Trowbridge-Reitz Sample, near normal, slope_x

Percentage Accurate: 99.0% → 99.0%

Time: 10.6s

Alternatives: 6

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: 99.0% 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: 99.0% 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 2: 94.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.15000000596046448:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + -19.739208802181317 \cdot \left(u2 \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.15000000596046448)
   (* (sqrt (/ u1 (- 1.0 u1))) (+ 1.0 (* -19.739208802181317 (* u2 u2))))
   (* (cos (* 6.28318530718 u2)) (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.15000000596046448f) {
		tmp = sqrtf((u1 / (1.0f - u1))) * (1.0f + (-19.739208802181317f * (u2 * u2)));
	} else {
		tmp = cosf((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) <= 0.15000000596046448e0) then
        tmp = sqrt((u1 / (1.0e0 - u1))) * (1.0e0 + ((-19.739208802181317e0) * (u2 * u2)))
    else
        tmp = cos((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(0.15000000596046448))
		tmp = Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * Float32(Float32(1.0) + Float32(Float32(-19.739208802181317) * Float32(u2 * u2))));
	else
		tmp = Float32(cos(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(0.15000000596046448))
		tmp = sqrt((u1 / (single(1.0) - u1))) * (single(1.0) + (single(-19.739208802181317) * (u2 * u2)));
	else
		tmp = cos((single(6.28318530718) * u2)) * sqrt(u1);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 314159265359/50000000000 u2) < 0.150000006

   1. Initial program 99.4%

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

      1. add-sqr-sqrt 99.4%

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

      2. pow1/2 99.4%

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

      3. pow1/2 99.4%

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

      4. pow-prod-down 99.4%

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

      5. swap-sqr 99.4%

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

      6. metadata-eval 99.4%

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

   3. Applied egg-rr 99.4%

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

      1. unpow1/2 99.4%

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

      2. unpow2 99.4%

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

      3. *-commutative 99.4%

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

      4. unpow2 99.4%

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

   5. Simplified 99.4%

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

      1. log1p-expm1-u 99.4%

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\sqrt{\left(u2 \cdot u2\right) \cdot 39.47841760436263}\right)\right)\right)} \]

      2. sqrt-prod 99.4%

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \color{blue}{\left(\sqrt{u2 \cdot u2} \cdot \sqrt{39.47841760436263}\right)}\right)\right) \]

      3. sqrt-prod 99.4%

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\color{blue}{\left(\sqrt{u2} \cdot \sqrt{u2}\right)} \cdot \sqrt{39.47841760436263}\right)\right)\right) \]

      4. metadata-eval 99.4%

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

      5. add-sqr-sqrt 99.4%

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

   7. Applied egg-rr 99.4%

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

   8. Taylor expanded in u2 around 0 99.0%

      \[\leadsto \color{blue}{-19.739208802181317 \cdot \left({u2}^{2} \cdot \sqrt{\frac{u1}{1 - u1}}\right) + \sqrt{\frac{u1}{1 - u1}}} \]
   9. Step-by-step derivation

      1. +-commutative 99.0%

         \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}} + -19.739208802181317 \cdot \left({u2}^{2} \cdot \sqrt{\frac{u1}{1 - u1}}\right)} \]

      2. *-lft-identity 99.0%

         \[\leadsto \color{blue}{1 \cdot \sqrt{\frac{u1}{1 - u1}}} + -19.739208802181317 \cdot \left({u2}^{2} \cdot \sqrt{\frac{u1}{1 - u1}}\right) \]

      3. associate-*r* 99.0%

         \[\leadsto 1 \cdot \sqrt{\frac{u1}{1 - u1}} + \color{blue}{\left(-19.739208802181317 \cdot {u2}^{2}\right) \cdot \sqrt{\frac{u1}{1 - u1}}} \]

      4. distribute-rgt-out 98.9%

         \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + -19.739208802181317 \cdot {u2}^{2}\right)} \]

