Trowbridge-Reitz Sample, near normal, slope_y

Percentage Accurate: 98.4% → 98.4%

Time: 13.0s

Alternatives: 11

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: 98.4% 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

\[\begin{array}{l} \\ \sqrt[3]{{\left(\frac{u1}{1 - u1}\right)}^{1.5} \cdot {\sin \left(6.28318530718 \cdot u2\right)}^{3}} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (cbrt (* (pow (/ u1 (- 1.0 u1)) 1.5) (pow (sin (* 6.28318530718 u2)) 3.0))))

C

float code(float cosTheta_i, float u1, float u2) {
	return cbrtf((powf((u1 / (1.0f - u1)), 1.5f) * powf(sinf((6.28318530718f * u2)), 3.0f)));
}

Julia

function code(cosTheta_i, u1, u2)
	return cbrt(Float32((Float32(u1 / Float32(Float32(1.0) - u1)) ^ Float32(1.5)) * (sin(Float32(Float32(6.28318530718) * u2)) ^ Float32(3.0))))
end

Derivation

1. Initial program 98.1%

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

   1. add-cbrt-cube 98.1%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\sqrt{\frac{u1}{1 - u1}} \cdot \sqrt{\frac{u1}{1 - u1}}\right) \cdot \sqrt{\frac{u1}{1 - u1}}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   2. add-cbrt-cube 98.0%

      \[\leadsto \sqrt[3]{\left(\sqrt{\frac{u1}{1 - u1}} \cdot \sqrt{\frac{u1}{1 - u1}}\right) \cdot \sqrt{\frac{u1}{1 - u1}}} \cdot \color{blue}{\sqrt[3]{\left(\sin \left(6.28318530718 \cdot u2\right) \cdot \sin \left(6.28318530718 \cdot u2\right)\right) \cdot \sin \left(6.28318530718 \cdot u2\right)}} \]

   3. cbrt-unprod 97.9%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\sqrt{\frac{u1}{1 - u1}} \cdot \sqrt{\frac{u1}{1 - u1}}\right) \cdot \sqrt{\frac{u1}{1 - u1}}\right) \cdot \left(\left(\sin \left(6.28318530718 \cdot u2\right) \cdot \sin \left(6.28318530718 \cdot u2\right)\right) \cdot \sin \left(6.28318530718 \cdot u2\right)\right)}} \]

   4. add-sqr-sqrt 98.1%

      \[\leadsto \sqrt[3]{\left(\color{blue}{\frac{u1}{1 - u1}} \cdot \sqrt{\frac{u1}{1 - u1}}\right) \cdot \left(\left(\sin \left(6.28318530718 \cdot u2\right) \cdot \sin \left(6.28318530718 \cdot u2\right)\right) \cdot \sin \left(6.28318530718 \cdot u2\right)\right)} \]

   5. pow1 98.1%

      \[\leadsto \sqrt[3]{\left(\color{blue}{{\left(\frac{u1}{1 - u1}\right)}^{1}} \cdot \sqrt{\frac{u1}{1 - u1}}\right) \cdot \left(\left(\sin \left(6.28318530718 \cdot u2\right) \cdot \sin \left(6.28318530718 \cdot u2\right)\right) \cdot \sin \left(6.28318530718 \cdot u2\right)\right)} \]

   6. pow1/2 98.1%

      \[\leadsto \sqrt[3]{\left({\left(\frac{u1}{1 - u1}\right)}^{1} \cdot \color{blue}{{\left(\frac{u1}{1 - u1}\right)}^{0.5}}\right) \cdot \left(\left(\sin \left(6.28318530718 \cdot u2\right) \cdot \sin \left(6.28318530718 \cdot u2\right)\right) \cdot \sin \left(6.28318530718 \cdot u2\right)\right)} \]

   7. pow-prod-up 98.2%

      \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{u1}{1 - u1}\right)}^{\left(1 + 0.5\right)}} \cdot \left(\left(\sin \left(6.28318530718 \cdot u2\right) \cdot \sin \left(6.28318530718 \cdot u2\right)\right) \cdot \sin \left(6.28318530718 \cdot u2\right)\right)} \]

