Trowbridge-Reitz Sample, near normal, slope_y

Percentage Accurate: 98.3% → 98.4%

Time: 14.5s

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: 98.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{\sin \left(\sqrt{{u2}^{2} \cdot 39.47841760436263}\right)}{\sqrt{\frac{1 - u1}{u1}}} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (/ (sin (sqrt (* (pow u2 2.0) 39.47841760436263))) (sqrt (/ (- 1.0 u1) u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sinf(sqrtf((powf(u2, 2.0f) * 39.47841760436263f))) / sqrtf(((1.0f - u1) / 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(sqrt(((u2 ** 2.0e0) * 39.47841760436263e0))) / sqrt(((1.0e0 - u1) / u1))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sin(sqrt(Float32((u2 ^ Float32(2.0)) * Float32(39.47841760436263)))) / sqrt(Float32(Float32(Float32(1.0) - u1) / u1)))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sin(sqrt(((u2 ^ single(2.0)) * single(39.47841760436263)))) / sqrt(((single(1.0) - u1) / 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. log1p-expm1-u 98.3%

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

4. Applied egg-rr 98.3%

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

   1. sqrt-div 97.9%

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

   2. clear-num 97.9%

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

   3. log1p-expm1-u 97.9%

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

   4. associate-*l/ 97.9%

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

   5. *-un-lft-identity 97.9%

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

   6. sqrt-undiv 98.4%

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

6. Applied egg-rr 98.4%

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

   1. add-sqr-sqrt 97.7%

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

   2. sqrt-unprod 98.4%

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

   3. *-commutative 98.4%

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

   4. *-commutative 98.4%

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

   5. swap-sqr 98.2%

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

   6. pow2 98.2%

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

   7. metadata-eval 98.5%

      \[\leadsto \frac{\sin \left(\sqrt{{u2}^{2} \cdot \color{blue}{39.47841760436263}}\right)}{\sqrt{\frac{1 - u1}{u1}}} \]

8. Applied egg-rr 98.5%

   \[\leadsto \frac{\sin \color{blue}{\left(\sqrt{{u2}^{2} \cdot 39.47841760436263}\right)}}{\sqrt{\frac{1 - u1}{u1}}} \]

9. Final simplification 98.5%

   \[\leadsto \frac{\sin \left(\sqrt{{u2}^{2} \cdot 39.47841760436263}\right)}{\sqrt{\frac{1 - u1}{u1}}} \]
10. Add Preprocessing

Alternative 2: 93.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.004000000189989805:\\ \;\;\;\;\frac{u2 \cdot 6.28318530718}{\sqrt{\frac{1 - u1}{u1}}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1 \cdot \left(1 + u1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 6.28318530718) 0.004000000189989805)
   (/ (* u2 6.28318530718) (sqrt (/ (- 1.0 u1) u1)))
   (* (sin (* u2 6.28318530718)) (sqrt (* u1 (+ 1.0 u1))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * 6.28318530718f) <= 0.004000000189989805f) {
		tmp = (u2 * 6.28318530718f) / sqrtf(((1.0f - u1) / u1));
	} else {
		tmp = sinf((u2 * 6.28318530718f)) * 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 ((u2 * 6.28318530718e0) <= 0.004000000189989805e0) then
        tmp = (u2 * 6.28318530718e0) / sqrt(((1.0e0 - u1) / u1))
    else
        tmp = sin((u2 * 6.28318530718e0)) * sqrt((u1 * (1.0e0 + u1)))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(u2 * Float32(6.28318530718)) <= Float32(0.004000000189989805))
		tmp = Float32(Float32(u2 * Float32(6.28318530718)) / sqrt(Float32(Float32(Float32(1.0) - u1) / u1)));
	else
		tmp = Float32(sin(Float32(u2 * Float32(6.28318530718))) * 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 ((u2 * single(6.28318530718)) <= single(0.004000000189989805))
		tmp = (u2 * single(6.28318530718)) / sqrt(((single(1.0) - u1) / u1));
	else
		tmp = sin((u2 * single(6.28318530718))) * 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) < 0.00400000019

