Trowbridge-Reitz Sample, near normal, slope_x

Percentage Accurate: 99.0% → 99.0%

Time: 10.2s

Alternatives: 10

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, 0.7× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - u1))) * cosf((6.28318530718f * (powf(u2, 2.0f) / 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 ** 2.0e0) / 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) * Float32((u2 ^ Float32(2.0)) / u2))))
end

MATLAB

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

Derivation

1. Initial program 99.2%

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

   1. expm1-log1p-u 99.2%

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

   2. expm1-undefine 99.2%

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

4. Applied egg-rr 99.2%

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

   1. expm1-define 99.2%

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

6. Simplified 99.2%

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

   1. expm1-undefine 99.2%

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

   2. log1p-undefine 99.2%

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

   3. rem-exp-log 99.2%

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

   4. +-commutative 99.2%

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

8. Applied egg-rr 99.2%

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

   1. associate--l+ 99.2%

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

   2. metadata-eval 99.2%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2 + \color{blue}{0}\right) \]

   3. flip-+ 99.2%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \color{blue}{\left(\frac{\left(6.28318530718 \cdot u2\right) \cdot \left(6.28318530718 \cdot u2\right) - 0 \cdot 0}{6.28318530718 \cdot u2 - 0}\right)} \]

   4. *-commutative 99.2%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(\frac{\color{blue}{\left(u2 \cdot 6.28318530718\right)} \cdot \left(6.28318530718 \cdot u2\right) - 0 \cdot 0}{6.28318530718 \cdot u2 - 0}\right) \]

   5. *-commutative 99.2%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(\frac{\left(u2 \cdot 6.28318530718\right) \cdot \color{blue}{\left(u2 \cdot 6.28318530718\right)} - 0 \cdot 0}{6.28318530718 \cdot u2 - 0}\right) \]

   6. swap-sqr 99.1%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(\frac{\color{blue}{\left(u2 \cdot u2\right) \cdot \left(6.28318530718 \cdot 6.28318530718\right)} - 0 \cdot 0}{6.28318530718 \cdot u2 - 0}\right) \]

   7. pow2 99.1%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(\frac{\color{blue}{{u2}^{2}} \cdot \left(6.28318530718 \cdot 6.28318530718\right) - 0 \cdot 0}{6.28318530718 \cdot u2 - 0}\right) \]

   8. metadata-eval 99.2%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(\frac{{u2}^{2} \cdot \color{blue}{39.47841760436263} - 0 \cdot 0}{6.28318530718 \cdot u2 - 0}\right) \]

   9. metadata-eval 99.2%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(\frac{{u2}^{2} \cdot 39.47841760436263 - \color{blue}{0}}{6.28318530718 \cdot u2 - 0}\right) \]

10. Applied egg-rr 99.2%

    \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \color{blue}{\left(\frac{{u2}^{2} \cdot 39.47841760436263 - 0}{6.28318530718 \cdot u2 - 0}\right)} \]
11. Step-by-step derivation

    1. --rgt-identity 99.2%

       \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(\frac{\color{blue}{{u2}^{2} \cdot 39.47841760436263}}{6.28318530718 \cdot u2 - 0}\right) \]

    2. *-commutative 99.2%

       \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(\frac{\color{blue}{39.47841760436263 \cdot {u2}^{2}}}{6.28318530718 \cdot u2 - 0}\right) \]

    3. --rgt-identity 99.2%

       \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(\frac{39.47841760436263 \cdot {u2}^{2}}{\color{blue}{6.28318530718 \cdot u2}}\right) \]

    4. times-frac 99.1%

       \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \color{blue}{\left(\frac{39.47841760436263}{6.28318530718} \cdot \frac{{u2}^{2}}{u2}\right)} \]