   8. metadata-eval 98.2%

      \[\leadsto \sqrt[3]{{\left(\frac{u1}{1 - u1}\right)}^{\color{blue}{1.5}} \cdot \left(\left(\sin \left(6.28318530718 \cdot u2\right) \cdot \sin \left(6.28318530718 \cdot u2\right)\right) \cdot \sin \left(6.28318530718 \cdot u2\right)\right)} \]

   9. pow3 98.2%

      \[\leadsto \sqrt[3]{{\left(\frac{u1}{1 - u1}\right)}^{1.5} \cdot \color{blue}{{\sin \left(6.28318530718 \cdot u2\right)}^{3}}} \]

4. Applied egg-rr 98.2%

   \[\leadsto \color{blue}{\sqrt[3]{{\left(\frac{u1}{1 - u1}\right)}^{1.5} \cdot {\sin \left(6.28318530718 \cdot u2\right)}^{3}}} \]

5. Final simplification 98.2%

   \[\leadsto \sqrt[3]{{\left(\frac{u1}{1 - u1}\right)}^{1.5} \cdot {\sin \left(6.28318530718 \cdot u2\right)}^{3}} \]
6. Add Preprocessing

Alternative 2: 98.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - u1))) * sinf(sqrtf((powf(u2, 2.0f) * 39.47841760436263f)));
}

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(sqrt(((u2 ** 2.0e0) * 39.47841760436263e0)))
end function

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((u1 / (single(1.0) - u1))) * sin(sqrt(((u2 ^ single(2.0)) * single(39.47841760436263))));
end

Derivation

1. Initial program 98.1%

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

   1. add-sqr-sqrt 97.4%

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

   2. sqrt-unprod 98.1%

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

   3. *-commutative 98.1%

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

   4. *-commutative 98.1%

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

   5. swap-sqr 97.8%

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

   6. pow2 97.8%

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

   7. metadata-eval 98.2%

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

4. Applied egg-rr 98.2%

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

5. Final simplification 98.2%

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

Alternative 3: 98.3% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

function code(cosTheta_i, u1, u2)
	return log1p(expm1(Float32(sin(Float32(Float32(6.28318530718) * u2)) * sqrt(Float32(u1 / Float32(Float32(1.0) - u1))))))
end

Derivation

1. Initial program 98.1%

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

   1. log1p-expm1-u 98.1%

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

4. Applied egg-rr 98.1%

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

5. Final simplification 98.1%

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

Alternative 4: 89.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.004999999888241291:\\ \;\;\;\;u2 \cdot \left(6.28318530718 \cdot \sqrt{\frac{u1 + 1}{\frac{1}{u1} - u1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.004999999888241291)
   (* u2 (* 6.28318530718 (sqrt (/ (+ u1 1.0) (- (/ 1.0 u1) u1)))))
   (* (sin (* 6.28318530718 u2)) (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.004999999888241291f) {
		tmp = u2 * (6.28318530718f * sqrtf(((u1 + 1.0f) / ((1.0f / u1) - u1))));
	} 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) <= 0.004999999888241291e0) then
        tmp = u2 * (6.28318530718e0 * sqrt(((u1 + 1.0e0) / ((1.0e0 / u1) - u1))))
    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(0.004999999888241291))
		tmp = Float32(u2 * Float32(Float32(6.28318530718) * sqrt(Float32(Float32(u1 + Float32(1.0)) / Float32(Float32(Float32(1.0) / u1) - u1)))));
	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(0.004999999888241291))
		tmp = u2 * (single(6.28318530718) * sqrt(((u1 + single(1.0)) / ((single(1.0) / u1) - u1))));
	else
		tmp = sin((single(6.28318530718) * u2)) * sqrt(u1);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 314159265359/50000000000 u2) < 0.00499999989

   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. flip-- 98.4%

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

      2. associate-/r/ 98.5%

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

      3. metadata-eval 98.5%

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

      4. pow2 98.5%

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

      5. +-commutative 98.5%

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

   4. Applied egg-rr 98.5%

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

   5. Taylor expanded in u2 around 0 97.2%

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

      1. *-commutative 97.2%

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

      2. *-commutative 97.2%

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

      3. associate-*l* 97.1%

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

      4. +-commutative 97.1%

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

      5. *-commutative 97.1%

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

      6. associate-*l/ 97.1%

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

      7. associate-/r/ 97.1%

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

      8. div-sub 97.1%

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

      9. unpow2 97.1%

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

      10. associate-/l* 97.1%

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

      11. *-inverses 97.1%

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

      12. *-rgt-identity 97.1%

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

   7. Simplified 97.1%

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

   if 0.00499999989 < (*.f32 314159265359/50000000000 u2)

