Trowbridge-Reitz Sample, near normal, slope_y

Percentage Accurate: 98.3% → 98.5%

Time: 13.1s

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: 98.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

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 = sqrt((u1 / (1.0e0 - u1))) * sin(sqrt((39.47841760436263e0 * (u2 ** 2.0e0))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.2%

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

   1. add-log-exp 63.4%

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

   2. exp-prod 63.4%

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

3. Applied egg-rr 63.4%

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

   1. pow-exp 63.4%

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

   2. add-log-exp 98.2%

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

   3. add-sqr-sqrt 97.4%

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

   4. sqrt-unprod 98.2%

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

   5. rem-log-exp 98.2%

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

   6. rem-log-exp 98.2%

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

   7. swap-sqr 98.0%

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

   8. rem-log-exp 98.0%

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

   9. rem-log-exp 98.0%

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

   10. metadata-eval 98.4%

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

   11. unpow2 98.4%

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

5. Applied egg-rr 98.4%

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

6. Final simplification 98.4%

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

Alternative 2: 98.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{u1}{1 - {u1}^{2}} \cdot \left(u1 + 1\right)} \cdot \sin \left(u2 \cdot 6.28318530718\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sqrt (* (/ u1 (- 1.0 (pow u1 2.0))) (+ u1 1.0)))
  (sin (* u2 6.28318530718))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(((u1 / (1.0f - powf(u1, 2.0f))) * (u1 + 1.0f))) * sinf((u2 * 6.28318530718f));
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = sqrt(((u1 / (1.0e0 - (u1 ** 2.0e0))) * (u1 + 1.0e0))) * sin((u2 * 6.28318530718e0))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(Float32(u1 / Float32(Float32(1.0) - (u1 ^ Float32(2.0)))) * Float32(u1 + Float32(1.0)))) * sin(Float32(u2 * Float32(6.28318530718))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt(((u1 / (single(1.0) - (u1 ^ single(2.0)))) * (u1 + single(1.0)))) * sin((u2 * single(6.28318530718)));
end

Derivation

1. Initial program 98.2%

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

   1. flip-- 98.3%

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

   2. associate-/r/ 98.3%

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

   3. metadata-eval 98.3%

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

   4. pow2 98.3%

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

   5. +-commutative 98.3%

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

3. Applied egg-rr 98.3%

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

4. Final simplification 98.3%

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

Alternative 3: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return sinf((u2 * 6.28318530718f)) * powf(((1.0f / u1) + -1.0f), -0.5f);
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = sin((u2 * 6.28318530718e0)) * (((1.0e0 / u1) + (-1.0e0)) ** (-0.5e0))
end function

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sin((u2 * single(6.28318530718))) * (((single(1.0) / u1) + single(-1.0)) ^ single(-0.5));
end

Derivation

1. Initial program 98.2%

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

   1. expm1-log1p-u 97.8%

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

   2. expm1-udef 41.8%

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

3. Applied egg-rr 41.8%

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

   1. expm1-def 97.8%

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

   2. expm1-log1p-u 98.2%

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

   3. clear-num 98.3%

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

   4. inv-pow 98.3%

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

   5. sqrt-pow1 98.3%

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

   6. div-sub 98.3%

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

   7. *-inverses 98.3%

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

   8. sub-neg 98.3%

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

   9. metadata-eval 98.3%

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

   10. metadata-eval 98.3%

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

5. Applied egg-rr 98.3%

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

6. Final simplification 98.3%

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

Alternative 4: 90.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.017999999225139618:\\ \;\;\;\;\left(u2 \cdot 6.28318530718\right) \cdot \sqrt{\frac{1}{\frac{1 - u1}{u1}}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 6.28318530718) 0.017999999225139618)
   (* (* u2 6.28318530718) (sqrt (/ 1.0 (/ (- 1.0 u1) u1))))
   (* (sin (* u2 6.28318530718)) (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * 6.28318530718f) <= 0.017999999225139618f) {
		tmp = (u2 * 6.28318530718f) * sqrtf((1.0f / ((1.0f - u1) / u1)));
	} else {
		tmp = sinf((u2 * 6.28318530718f)) * sqrtf(u1);
	}
	return tmp;
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    real(4) :: tmp
    if ((u2 * 6.28318530718e0) <= 0.017999999225139618e0) then
        tmp = (u2 * 6.28318530718e0) * sqrt((1.0e0 / ((1.0e0 - u1) / u1)))
    else
        tmp = sin((u2 * 6.28318530718e0)) * sqrt(u1)
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(u2 * Float32(6.28318530718)) <= Float32(0.017999999225139618))
		tmp = Float32(Float32(u2 * Float32(6.28318530718)) * sqrt(Float32(Float32(1.0) / Float32(Float32(Float32(1.0) - u1) / u1))));
	else
		tmp = Float32(sin(Float32(u2 * Float32(6.28318530718))) * sqrt(u1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((u2 * single(6.28318530718)) <= single(0.017999999225139618))
		tmp = (u2 * single(6.28318530718)) * sqrt((single(1.0) / ((single(1.0) - u1) / u1)));
	else
		tmp = sin((u2 * single(6.28318530718))) * sqrt(u1);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 314159265359/50000000000 u2) < 0.0179999992

   1. Initial program 98.7%

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

      1. expm1-log1p-u 98.7%

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

   3. Applied egg-rr 98.7%

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

      1. expm1-log1p-u 98.7%

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

      2. clear-num 98.7%

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

   5. Applied egg-rr 98.7%

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

   6. Taylor expanded in u2 around 0 95.1%

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

   if 0.0179999992 < (*.f32 314159265359/50000000000 u2)

