Trowbridge-Reitz Sample, near normal, slope_x

Percentage Accurate: 99.0% → 98.9%

Time: 15.8s

Alternatives: 7

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: 98.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ {\left(\frac{1}{u1} + -1\right)}^{-0.5} \cdot \cos \left(\sqrt{{u2}^{2} \cdot 39.47841760436263}\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (pow (+ (/ 1.0 u1) -1.0) -0.5)
  (cos (sqrt (* (pow u2 2.0) 39.47841760436263)))))

C

float code(float cosTheta_i, float u1, float u2) {
	return powf(((1.0f / u1) + -1.0f), -0.5f) * cosf(sqrtf((powf(u2, 2.0f) * 39.47841760436263f)));
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = (((1.0e0 / u1) + (-1.0e0)) ** (-0.5e0)) * cos(sqrt(((u2 ** 2.0e0) * 39.47841760436263e0)))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32((Float32(Float32(Float32(1.0) / u1) + Float32(-1.0)) ^ Float32(-0.5)) * cos(sqrt(Float32((u2 ^ Float32(2.0)) * Float32(39.47841760436263)))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = (((single(1.0) / u1) + single(-1.0)) ^ single(-0.5)) * cos(sqrt(((u2 ^ single(2.0)) * single(39.47841760436263))));
end

Derivation

1. Initial program 98.8%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
2. Add Preprocessing
3. 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. sqrt-unprod 98.8%

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

   3. *-commutative 98.8%

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

   4. *-commutative 98.8%

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

   5. swap-sqr 98.8%

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

   6. pow2 98.8%

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

   7. metadata-eval 98.8%

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

4. Applied egg-rr 98.8%

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

   1. sqrt-prod 98.8%

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

   2. metadata-eval 98.8%

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

   3. unpow2 98.8%

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

   4. sqrt-prod 98.7%

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

   5. add-sqr-sqrt 98.8%

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

   6. *-commutative 98.8%

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

   7. expm1-log1p-u 98.7%

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

   8. expm1-udef 80.5%

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

   9. *-commutative 80.5%

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

6. Applied egg-rr 80.5%

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

   1. expm1-def 98.7%

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

   2. expm1-log1p-u 98.8%

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

   3. clear-num 98.7%

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

   4. inv-pow 98.7%

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

   5. sqrt-pow1 98.9%

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

   6. div-sub 98.8%

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

   7. *-inverses 98.8%

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

   8. sub-neg 98.8%

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

   9. metadata-eval 98.8%

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

   10. metadata-eval 98.8%

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

   11. *-commutative 98.8%

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

8. Applied egg-rr 98.8%

   \[\leadsto \color{blue}{{\left(\frac{1}{u1} + -1\right)}^{-0.5} \cdot \cos \left(6.28318530718 \cdot u2\right)} \]
9. 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. sqrt-unprod 98.8%

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

   3. *-commutative 98.8%

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

   4. *-commutative 98.8%

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

   5. swap-sqr 98.8%

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

   6. pow2 98.8%

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

   7. metadata-eval 98.8%

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

10. Applied egg-rr 98.9%

    \[\leadsto {\left(\frac{1}{u1} + -1\right)}^{-0.5} \cdot \cos \color{blue}{\left(\sqrt{{u2}^{2} \cdot 39.47841760436263}\right)} \]

11. Final simplification 98.9%

    \[\leadsto {\left(\frac{1}{u1} + -1\right)}^{-0.5} \cdot \cos \left(\sqrt{{u2}^{2} \cdot 39.47841760436263}\right) \]
12. Add Preprocessing

Alternative 2: 90.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.011149999685585499:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{\frac{1}{\frac{1}{u1}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 6.28318530718) 0.011149999685585499)
   (sqrt (/ u1 (- 1.0 u1)))
   (* (cos (* u2 6.28318530718)) (sqrt (/ 1.0 (/ 1.0 u1))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * 6.28318530718f) <= 0.011149999685585499f) {
		tmp = sqrtf((u1 / (1.0f - u1)));
	} else {
		tmp = cosf((u2 * 6.28318530718f)) * sqrtf((1.0f / (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) :: tmp
    if ((u2 * 6.28318530718e0) <= 0.011149999685585499e0) then
        tmp = sqrt((u1 / (1.0e0 - u1)))
    else
        tmp = cos((u2 * 6.28318530718e0)) * sqrt((1.0e0 / (1.0e0 / 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(0.011149999685585499))
		tmp = sqrt(Float32(u1 / Float32(Float32(1.0) - u1)));
	else
		tmp = Float32(cos(Float32(u2 * Float32(6.28318530718))) * sqrt(Float32(Float32(1.0) / Float32(Float32(1.0) / u1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((u2 * single(6.28318530718)) <= single(0.011149999685585499))
		tmp = sqrt((u1 / (single(1.0) - u1)));
	else
		tmp = cos((u2 * single(6.28318530718))) * sqrt((single(1.0) / (single(1.0) / u1)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 314159265359/50000000000 u2) < 0.0111499997