   1. Initial program 97.5%

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

   3. Taylor expanded in u1 around 0 75.2%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.004999999888241291:\\ \;\;\;\;u2 \cdot \left(6.28318530718 \cdot \sqrt{\frac{u1 + 1}{\frac{1}{u1} - u1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.4% 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 98.1%

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

3. Final simplification 98.1%

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

Alternative 6: 81.9% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

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

   1. flip-- 98.0%

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

   2. associate-/r/ 98.1%

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

   3. metadata-eval 98.1%

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

   4. pow2 98.1%

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

   5. +-commutative 98.1%

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

4. Applied egg-rr 98.1%

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

5. Taylor expanded in u2 around 0 78.7%

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

   1. *-commutative 78.7%

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

   2. *-commutative 78.7%

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

   3. associate-*l* 78.6%

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

   4. +-commutative 78.6%

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

   5. *-commutative 78.6%

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

   6. associate-*l/ 78.6%

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

   7. associate-/r/ 78.6%

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

   8. div-sub 78.6%

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

   9. unpow2 78.6%

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

   10. associate-/l* 78.6%

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

   11. *-inverses 78.6%

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

   12. *-rgt-identity 78.6%

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

7. Simplified 78.6%

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

8. Final simplification 78.6%

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

Alternative 7: 81.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

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 u2 around 0 78.6%

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

   1. clear-num 78.6%

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

   2. sqrt-div 78.6%

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

   3. metadata-eval 78.6%

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

5. Applied egg-rr 78.6%

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

   1. div-sub 78.6%

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

   2. sub-neg 78.6%

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

   3. *-inverses 78.6%

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

   4. metadata-eval 78.6%

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

7. Simplified 78.6%

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

8. Final simplification 78.6%

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

Alternative 8: 82.0% 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 98.1%

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

3. Taylor expanded in u2 around 0 78.6%

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

4. Final simplification 78.6%

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

Alternative 9: 81.9% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

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

3. Taylor expanded in u2 around 0 78.6%

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

   1. associate-*r* 78.6%

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

   2. *-commutative 78.6%

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

5. Simplified 78.6%

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

6. Final simplification 78.6%

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

Alternative 10: 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 98.1%

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

3. Taylor expanded in u2 around 0 78.6%

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

4. Taylor expanded in u1 around 0 61.3%

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

5. Final simplification 61.3%

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

Alternative 11: -0.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ u2 \cdot \sqrt{-39.47841760436263} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* u2 (sqrt -39.47841760436263)))

C

float code(float cosTheta_i, float u1, float u2) {
	return u2 * sqrtf(-39.47841760436263f);
}

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((-39.47841760436263e0))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(u2 * sqrt(Float32(-39.47841760436263)))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = u2 * sqrt(single(-39.47841760436263));
end

Derivation

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 u2 around 0 78.6%

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

   1. add-sqr-sqrt 78.4%

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

   2. sqrt-unprod 78.6%

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

   3. *-commutative 78.6%

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

   4. *-commutative 78.6%

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

   5. swap-sqr 78.4%

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

   6. swap-sqr 78.5%

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

   7. add-sqr-sqrt 78.7%

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

   8. pow2 78.7%

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

   9. metadata-eval 79.1%

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

5. Applied egg-rr 79.1%

   \[\leadsto \color{blue}{\sqrt{\left(\frac{u1}{1 - u1} \cdot {u2}^{2}\right) \cdot 39.47841760436263}} \]
6. Step-by-step derivation

   1. associate-*l* 79.2%

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

7. Simplified 79.2%

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

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

   \[\leadsto \color{blue}{u2 \cdot \sqrt{-39.47841760436263}} \]

9. Final simplification -0.0%

   \[\leadsto u2 \cdot \sqrt{-39.47841760436263} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024041 
(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))))