Trowbridge-Reitz Sample, near normal, slope_x

Percentage Accurate: 98.9% → 98.8%

Time: 12.0s

Alternatives: 9

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(e^{\mathsf{log1p}\left(6.28318530718 \cdot u2\right)} + -1\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sqrt (/ u1 (- 1.0 u1)))
  (cos (+ (exp (log1p (* 6.28318530718 u2))) -1.0))))

C

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

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * cos(Float32(exp(log1p(Float32(Float32(6.28318530718) * u2))) + Float32(-1.0))))
end

Derivation

1. Initial program 99.0%

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

   1. expm1-log1p-u 99.0%

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

   2. expm1-udef 99.0%

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

3. Applied egg-rr 99.0%

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

4. Final simplification 99.0%

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

Alternative 2: 98.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(\frac{1}{\frac{1 + \left(1 + 6.28318530718 \cdot u2\right)}{6.28318530718 \cdot \left(u2 \cdot \mathsf{fma}\left(u2, 6.28318530718, 2\right)\right)}}\right) \end{array} \]

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.0%

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

   1. expm1-log1p-u 99.0%

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

   2. expm1-udef 99.0%

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

3. Applied egg-rr 99.0%

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

   1. flip-- 99.0%

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

   2. clear-num 99.0%

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

   3. log1p-udef 98.9%

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

   4. rem-exp-log 98.9%

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

   5. +-commutative 98.9%

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

   6. metadata-eval 98.9%

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

   7. sub-neg 98.9%

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

   8. pow2 98.9%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(\frac{1}{\frac{\left(6.28318530718 \cdot u2 + 1\right) + 1}{\color{blue}{{\left(e^{\mathsf{log1p}\left(6.28318530718 \cdot u2\right)}\right)}^{2}} + \left(-1\right)}}\right) \]

   9. log1p-udef 98.9%

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

   10. rem-exp-log 98.9%

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

   11. +-commutative 98.9%

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

   12. metadata-eval 98.9%

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

5. Applied egg-rr 98.9%

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

   1. unpow2 98.9%

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

   2. difference-of-sqr--1 98.9%

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

   3. associate-+l+ 98.9%

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

   4. metadata-eval 98.9%

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

   5. fma-def 98.9%

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

   6. +-commutative 98.9%

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

   7. add-exp-log 98.9%

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

   8. log1p-udef 99.0%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(\frac{1}{\frac{\left(6.28318530718 \cdot u2 + 1\right) + 1}{\mathsf{fma}\left(6.28318530718, u2, 2\right) \cdot \left(e^{\color{blue}{\mathsf{log1p}\left(6.28318530718 \cdot u2\right)}} - 1\right)}}\right) \]

   9. expm1-udef 99.0%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(\frac{1}{\frac{\left(6.28318530718 \cdot u2 + 1\right) + 1}{\mathsf{fma}\left(6.28318530718, u2, 2\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(6.28318530718 \cdot u2\right)\right)}}}\right) \]

   10. expm1-log1p-u 99.0%

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

7. Applied egg-rr 99.0%

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

   1. *-commutative 99.0%

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

   2. associate-*l* 99.0%

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

   3. fma-udef 99.0%

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

   4. *-commutative 99.0%

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

   5. fma-def 99.0%

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

9. Simplified 99.0%

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

10. Final simplification 99.0%

    \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(\frac{1}{\frac{1 + \left(1 + 6.28318530718 \cdot u2\right)}{6.28318530718 \cdot \left(u2 \cdot \mathsf{fma}\left(u2, 6.28318530718, 2\right)\right)}}\right) \]

Alternative 3: 90.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(6.28318530718 \cdot u2\right)\\ \mathbf{if}\;t_0 \leq 0.9999719858169556:\\ \;\;\;\;t_0 \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* 6.28318530718 u2))))
   (if (<= t_0 0.9999719858169556)
     (* t_0 (sqrt u1))
     (sqrt (/ u1 (- 1.0 u1))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = cosf((6.28318530718f * u2));
	float tmp;
	if (t_0 <= 0.9999719858169556f) {
		tmp = t_0 * sqrtf(u1);
	} else {
		tmp = sqrtf((u1 / (1.0f - u1)));
	}
	return tmp;
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    real(4) :: t_0
    real(4) :: tmp
    t_0 = cos((6.28318530718e0 * u2))
    if (t_0 <= 0.9999719858169556e0) then
        tmp = t_0 * sqrt(u1)
    else
        tmp = sqrt((u1 / (1.0e0 - u1)))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	t_0 = cos(Float32(Float32(6.28318530718) * u2))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.9999719858169556))
		tmp = Float32(t_0 * sqrt(u1));
	else
		tmp = sqrt(Float32(u1 / Float32(Float32(1.0) - u1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	t_0 = cos((single(6.28318530718) * u2));
	tmp = single(0.0);
	if (t_0 <= single(0.9999719858169556))
		tmp = t_0 * sqrt(u1);
	else
		tmp = sqrt((u1 / (single(1.0) - u1)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 314159265359/50000000000 u2)) < 0.999971986