   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. Step-by-step derivation

      1. sqrt-div 98.2%

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

      2. clear-num 98.2%

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

      3. log1p-expm1-u 98.2%

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

      4. associate-*l/ 98.3%

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

      5. *-un-lft-identity 98.3%

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

      6. sqrt-undiv 98.7%

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

   6. Applied egg-rr 98.7%

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

   7. Taylor expanded in u2 around 0 97.4%

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

   if 0.00400000019 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 97.9%

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

   3. Taylor expanded in u1 around 0 86.1%

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

      1. +-commutative 51.3%

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

   5. Simplified 86.1%

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

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.004000000189989805:\\ \;\;\;\;\frac{u2 \cdot 6.28318530718}{\sqrt{\frac{1 - u1}{u1}}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1 \cdot \left(1 + u1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.009800000116229057:\\ \;\;\;\;\frac{u2 \cdot 6.28318530718}{\sqrt{\frac{1 - u1}{u1}}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 6.28318530718) 0.009800000116229057)
   (/ (* u2 6.28318530718) (sqrt (/ (- 1.0 u1) u1)))
   (* (sin (* u2 6.28318530718)) (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * 6.28318530718f) <= 0.009800000116229057f) {
		tmp = (u2 * 6.28318530718f) / sqrtf(((1.0f - u1) / u1));
	} else {
		tmp = sinf((u2 * 6.28318530718f)) * 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 ((u2 * 6.28318530718e0) <= 0.009800000116229057e0) then
        tmp = (u2 * 6.28318530718e0) / sqrt(((1.0e0 - u1) / u1))
    else
        tmp = sin((u2 * 6.28318530718e0)) * sqrt(u1)
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(u2 * Float32(6.28318530718)) <= Float32(0.009800000116229057))
		tmp = Float32(Float32(u2 * Float32(6.28318530718)) / sqrt(Float32(Float32(Float32(1.0) - u1) / u1)));
	else
		tmp = Float32(sin(Float32(u2 * Float32(6.28318530718))) * sqrt(u1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((u2 * single(6.28318530718)) <= single(0.009800000116229057))
		tmp = (u2 * single(6.28318530718)) / sqrt(((single(1.0) - u1) / u1));
	else
		tmp = sin((u2 * single(6.28318530718))) * sqrt(u1);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.00980000012

   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. Step-by-step derivation

      1. sqrt-div 98.1%

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

      2. clear-num 98.1%

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

      3. log1p-expm1-u 98.1%

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

      4. associate-*l/ 98.2%

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

      5. *-un-lft-identity 98.2%

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

      6. sqrt-undiv 98.6%

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

   6. Applied egg-rr 98.6%

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

   7. Taylor expanded in u2 around 0 96.7%

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

   if 0.00980000012 < (*.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 75.1%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.009800000116229057:\\ \;\;\;\;\frac{u2 \cdot 6.28318530718}{\sqrt{\frac{1 - u1}{u1}}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 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 5: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sin((u2 * single(6.28318530718))) / sqrt(((single(1.0) - u1) / 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. log1p-expm1-u 98.3%

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

4. Applied egg-rr 98.3%

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

   1. sqrt-div 97.9%

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

   2. clear-num 97.9%

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

   3. log1p-expm1-u 97.9%

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

   4. associate-*l/ 97.9%

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

   5. *-un-lft-identity 97.9%

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

   6. sqrt-undiv 98.4%

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

6. Applied egg-rr 98.4%

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

7. Final simplification 98.4%

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

Alternative 6: 73.4% 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 98.3%

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

3. Taylor expanded in u2 around 0 81.7%

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

4. Taylor expanded in u1 around 0 72.9%

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

   1. +-commutative 72.9%

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

6. Simplified 72.9%

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

7. Final simplification 72.9%

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

Alternative 7: 81.8% 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 98.3%

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

3. Taylor expanded in u2 around 0 81.7%

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

4. Final simplification 81.7%

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

Alternative 8: 81.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = (u2 * single(6.28318530718)) / sqrt(((single(1.0) - u1) / 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. log1p-expm1-u 98.3%

