{2}, 6.28318530718\right)\right)} \]
6. Step-by-step derivation

   1. fma-undefine 88.4%

      \[\leadsto u2 \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\left(-41.341702240407926 \cdot {u2}^{2} + 6.28318530718\right)}\right) \]

7. Applied egg-rr 88.4%

   \[\leadsto u2 \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\left(-41.341702240407926 \cdot {u2}^{2} + 6.28318530718\right)}\right) \]
8. Step-by-step derivation

   1. unpow2 88.4%

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

   2. associate-*r* 88.4%

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

9. Applied egg-rr 88.4%

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

10. Final simplification 88.4%

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

Alternative 6: 82.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = single(6.28318530718) * (sqrt((u1 / (single(1.0) - u1))) * 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 81.1%

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

Alternative 7: 73.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return 6.28318530718f * (u2 * sqrtf((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 = 6.28318530718e0 * (u2 * sqrt((u1 * (u1 + 1.0e0))))
end function

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = single(6.28318530718) * (u2 * sqrt((u1 * (u1 + single(1.0)))));
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 81.1%

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

4. Taylor expanded in u1 around 0 72.5%

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

5. Final simplification 72.5%

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

Alternative 8: 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 98.5%

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

3. Taylor expanded in u2 around 0 81.1%

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

4. Taylor expanded in u1 around 0 64.3%

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

   1. *-commutative 64.3%

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

   2. associate-*r* 64.4%

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

6. Applied egg-rr 64.4%

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

Alternative 9: 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 98.5%

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

3. Taylor expanded in u2 around 0 81.1%

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

4. Taylor expanded in u1 around 0 64.3%

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

5. Final simplification 64.3%

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

Reproduce

herbie shell --seed 2024096 
(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))))