   1. Initial program 97.9%

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

   2. Taylor expanded in u1 around 0 76.2%

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

   if 0.999971986 < (cos.f32 (*.f32 314159265359/50000000000 u2))

   1. Initial program 99.5%

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

   2. Taylor expanded in u2 around 0 96.1%

      \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(6.28318530718 \cdot u2\right) \leq 0.9999719858169556:\\ \;\;\;\;\cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}}\\ \end{array} \]

Alternative 4: 98.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - u1))) * cosf((1.0f / ((1.0f + (1.0f + (6.28318530718f * u2))) / (6.28318530718f * ((u2 * (6.28318530718f * u2)) + (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))) * cos((1.0e0 / ((1.0e0 + (1.0e0 + (6.28318530718e0 * u2))) / (6.28318530718e0 * ((u2 * (6.28318530718e0 * u2)) + (u2 * 2.0e0))))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

   1. expm1-log1p-u 99.0%

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

   2. expm1-udef 99.0%

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

3. Applied egg-rr 99.0%

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

   1. flip-- 99.0%

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

   2. clear-num 99.0%

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

   3. log1p-udef 98.9%

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

   4. rem-exp-log 98.9%

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

   5. +-commutative 98.9%

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

   6. metadata-eval 98.9%

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

   7. sub-neg 98.9%

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

   8. pow2 98.9%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(\frac{1}{\frac{\left(6.28318530718 \cdot u2 + 1\right) + 1}{\color{blue}{{\left(e^{\mathsf{log1p}\left(6.28318530718 \cdot u2\right)}\right)}^{2}} + \left(-1\right)}}\right) \]

   9. log1p-udef 98.9%

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

   10. rem-exp-log 98.9%

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

   11. +-commutative 98.9%

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

   12. metadata-eval 98.9%

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

5. Applied egg-rr 98.9%

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

   1. unpow2 98.9%

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

   2. difference-of-sqr--1 98.9%

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

   3. associate-+l+ 98.9%

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

   4. metadata-eval 98.9%

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

   5. fma-def 98.9%

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

   6. +-commutative 98.9%

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

   7. add-exp-log 98.9%

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

   8. log1p-udef 99.0%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(\frac{1}{\frac{\left(6.28318530718 \cdot u2 + 1\right) + 1}{\mathsf{fma}\left(6.28318530718, u2, 2\right) \cdot \left(e^{\color{blue}{\mathsf{log1p}\left(6.28318530718 \cdot u2\right)}} - 1\right)}}\right) \]

   9. expm1-udef 99.0%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(\frac{1}{\frac{\left(6.28318530718 \cdot u2 + 1\right) + 1}{\mathsf{fma}\left(6.28318530718, u2, 2\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(6.28318530718 \cdot u2\right)\right)}}}\right) \]

   10. expm1-log1p-u 99.0%

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

7. Applied egg-rr 99.0%

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

   1. *-commutative 99.0%

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

   2. associate-*l* 99.0%

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

   3. fma-udef 99.0%

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

   4. *-commutative 99.0%

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

   5. fma-def 99.0%

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

9. Simplified 99.0%

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

    1. fma-udef 99.0%

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

    2. *-commutative 99.0%

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

    3. distribute-rgt-in 99.0%

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

11. Applied egg-rr 99.0%

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

12. Final simplification 99.0%

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

Alternative 5: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

2. Final simplification 99.0%

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

Alternative 6: 71.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

2. Taylor expanded in u2 around 0 79.2%

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

3. Taylor expanded in u1 around 0 72.9%

   \[\leadsto \sqrt{\color{blue}{u1 + \left({u1}^{2} + {u1}^{3}\right)}} \]
4. Step-by-step derivation