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

4. Applied egg-rr 98.3%

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

   1. sqrt-div 97.9%

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

   2. clear-num 97.9%

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

   3. log1p-expm1-u 97.9%

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

   4. associate-*l/ 97.9%

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

   5. *-un-lft-identity 97.9%

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

   6. sqrt-undiv 98.4%

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

6. Applied egg-rr 98.4%

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

7. Taylor expanded in u2 around 0 81.7%

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

8. Final simplification 81.7%

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

Alternative 9: 65.1% 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 98.3%

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

3. Taylor expanded in u2 around 0 81.7%

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

4. Taylor expanded in u1 around 0 65.9%

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

5. Final simplification 65.9%

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

Alternative 10: 65.1% accurate, 2.0× speedup

Math

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

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(u2 * Float32(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 81.7%

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

4. Taylor expanded in u1 around 0 65.9%

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

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. associate-*r* -0.0%

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

   3. *-commutative -0.0%

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

   4. unpow2 -0.0%

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

   5. rem-square-sqrt 65.9%

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

   6. associate-*l* 65.9%

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

   7. metadata-eval 65.9%

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

   8. associate-*l* 65.9%

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

7. Simplified 65.9%

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

8. Final simplification 65.9%

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

Alternative 11: 20.5% accurate, 19.0× speedup

Math

\[\begin{array}{l} \\ u1 \cdot \left(u2 \cdot 6.28318530718 + 3.14159265359 \cdot \frac{u2}{u1}\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* u1 (+ (* u2 6.28318530718) (* 3.14159265359 (/ u2 u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	return u1 * ((u2 * 6.28318530718f) + (3.14159265359f * (u2 / 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 = u1 * ((u2 * 6.28318530718e0) + (3.14159265359e0 * (u2 / u1)))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(u1 * Float32(Float32(u2 * Float32(6.28318530718)) + Float32(Float32(3.14159265359) * Float32(u2 / u1))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = u1 * ((u2 * single(6.28318530718)) + (single(3.14159265359) * (u2 / 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 u1 around 0 85.5%

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

   1. distribute-lft-in 85.4%

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

   2. *-rgt-identity 85.4%

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

   3. unpow2 85.4%

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

5. Simplified 85.4%

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

6. Taylor expanded in u2 around 0 73.0%

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

   1. associate-*r* 72.9%

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

   2. *-commutative 72.9%

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

   3. +-commutative 72.9%

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

   4. unpow2 72.9%

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

   5. fma-undefine 72.9%

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

   6. rem-square-sqrt 72.8%

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

   7. fma-define 72.8%

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

   8. hypot-undefine 72.9%

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

8. Simplified 72.9%

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

9. Taylor expanded in u1 around inf 20.6%

   \[\leadsto \color{blue}{u1 \cdot \left(3.14159265359 \cdot \frac{u2}{u1} + 6.28318530718 \cdot u2\right)} \]

10. Final simplification 20.6%

    \[\leadsto u1 \cdot \left(u2 \cdot 6.28318530718 + 3.14159265359 \cdot \frac{u2}{u1}\right) \]
11. Add Preprocessing

Alternative 12: 19.4% accurate, 41.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = single(6.28318530718) * (u2 * 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 u1 around 0 85.5%

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

   1. distribute-lft-in 85.4%

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

   2. *-rgt-identity 85.4%

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

   3. unpow2 85.4%

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

5. Simplified 85.4%

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

6. Taylor expanded in u2 around 0 73.0%

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

   1. associate-*r* 72.9%

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

   2. *-commutative 72.9%

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

   3. +-commutative 72.9%

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

   4. unpow2 72.9%

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

   5. fma-undefine 72.9%

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

   6. rem-square-sqrt 72.8%

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

   7. fma-define 72.8%

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

   8. hypot-undefine 72.9%

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

8. Simplified 72.9%

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

9. Taylor expanded in u1 around inf 19.5%

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

10. Final simplification 19.5%

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

Reproduce

herbie shell --seed 2024061 
(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))))