Trowbridge-Reitz Sample, near normal, slope_y

Percentage Accurate: 98.3% → 98.3%

Time: 14.2s

Alternatives: 14

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, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

   1. add-exp-log 96.1%

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

3. Applied egg-rr 96.1%

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

   1. add-exp-log 98.3%

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

   2. clear-num 98.3%

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

5. Applied egg-rr 98.3%

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

6. Final simplification 98.3%

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

Alternative 2: 94.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.006000000052154064:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1} \cdot \left(u2 \cdot \left(u2 \cdot 39.47841760436263\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1 + u1 \cdot u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.006000000052154064)
   (sqrt (* (/ u1 (- 1.0 u1)) (* u2 (* u2 39.47841760436263))))
   (* (sin (* 6.28318530718 u2)) (sqrt (+ u1 (* u1 u1))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.006000000052154064f) {
		tmp = sqrtf(((u1 / (1.0f - u1)) * (u2 * (u2 * 39.47841760436263f))));
	} else {
		tmp = sinf((6.28318530718f * u2)) * sqrtf((u1 + (u1 * 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) <= 0.006000000052154064e0) then
        tmp = sqrt(((u1 / (1.0e0 - u1)) * (u2 * (u2 * 39.47841760436263e0))))
    else
        tmp = sin((6.28318530718e0 * u2)) * sqrt((u1 + (u1 * 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(0.006000000052154064))
		tmp = sqrt(Float32(Float32(u1 / Float32(Float32(1.0) - u1)) * Float32(u2 * Float32(u2 * Float32(39.47841760436263)))));
	else
		tmp = Float32(sin(Float32(Float32(6.28318530718) * u2)) * sqrt(Float32(u1 + Float32(u1 * u1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((single(6.28318530718) * u2) <= single(0.006000000052154064))
		tmp = sqrt(((u1 / (single(1.0) - u1)) * (u2 * (u2 * single(39.47841760436263)))));
	else
		tmp = sin((single(6.28318530718) * u2)) * sqrt((u1 + (u1 * u1)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 314159265359/50000000000 u2) < 0.00600000005

   1. Initial program 98.6%

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

   2. Taylor expanded in u2 around 0 97.3%

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

      1. add-sqr-sqrt 96.7%

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

      2. sqrt-unprod 97.3%

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

      3. swap-sqr 97.2%

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

      4. metadata-eval 97.5%

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

      5. *-commutative 97.5%

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

      6. *-commutative 97.5%

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

      7. swap-sqr 97.5%

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

      8. add-sqr-sqrt 97.8%

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

   4. Applied egg-rr 97.8%

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

      1. *-commutative 97.8%

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

      2. unpow2 97.8%

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

      3. associate-*l* 97.9%

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

      4. *-commutative 97.9%

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

      5. unpow2 97.9%

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

      6. associate-*r* 98.0%

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

   6. Simplified 98.0%

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

   if 0.00600000005 < (*.f32 314159265359/50000000000 u2)

   1. Initial program 97.7%

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

      1. add-exp-log 95.7%

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

   3. Applied egg-rr 95.7%

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

      1. add-exp-log 97.7%

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

      2. clear-num 97.8%

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

   5. Applied egg-rr 97.8%

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

      1. inv-pow 97.8%

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

      2. div-sub 97.6%

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

      3. *-inverses 97.6%

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

      4. sub-neg 97.6%

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

      5. inv-pow 97.6%

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

      6. metadata-eval 97.6%

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

   7. Applied egg-rr 97.6%

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

      1. unpow-1 97.6%

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

      2. +-commutative 97.6%

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

      3. unpow-1 97.6%

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

   9. Simplified 97.6%

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

   10. Taylor expanded in u1 around 0 85.5%

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

       1. +-commutative 85.5%

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

       2. unpow2 85.5%

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

   12. Simplified 85.5%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.006000000052154064:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1} \cdot \left(u2 \cdot \left(u2 \cdot 39.47841760436263\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1 + u1 \cdot u1}\\ \end{array} \]