   1. +-commutative 72.9%

      \[\leadsto \sqrt{\color{blue}{\left({u1}^{2} + {u1}^{3}\right) + u1}} \]

   2. unpow2 72.9%

      \[\leadsto \sqrt{\left(\color{blue}{u1 \cdot u1} + {u1}^{3}\right) + u1} \]

   3. cube-mult 72.9%

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

   4. unpow2 72.9%

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

   5. distribute-lft-out 72.9%

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

   6. unpow2 72.9%

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

   7. distribute-rgt1-in 72.9%

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

   8. *-commutative 72.9%

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

   9. *-rgt-identity 72.9%

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

   10. distribute-lft-in 72.9%

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

   11. fma-udef 72.9%

       \[\leadsto \sqrt{u1 \cdot \color{blue}{\mathsf{fma}\left(u1, u1 + 1, 1\right)}} \]

5. Simplified 72.9%

   \[\leadsto \sqrt{\color{blue}{u1 \cdot \mathsf{fma}\left(u1, u1 + 1, 1\right)}} \]

6. Taylor expanded in u1 around 0 70.4%

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

7. Final simplification 70.4%

   \[\leadsto \sqrt{u1 \cdot \left(u1 + 1\right)} \]

Alternative 7: 79.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

2. Taylor expanded in u2 around 0 79.2%

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

3. Final simplification 79.2%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \]

Alternative 8: 63.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{u1} \end{array} \]

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return 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 = sqrt(u1)
end function

Julia

function code(cosTheta_i, u1, u2)
	return sqrt(u1)
end

MATLAB

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

Derivation

1. Initial program 99.0%

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

2. Taylor expanded in u2 around 0 79.2%

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

3. Taylor expanded in u1 around 0 63.6%

   \[\leadsto \sqrt{\color{blue}{u1}} \]

4. Final simplification 63.6%

   \[\leadsto \sqrt{u1} \]

Alternative 9: 20.2% accurate, 69.7× speedup

Math

\[\begin{array}{l} \\ u1 + 0.5 \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2) :precision binary32 (+ u1 0.5))

C

float code(float cosTheta_i, float u1, float u2) {
	return u1 + 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 = u1 + 0.5e0
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(u1 + Float32(0.5))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = u1 + single(0.5);
end

Derivation

1. Initial program 99.0%

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

2. Taylor expanded in u2 around 0 79.2%

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

3. Taylor expanded in u1 around 0 72.9%

   \[\leadsto \sqrt{\color{blue}{u1 + \left({u1}^{2} + {u1}^{3}\right)}} \]
4. Step-by-step derivation

   1. +-commutative 72.9%

      \[\leadsto \sqrt{\color{blue}{\left({u1}^{2} + {u1}^{3}\right) + u1}} \]

   2. unpow2 72.9%

      \[\leadsto \sqrt{\left(\color{blue}{u1 \cdot u1} + {u1}^{3}\right) + u1} \]

   3. cube-mult 72.9%

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

   4. unpow2 72.9%

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

   5. distribute-lft-out 72.9%

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

   6. unpow2 72.9%

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

   7. distribute-rgt1-in 72.9%

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

   8. *-commutative 72.9%

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

   9. *-rgt-identity 72.9%

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

   10. distribute-lft-in 72.9%

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

   11. fma-udef 72.9%

       \[\leadsto \sqrt{u1 \cdot \color{blue}{\mathsf{fma}\left(u1, u1 + 1, 1\right)}} \]

5. Simplified 72.9%

   \[\leadsto \sqrt{\color{blue}{u1 \cdot \mathsf{fma}\left(u1, u1 + 1, 1\right)}} \]

6. Taylor expanded in u1 around 0 70.4%

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

7. Taylor expanded in u1 around inf 20.0%

   \[\leadsto \color{blue}{0.5 + u1} \]
8. Step-by-step derivation

   1. +-commutative 20.0%

      \[\leadsto \color{blue}{u1 + 0.5} \]

9. Simplified 20.0%

   \[\leadsto \color{blue}{u1 + 0.5} \]

10. Final simplification 20.0%

    \[\leadsto u1 + 0.5 \]

Reproduce

herbie shell --seed 2023334 
(FPCore (cosTheta_i u1 u2)
  :name "Trowbridge-Reitz Sample, near normal, slope_x"
  :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))) (cos (* 6.28318530718 u2))))