   1. Initial program 97.1%

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

   2. Taylor expanded in u1 around 0 73.3%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.017999999225139618:\\ \;\;\;\;\left(u2 \cdot 6.28318530718\right) \cdot \sqrt{\frac{1}{\frac{1 - u1}{u1}}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1}\\ \end{array} \]

Alternative 5: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.2%

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

2. Final simplification 98.2%

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

Alternative 6: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.2%

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

   1. expm1-log1p-u 98.2%

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

3. Applied egg-rr 98.2%

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

   1. expm1-log1p-u 98.2%

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

   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. Step-by-step derivation

   1. *-commutative 98.3%

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

   2. sqrt-div 98.1%

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

   3. metadata-eval 98.1%

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

   4. un-div-inv 98.3%

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

   5. div-sub 98.2%

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

   6. *-inverses 98.2%

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

   7. sub-neg 98.2%

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

   8. metadata-eval 98.2%

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

7. Applied egg-rr 98.2%

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

8. Final simplification 98.2%

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

Alternative 7: 81.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.2%

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

   1. expm1-log1p-u 97.8%

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

   2. expm1-udef 41.8%

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

3. Applied egg-rr 41.8%

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

   1. expm1-def 97.8%

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

   2. expm1-log1p-u 98.2%

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

   3. clear-num 98.3%

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

   4. inv-pow 98.3%

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

   5. sqrt-pow1 98.3%

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

   6. div-sub 98.3%

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

   7. *-inverses 98.3%

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

   8. sub-neg 98.3%

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

   9. metadata-eval 98.3%

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

   10. metadata-eval 98.3%

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

5. Applied egg-rr 98.3%

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

6. Taylor expanded in u2 around 0 80.0%

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

   1. associate-*r* 80.0%

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

   2. *-commutative 80.0%

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

   3. associate-*l* 80.0%

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

   4. sub-neg 80.0%

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

   5. metadata-eval 80.0%

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

   6. +-commutative 80.0%

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

8. Simplified 80.0%

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

9. Final simplification 80.0%

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

Alternative 8: 81.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.2%

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

   1. expm1-log1p-u 98.2%

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

3. Applied egg-rr 98.2%

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

   1. expm1-log1p-u 98.2%

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

   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. Taylor expanded in u2 around 0 80.1%

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

7. Final simplification 80.1%

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

Alternative 9: 81.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return (u2 * 6.28318530718f) * powf(((1.0f / u1) + -1.0f), -0.5f);
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = (u2 * 6.28318530718e0) * (((1.0e0 / u1) + (-1.0e0)) ** (-0.5e0))
end function

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = (u2 * single(6.28318530718)) * (((single(1.0) / u1) + single(-1.0)) ^ single(-0.5));
end

Derivation

1. Initial program 98.2%

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

   1. expm1-log1p-u 97.8%

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

   2. expm1-udef 41.8%

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

3. Applied egg-rr 41.8%

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

   1. expm1-def 97.8%

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

   2. expm1-log1p-u 98.2%

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

   3. clear-num 98.3%

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

   4. inv-pow 98.3%

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

   5. sqrt-pow1 98.3%

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

   6. div-sub 98.3%

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

   7. *-inverses 98.3%

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

   8. sub-neg 98.3%

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

   9. metadata-eval 98.3%

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

   10. metadata-eval 98.3%

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

5. Applied egg-rr 98.3%

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

6. Taylor expanded in u2 around 0 80.0%

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

7. Final simplification 80.0%

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

Alternative 10: 81.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.2%

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

2. Taylor expanded in u2 around 0 80.0%

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

3. Final simplification 80.0%

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

Alternative 11: 81.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.2%

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

2. Taylor expanded in u2 around 0 80.0%

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

   1. *-commutative 80.0%

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

   2. associate-*r* 80.0%

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

4. Simplified 80.0%

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

5. Final simplification 80.0%

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

Alternative 12: 64.6% 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 98.2%

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

2. Taylor expanded in u2 around 0 80.0%

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

3. Taylor expanded in u1 around 0 63.8%

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

4. Final simplification 63.8%

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

Alternative 13: 64.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.2%

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

2. Taylor expanded in u2 around 0 80.0%

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

3. Taylor expanded in u1 around 0 63.8%

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

   1. expm1-log1p-u 63.8%

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

   2. expm1-udef 29.3%

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

   3. *-commutative 29.3%

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

5. Applied egg-rr 29.3%

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

   1. expm1-def 63.8%

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

   2. expm1-log1p 63.8%

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

   3. associate-*r* 63.8%

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

   4. *-commutative 63.8%

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

   5. associate-*l* 63.8%

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

7. Simplified 63.8%

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

8. Final simplification 63.8%

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

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 98.2%

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

2. Taylor expanded in u2 around 0 80.0%

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

   1. add-sqr-sqrt 79.7%

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

   2. sqrt-unprod 80.0%

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

   3. *-commutative 80.0%

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

   4. *-commutative 80.0%

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

   5. swap-sqr 79.9%

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

   6. swap-sqr 79.9%

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

   7. add-sqr-sqrt 79.9%

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

   8. pow2 79.9%

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

   9. metadata-eval 80.3%

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

4. Applied egg-rr 80.3%

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

   1. *-commutative 80.3%

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

   2. associate-*l/ 80.2%

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

6. Simplified 80.2%

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

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

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

8. Final simplification -0.0%

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

Reproduce

herbie shell --seed 2023326 
(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))))