Alternative 3: 90.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.011500000022351742:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1} \cdot \left(u2 \cdot \left(u2 \cdot 39.47841760436263\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.011500000022351742)
   (sqrt (* (/ u1 (- 1.0 u1)) (* u2 (* u2 39.47841760436263))))
   (* (sin (* 6.28318530718 u2)) (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.011500000022351742f) {
		tmp = sqrtf(((u1 / (1.0f - u1)) * (u2 * (u2 * 39.47841760436263f))));
	} 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) <= 0.011500000022351742e0) then
        tmp = sqrt(((u1 / (1.0e0 - u1)) * (u2 * (u2 * 39.47841760436263e0))))
    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(0.011500000022351742))
		tmp = sqrt(Float32(Float32(u1 / Float32(Float32(1.0) - u1)) * Float32(u2 * Float32(u2 * Float32(39.47841760436263)))));
	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(0.011500000022351742))
		tmp = sqrt(((u1 / (single(1.0) - u1)) * (u2 * (u2 * single(39.47841760436263)))));
	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) < 0.0115

   1. Initial program 98.5%

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

   2. Taylor expanded in u2 around 0 96.8%

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

      1. add-sqr-sqrt 96.2%

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

      2. sqrt-unprod 96.8%

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

      3. swap-sqr 96.8%

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

      4. metadata-eval 97.0%

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

      5. *-commutative 97.0%

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

      6. *-commutative 97.0%

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

      7. swap-sqr 97.1%

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

      8. add-sqr-sqrt 97.3%

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

   4. Applied egg-rr 97.3%

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

      1. *-commutative 97.3%

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

      2. unpow2 97.3%

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

      3. associate-*l* 97.4%

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

      4. *-commutative 97.4%

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

      5. unpow2 97.4%

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

      6. associate-*r* 97.5%

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

   6. Simplified 97.5%

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

   if 0.0115 < (*.f32 314159265359/50000000000 u2)

   1. Initial program 97.8%

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

   2. Taylor expanded in u1 around 0 75.4%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.011500000022351742:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1} \cdot \left(u2 \cdot \left(u2 \cdot 39.47841760436263\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \]

Alternative 4: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

2. Final simplification 98.3%

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

Alternative 5: 82.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

2. Taylor expanded in u2 around 0 83.5%

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

   1. add-sqr-sqrt 83.0%

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

   2. sqrt-unprod 83.5%

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

   3. swap-sqr 83.4%

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

   4. metadata-eval 83.6%

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

   5. *-commutative 83.6%

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

   6. *-commutative 83.6%

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

   7. swap-sqr 83.6%

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

   8. add-sqr-sqrt 83.8%

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

4. Applied egg-rr 83.8%

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

5. Final simplification 83.8%

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

Alternative 6: 82.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \sqrt{39.47841760436263 \cdot \frac{u2}{\frac{1 - u1}{u1 \cdot u2}}} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

   1. add-sqr-sqrt 97.9%

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

   2. pow1/2 97.9%

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

   3. pow1/2 97.9%

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

   4. pow-prod-down 98.3%

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

   5. swap-sqr 98.1%

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

   6. metadata-eval 98.3%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left({\left(\color{blue}{39.47841760436263} \cdot \left(u2 \cdot u2\right)\right)}^{0.5}\right) \]

3. Applied egg-rr 98.3%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \sin \color{blue}{\left({\left(39.47841760436263 \cdot \left(u2 \cdot u2\right)\right)}^{0.5}\right)} \]
4. Step-by-step derivation

   1. unpow1/2 98.3%

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

5. Simplified 98.3%

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

   1. sqrt-div 98.2%

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

   2. associate-*l/ 98.1%

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

   3. *-commutative 98.1%

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

   4. sqrt-prod 97.9%

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

   5. sqrt-prod 97.5%

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

   6. add-sqr-sqrt 97.9%

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

   7. metadata-eval 97.9%

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

   8. *-commutative 97.9%

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

   9. *-commutative 97.9%

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

   10. associate-*l/ 97.9%

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

   11. add-sqr-sqrt 95.3%

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

   12. sqrt-unprod 95.7%

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

7. Applied egg-rr 96.2%

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

   1. associate-*r/ 96.2%

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

   2. associate-/l* 96.2%

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

   3. *-commutative 96.2%

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

9. Simplified 96.2%

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

10. Taylor expanded in u2 around 0 83.9%

    \[\leadsto \sqrt{\color{blue}{39.47841760436263 \cdot \frac{{u2}^{2} \cdot u1}{1 - u1}}} \]
11. Step-by-step derivation