   1. Initial program 99.4%

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

   3. Taylor expanded in u2 around 0 95.8%

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

   if 0.0111499997 < (*.f32 314159265359/50000000000 u2)

   1. Initial program 97.3%

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

      1. add-log-exp 57.2%

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

   4. Applied egg-rr 57.2%

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

      1. rem-log-exp 97.3%

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

      2. clear-num 97.3%

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

   6. Applied egg-rr 97.3%

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

   7. Taylor expanded in u1 around 0 72.3%

      \[\leadsto \sqrt{\frac{1}{\color{blue}{\frac{1}{u1}}}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.011149999685585499:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{\frac{1}{\frac{1}{u1}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.011149999685585499:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 6.28318530718) 0.011149999685585499)
   (sqrt (/ u1 (- 1.0 u1)))
   (* (cos (* u2 6.28318530718)) (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * 6.28318530718f) <= 0.011149999685585499f) {
		tmp = sqrtf((u1 / (1.0f - u1)));
	} else {
		tmp = cosf((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) <= 0.011149999685585499e0) then
        tmp = sqrt((u1 / (1.0e0 - u1)))
    else
        tmp = cos((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(0.011149999685585499))
		tmp = sqrt(Float32(u1 / Float32(Float32(1.0) - u1)));
	else
		tmp = Float32(cos(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(0.011149999685585499))
		tmp = sqrt((u1 / (single(1.0) - u1)));
	else
		tmp = cos((u2 * single(6.28318530718))) * sqrt(u1);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 314159265359/50000000000 u2) < 0.0111499997

   1. Initial program 99.4%

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

   3. Taylor expanded in u2 around 0 95.8%

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

   if 0.0111499997 < (*.f32 314159265359/50000000000 u2)

   1. Initial program 97.3%

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

   3. Taylor expanded in u1 around 0 72.3%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.011149999685585499:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return powf(((1.0f / u1) + -1.0f), -0.5f) * cosf((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 = (((1.0e0 / u1) + (-1.0e0)) ** (-0.5e0)) * cos((u2 * 6.28318530718e0))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.8%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
2. Add Preprocessing
3. 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. sqrt-unprod 98.8%

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

   3. *-commutative 98.8%

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

   4. *-commutative 98.8%

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

   5. swap-sqr 98.8%

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

   6. pow2 98.8%

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

   7. metadata-eval 98.8%

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

4. Applied egg-rr 98.8%

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

   1. sqrt-prod 98.8%

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

   2. metadata-eval 98.8%

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

   3. unpow2 98.8%

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

   4. sqrt-prod 98.7%

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

   5. add-sqr-sqrt 98.8%

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

   6. *-commutative 98.8%

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

   7. expm1-log1p-u 98.7%

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

   8. expm1-udef 80.5%

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

   9. *-commutative 80.5%

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

6. Applied egg-rr 80.5%

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

   1. expm1-def 98.7%

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

   2. expm1-log1p-u 98.8%

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

   3. clear-num 98.7%

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

   4. inv-pow 98.7%

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

   5. sqrt-pow1 98.9%

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

   6. div-sub 98.8%

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

   7. *-inverses 98.8%

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

   8. sub-neg 98.8%

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

   9. metadata-eval 98.8%

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

   10. metadata-eval 98.8%

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

   11. *-commutative 98.8%

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

8. Applied egg-rr 98.8%

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

9. Final simplification 98.8%

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

Alternative 5: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.8%

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

3. Final simplification 98.8%

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

Alternative 6: 80.3% 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 98.8%

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

3. Taylor expanded in u2 around 0 80.5%

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

4. Final simplification 80.5%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \]
5. Add Preprocessing

Alternative 7: 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 98.8%

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

3. Taylor expanded in u2 around 0 80.5%

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

4. Taylor expanded in u1 around 0 62.3%

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

5. Final simplification 62.3%

   \[\leadsto \sqrt{u1} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024017 
(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))))