      5. unpow2 98.9%

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + -19.739208802181317 \cdot \color{blue}{\left(u2 \cdot u2\right)}\right) \]

   10. Simplified 98.9%

       \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + -19.739208802181317 \cdot \left(u2 \cdot u2\right)\right)} \]

   if 0.150000006 < (*.f32 314159265359/50000000000 u2)

   1. Initial program 97.1%

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

   2. Taylor expanded in u1 around 0 74.6%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.15000000596046448:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + -19.739208802181317 \cdot \left(u2 \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \]

Alternative 3: 88.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + -19.739208802181317 \cdot \left(u2 \cdot u2\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (/ u1 (- 1.0 u1))) (+ 1.0 (* -19.739208802181317 (* u2 u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - u1))) * (1.0f + (-19.739208802181317f * (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((u1 / (1.0e0 - u1))) * (1.0e0 + ((-19.739208802181317e0) * (u2 * u2)))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * Float32(Float32(1.0) + Float32(Float32(-19.739208802181317) * Float32(u2 * u2))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((u1 / (single(1.0) - u1))) * (single(1.0) + (single(-19.739208802181317) * (u2 * u2)));
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. add-sqr-sqrt 98.7%

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

   2. pow1/2 98.7%

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

   3. pow1/2 98.7%

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

   4. pow-prod-down 99.0%

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

   5. swap-sqr 99.0%

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

   6. metadata-eval 99.0%

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

3. Applied egg-rr 99.0%

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

   1. unpow1/2 99.0%

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

   2. unpow2 99.0%

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

   3. *-commutative 99.0%

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

   4. unpow2 99.0%

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

5. Simplified 99.0%

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

   1. log1p-expm1-u 98.9%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\sqrt{\left(u2 \cdot u2\right) \cdot 39.47841760436263}\right)\right)\right)} \]

   2. sqrt-prod 98.8%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \color{blue}{\left(\sqrt{u2 \cdot u2} \cdot \sqrt{39.47841760436263}\right)}\right)\right) \]

   3. sqrt-prod 98.7%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\color{blue}{\left(\sqrt{u2} \cdot \sqrt{u2}\right)} \cdot \sqrt{39.47841760436263}\right)\right)\right) \]

   4. metadata-eval 98.9%

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

   5. add-sqr-sqrt 99.0%

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

7. Applied egg-rr 99.0%

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

8. Taylor expanded in u2 around 0 88.6%

   \[\leadsto \color{blue}{-19.739208802181317 \cdot \left({u2}^{2} \cdot \sqrt{\frac{u1}{1 - u1}}\right) + \sqrt{\frac{u1}{1 - u1}}} \]
9. Step-by-step derivation

   1. +-commutative 88.6%

      \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}} + -19.739208802181317 \cdot \left({u2}^{2} \cdot \sqrt{\frac{u1}{1 - u1}}\right)} \]

   2. *-lft-identity 88.6%

      \[\leadsto \color{blue}{1 \cdot \sqrt{\frac{u1}{1 - u1}}} + -19.739208802181317 \cdot \left({u2}^{2} \cdot \sqrt{\frac{u1}{1 - u1}}\right) \]

   3. associate-*r* 88.6%

      \[\leadsto 1 \cdot \sqrt{\frac{u1}{1 - u1}} + \color{blue}{\left(-19.739208802181317 \cdot {u2}^{2}\right) \cdot \sqrt{\frac{u1}{1 - u1}}} \]

   4. distribute-rgt-out 88.6%

      \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + -19.739208802181317 \cdot {u2}^{2}\right)} \]

   5. unpow2 88.6%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + -19.739208802181317 \cdot \color{blue}{\left(u2 \cdot u2\right)}\right) \]

10. Simplified 88.6%

    \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + -19.739208802181317 \cdot \left(u2 \cdot u2\right)\right)} \]