    1. *-commutative 83.9%

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

    2. unpow2 83.9%

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

    3. associate-*r* 83.8%

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

    4. *-commutative 83.8%

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

12. Simplified 83.8%

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

13. Taylor expanded in u2 around 0 83.9%

    \[\leadsto \sqrt{\color{blue}{39.47841760436263 \cdot \frac{{u2}^{2} \cdot u1}{1 - u1}}} \]
14. Step-by-step derivation

    1. associate-/l* 83.8%

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

    2. unpow2 83.8%

       \[\leadsto \sqrt{39.47841760436263 \cdot \frac{\color{blue}{u2 \cdot u2}}{\frac{1 - u1}{u1}}} \]

    3. associate-/l* 83.8%

       \[\leadsto \sqrt{39.47841760436263 \cdot \color{blue}{\frac{u2}{\frac{\frac{1 - u1}{u1}}{u2}}}} \]

    4. associate-/r* 83.8%

       \[\leadsto \sqrt{39.47841760436263 \cdot \frac{u2}{\color{blue}{\frac{1 - u1}{u1 \cdot u2}}}} \]

    5. *-commutative 83.8%

       \[\leadsto \sqrt{39.47841760436263 \cdot \frac{u2}{\frac{1 - u1}{\color{blue}{u2 \cdot u1}}}} \]

15. Simplified 83.8%

    \[\leadsto \sqrt{\color{blue}{39.47841760436263 \cdot \frac{u2}{\frac{1 - u1}{u2 \cdot u1}}}} \]

16. Final simplification 83.8%

    \[\leadsto \sqrt{39.47841760436263 \cdot \frac{u2}{\frac{1 - u1}{u1 \cdot u2}}} \]

Alternative 7: 82.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((single(39.47841760436263) * ((u2 * (u1 * u2)) / (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. Step-by-step derivation

   1. add-sqr-sqrt 97.9%

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

   2. pow1/2 97.9%

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

   3. pow1/2 97.9%

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

   4. pow-prod-down 98.3%

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

   5. swap-sqr 98.1%

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

   6. metadata-eval 98.3%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left({\left(\color{blue}{39.47841760436263} \cdot \left(u2 \cdot u2\right)\right)}^{0.5}\right) \]

3. Applied egg-rr 98.3%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \sin \color{blue}{\left({\left(39.47841760436263 \cdot \left(u2 \cdot u2\right)\right)}^{0.5}\right)} \]
4. Step-by-step derivation

   1. unpow1/2 98.3%

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

5. Simplified 98.3%

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

   1. sqrt-div 98.2%

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

   2. associate-*l/ 98.1%

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

   3. *-commutative 98.1%

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

   4. sqrt-prod 97.9%

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

   5. sqrt-prod 97.5%

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

   6. add-sqr-sqrt 97.9%

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

   7. metadata-eval 97.9%

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

   8. *-commutative 97.9%

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

   9. *-commutative 97.9%

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

   10. associate-*l/ 97.9%

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

   11. add-sqr-sqrt 95.3%

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

   12. sqrt-unprod 95.7%

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

7. Applied egg-rr 96.2%

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

   1. associate-*r/ 96.2%

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

   2. associate-/l* 96.2%

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

   3. *-commutative 96.2%

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

9. Simplified 96.2%

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

10. Taylor expanded in u2 around 0 83.9%

    \[\leadsto \sqrt{\color{blue}{39.47841760436263 \cdot \frac{{u2}^{2} \cdot u1}{1 - u1}}} \]
11. Step-by-step derivation