11. Final simplification 88.6%

    \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + -19.739208802181317 \cdot \left(u2 \cdot u2\right)\right) \]

Alternative 4: 71.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((u1 + (u1 * 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 81.3%

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

   1. add-sqr-sqrt 81.3%

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

   2. pow1/2 81.3%

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

   3. pow1/2 81.3%

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

   4. pow-prod-down 81.3%

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

   5. clear-num 81.1%

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

   6. clear-num 81.0%

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

   7. inv-pow 81.0%

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

   8. inv-pow 81.0%

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

   9. pow-prod-up 81.0%

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

   10. div-sub 81.0%

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

   11. inv-pow 81.0%

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

   12. pow1 81.0%

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

   13. pow1 81.0%

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

   14. pow-div 81.0%

       \[\leadsto \sqrt{{\left({\left({u1}^{-1} - \color{blue}{{u1}^{\left(1 - 1\right)}}\right)}^{\left(-1 + -1\right)}\right)}^{0.5}} \]

   15. metadata-eval 81.0%

       \[\leadsto \sqrt{{\left({\left({u1}^{-1} - {u1}^{\color{blue}{0}}\right)}^{\left(-1 + -1\right)}\right)}^{0.5}} \]

   16. metadata-eval 81.0%

       \[\leadsto \sqrt{{\left({\left({u1}^{-1} - \color{blue}{1}\right)}^{\left(-1 + -1\right)}\right)}^{0.5}} \]

   17. metadata-eval 81.0%

       \[\leadsto \sqrt{{\left({\left({u1}^{-1} - 1\right)}^{\color{blue}{-2}}\right)}^{0.5}} \]

4. Applied egg-rr 81.0%

   \[\leadsto \sqrt{\color{blue}{{\left({\left({u1}^{-1} - 1\right)}^{-2}\right)}^{0.5}}} \]
5. Step-by-step derivation

   1. exp-to-pow 79.5%

      \[\leadsto \sqrt{\color{blue}{e^{\log \left({\left({u1}^{-1} - 1\right)}^{-2}\right) \cdot 0.5}}} \]

   2. *-commutative 79.5%

      \[\leadsto \sqrt{e^{\color{blue}{0.5 \cdot \log \left({\left({u1}^{-1} - 1\right)}^{-2}\right)}}} \]

   3. log-pow 79.4%

      \[\leadsto \sqrt{e^{0.5 \cdot \color{blue}{\left(-2 \cdot \log \left({u1}^{-1} - 1\right)\right)}}} \]

   4. associate-*r* 79.4%

      \[\leadsto \sqrt{e^{\color{blue}{\left(0.5 \cdot -2\right) \cdot \log \left({u1}^{-1} - 1\right)}}} \]

   5. metadata-eval 79.4%

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

   6. *-commutative 79.4%

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

   7. exp-to-pow 81.0%

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

   8. unpow-1 81.0%

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

   9. unpow-1 81.0%

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

   10. sub-neg 81.0%

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

   11. metadata-eval 81.0%

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

   12. +-commutative 81.0%

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

6. Simplified 81.0%

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

7. Taylor expanded in u1 around 0 72.4%

   \[\leadsto \sqrt{\color{blue}{{u1}^{2} + u1}} \]
8. Step-by-step derivation

   1. +-commutative 72.4%

      \[\leadsto \sqrt{\color{blue}{u1 + {u1}^{2}}} \]

   2. unpow2 72.4%

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

9. Simplified 72.4%

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

10. Final simplification 72.4%

    \[\leadsto \sqrt{u1 + u1 \cdot u1} \]

Alternative 5: 80.1% 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 81.3%

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

3. Final simplification 81.3%

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

Alternative 6: 63.5% 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 81.3%

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

3. Taylor expanded in u1 around 0 62.9%

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

4. Final simplification 62.9%

   \[\leadsto \sqrt{u1} \]

Reproduce

herbie shell --seed 2023194 
(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))))