    1. *-commutative 83.9%

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

    2. unpow2 83.9%

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

    3. associate-*r* 83.8%

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

    4. *-commutative 83.8%

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

12. Simplified 83.8%

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

13. Final simplification 83.8%

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

Alternative 8: 82.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

2. Taylor expanded in u2 around 0 83.5%

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

   1. add-sqr-sqrt 83.0%

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

   2. sqrt-unprod 83.5%

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

   3. swap-sqr 83.4%

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

   4. metadata-eval 83.6%

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

   5. *-commutative 83.6%

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

   6. *-commutative 83.6%

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

   7. swap-sqr 83.6%

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

   8. add-sqr-sqrt 83.8%

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

4. Applied egg-rr 83.8%

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

   1. *-commutative 83.8%

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

   2. unpow2 83.8%

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

   3. associate-*l* 83.9%

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

   4. *-commutative 83.9%

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

   5. unpow2 83.9%

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

   6. associate-*r* 84.0%

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

6. Simplified 84.0%

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

7. Final simplification 84.0%

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

Alternative 9: 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 98.3%

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

2. Taylor expanded in u2 around 0 83.5%

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

3. Final simplification 83.5%

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

Alternative 10: 82.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = single(6.28318530718) * (u2 / sqrt((single(-1.0) + (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. Taylor expanded in u2 around 0 83.5%

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

   1. associate-*r* 83.5%

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

   2. add-sqr-sqrt 83.1%

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

   3. metadata-eval 83.1%

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

   4. sqrt-prod 83.5%

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

   5. sqrt-prod 83.5%

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

   6. sqrt-div 83.4%

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

   7. associate-*r/ 83.4%

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

   8. *-commutative 83.4%

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

   9. sqrt-prod 83.4%

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

   10. sqrt-prod 83.1%

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

   11. add-sqr-sqrt 83.4%

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

   12. metadata-eval 83.2%

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

4. Applied egg-rr 83.2%

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

   1. associate-/l* 83.3%

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

   2. *-commutative 83.3%

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

   3. associate-/l* 83.4%

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

6. Simplified 83.4%

   \[\leadsto \color{blue}{\frac{6.28318530718}{\frac{\frac{\sqrt{1 - u1}}{\sqrt{u1}}}{u2}}} \]
7. Step-by-step derivation

   1. expm1-log1p-u 83.4%

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

   2. expm1-udef 29.0%

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

   3. div-inv 29.0%

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

   4. clear-num 29.0%

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

   5. sqrt-undiv 29.0%

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

   6. div-sub 29.0%

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

   7. *-inverses 29.0%

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

   8. sub-neg 29.0%

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

   9. inv-pow 29.0%

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

   10. metadata-eval 29.0%

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

8. Applied egg-rr 29.0%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(6.28318530718 \cdot \frac{u2}{\sqrt{{u1}^{-1} + -1}}\right)} - 1} \]
9. Step-by-step derivation

   1. expm1-def 83.5%

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

   2. expm1-log1p 83.5%

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

   3. +-commutative 83.5%

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

   4. unpow-1 83.5%

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

10. Simplified 83.5%

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

11. Final simplification 83.5%

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

Alternative 11: 82.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

2. Taylor expanded in u2 around 0 83.5%

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

   1. associate-*r* 83.5%

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

   2. add-sqr-sqrt 83.1%

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

   3. metadata-eval 83.1%

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

   4. sqrt-prod 83.5%

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

   5. sqrt-prod 83.5%

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

   6. sqrt-div 83.4%

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

   7. associate-*r/ 83.4%

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

   8. *-commutative 83.4%

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

   9. sqrt-prod 83.4%

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

   10. sqrt-prod 83.1%

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

   11. add-sqr-sqrt 83.4%

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

   12. metadata-eval 83.2%

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

4. Applied egg-rr 83.2%

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

   1. associate-/l* 83.3%

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

   2. *-commutative 83.3%

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

   3. associate-/l* 83.4%

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

6. Simplified 83.4%

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

7. Taylor expanded in u2 around 0 83.4%

   \[\leadsto \frac{6.28318530718}{\color{blue}{\frac{1}{u2} \cdot \sqrt{\frac{1 - u1}{u1}}}} \]
8. Step-by-step derivation

   1. associate-*l/ 83.5%

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

   2. *-lft-identity 83.5%

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

   3. div-sub 83.6%

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

   4. unpow-1 83.6%

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

   5. sub-neg 83.6%

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

   6. *-inverses 83.6%

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

   7. metadata-eval 83.6%

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

   8. +-commutative 83.6%

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

   9. unpow-1 83.6%

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

9. Simplified 83.6%

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

10. Final simplification 83.6%

    \[\leadsto \frac{6.28318530718}{\frac{\sqrt{-1 + \frac{1}{u1}}}{u2}} \]

Alternative 12: 65.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{u1 \cdot \left(u2 \cdot \left(u2 \cdot 39.47841760436263\right)\right)} \end{array} \]

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 * (u2 * (u2 * 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 * (u2 * (u2 * 39.47841760436263e0))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

2. Taylor expanded in u2 around 0 83.5%

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

3. Taylor expanded in u1 around 0 66.1%

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

   1. add-sqr-sqrt 66.0%

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

   2. sqrt-unprod 66.1%

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

   3. associate-*r* 66.1%

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

   4. associate-*r* 66.1%

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

   5. swap-sqr 66.1%

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

   6. *-commutative 66.1%

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

   7. *-commutative 66.1%

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

   8. swap-sqr 66.2%

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

   9. metadata-eval 66.1%

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

   10. add-sqr-sqrt 66.1%

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

5. Applied egg-rr 66.1%

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

   1. *-commutative 66.1%

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

   2. associate-*l* 66.2%

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

7. Simplified 66.2%

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

8. Final simplification 66.2%

   \[\leadsto \sqrt{u1 \cdot \left(u2 \cdot \left(u2 \cdot 39.47841760436263\right)\right)} \]

Alternative 13: 65.2% 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. Taylor expanded in u2 around 0 83.5%

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

3. Taylor expanded in u1 around 0 66.1%

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

4. Final simplification 66.1%

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

Alternative 14: -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 98.3%

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

   1. add-sqr-sqrt 97.9%

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

   2. pow1/2 97.9%

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

   3. pow1/2 97.9%

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

   4. pow-prod-down 98.3%

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

   5. swap-sqr 98.1%

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

   6. metadata-eval 98.3%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left({\left(\color{blue}{39.47841760436263} \cdot \left(u2 \cdot u2\right)\right)}^{0.5}\right) \]

3. Applied egg-rr 98.3%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \sin \color{blue}{\left({\left(39.47841760436263 \cdot \left(u2 \cdot u2\right)\right)}^{0.5}\right)} \]
4. Step-by-step derivation

   1. unpow1/2 98.3%

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

5. Simplified 98.3%

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

   1. sqrt-div 98.2%

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

   2. associate-*l/ 98.1%

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

   3. *-commutative 98.1%

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

   4. sqrt-prod 97.9%

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

   5. sqrt-prod 97.5%

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

   6. add-sqr-sqrt 97.9%

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

   7. metadata-eval 97.9%

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

   8. *-commutative 97.9%

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

   9. *-commutative 97.9%

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

   10. associate-*l/ 97.9%

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

   11. add-sqr-sqrt 95.3%

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

   12. sqrt-unprod 95.7%

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

7. Applied egg-rr 96.2%

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

   1. associate-*r/ 96.2%

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

   2. associate-/l* 96.2%

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

   3. *-commutative 96.2%

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

9. Simplified 96.2%

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

10. Taylor expanded in u2 around 0 83.9%

    \[\leadsto \sqrt{\color{blue}{39.47841760436263 \cdot \frac{{u2}^{2} \cdot u1}{1 - u1}}} \]
11. Step-by-step derivation

    1. *-commutative 83.9%

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

    2. unpow2 83.9%

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

    3. associate-*r* 83.8%

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

    4. *-commutative 83.8%

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

12. Simplified 83.8%

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

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

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

14. Final simplification -0.0%

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

Reproduce

herbie shell --seed 2023230 
(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))))