    5. metadata-eval 99.2%

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

12. Simplified 99.2%

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

Alternative 2: 99.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.2%

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

   1. expm1-log1p-u 99.2%

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

   2. expm1-undefine 99.2%

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

4. Applied egg-rr 99.2%

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

   1. expm1-define 99.2%

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

6. Simplified 99.2%

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

   1. expm1-undefine 99.2%

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

   2. log1p-undefine 99.2%

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

   3. rem-exp-log 99.2%

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

   4. +-commutative 99.2%

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

8. Applied egg-rr 99.2%

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

   1. +-commutative 99.2%

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

   2. associate--l+ 99.2%

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

   3. fma-neg 99.2%

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

   4. metadata-eval 99.2%

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

10. Applied egg-rr 99.2%

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

Alternative 3: 93.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.000859999970998615:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1 \cdot \left(u1 + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.000859999970998615)
   (sqrt (/ u1 (- 1.0 u1)))
   (* (cos (* 6.28318530718 u2)) (sqrt (* u1 (+ u1 1.0))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.000859999970998615f) {
		tmp = sqrtf((u1 / (1.0f - u1)));
	} else {
		tmp = cosf((6.28318530718f * u2)) * sqrtf((u1 * (u1 + 1.0f)));
	}
	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) <= 0.000859999970998615e0) then
        tmp = sqrt((u1 / (1.0e0 - u1)))
    else
        tmp = cos((6.28318530718e0 * u2)) * sqrt((u1 * (u1 + 1.0e0)))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(6.28318530718) * u2) <= Float32(0.000859999970998615))
		tmp = sqrt(Float32(u1 / Float32(Float32(1.0) - u1)));
	else
		tmp = Float32(cos(Float32(Float32(6.28318530718) * u2)) * sqrt(Float32(u1 * Float32(u1 + Float32(1.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((single(6.28318530718) * u2) <= single(0.000859999970998615))
		tmp = sqrt((u1 / (single(1.0) - u1)));
	else
		tmp = cos((single(6.28318530718) * u2)) * sqrt((u1 * (u1 + single(1.0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 8.59999971e-4

   1. Initial program 99.6%

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

   3. Taylor expanded in u2 around 0 99.4%

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

   if 8.59999971e-4 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 98.5%

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

   3. Taylor expanded in u1 around 0 87.2%

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

      1. +-commutative 87.2%

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

   5. Simplified 87.2%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.000859999970998615:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1 \cdot \left(u1 + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.004000000189989805:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.004000000189989805)
   (sqrt (/ u1 (- 1.0 u1)))
   (* (cos (* 6.28318530718 u2)) (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.004000000189989805f) {
		tmp = sqrtf((u1 / (1.0f - u1)));
	} 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) <= 0.004000000189989805e0) then
        tmp = sqrt((u1 / (1.0e0 - u1)))
    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(0.004000000189989805))
		tmp = sqrt(Float32(u1 / Float32(Float32(1.0) - u1)));
	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(0.004000000189989805))
		tmp = sqrt((u1 / (single(1.0) - u1)));
	else
		tmp = cos((single(6.28318530718) * u2)) * sqrt(u1);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.00400000019

   1. Initial program 99.6%

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

   3. Taylor expanded in u2 around 0 98.7%

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

   if 0.00400000019 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 98.4%

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

   3. Taylor expanded in u1 around 0 74.2%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.004000000189989805:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 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 99.2%

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

Alternative 6: 80.4% 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 99.2%

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

3. Taylor expanded in u2 around 0 83.1%

   \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]
4. Add Preprocessing

Alternative 7: 72.0% 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 99.2%

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

3. Taylor expanded in u2 around 0 83.1%

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

4. Taylor expanded in u1 around 0 74.8%

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

   1. +-commutative 87.7%

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

6. Simplified 74.8%

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

Alternative 8: 63.7% 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 99.2%

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

3. Taylor expanded in u2 around 0 83.1%

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

4. Taylor expanded in u1 around 0 65.7%

   \[\leadsto \sqrt{\color{blue}{u1}} \]
5. Add Preprocessing

Alternative 9: 20.3% 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 99.2%

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

3. Taylor expanded in u2 around 0 83.1%

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

4. Taylor expanded in u1 around 0 74.8%

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

   1. +-commutative 74.8%

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

   2. *-commutative 74.8%

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

   3. distribute-lft1-in 74.9%

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

   4. fma-define 74.9%

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

6. Simplified 74.9%

   \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(u1, u1, u1\right)}} \]
7. Step-by-step derivation

   1. expm1-log1p-u 74.9%

      \[\leadsto \sqrt{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(u1, u1, u1\right)\right)\right)}} \]

   2. expm1-undefine 42.0%

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

   3. log1p-undefine 42.0%

      \[\leadsto \sqrt{e^{\color{blue}{\log \left(1 + \mathsf{fma}\left(u1, u1, u1\right)\right)}} - 1} \]

   4. add-exp-log 42.0%

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

8. Applied egg-rr 42.0%

   \[\leadsto \sqrt{\color{blue}{\left(1 + \mathsf{fma}\left(u1, u1, u1\right)\right) - 1}} \]
9. Step-by-step derivation

   1. associate--l+ 44.2%

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

10. Simplified 44.2%

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

11. Taylor expanded in u1 around inf 20.7%

    \[\leadsto \color{blue}{u1 \cdot \left(1 + 0.5 \cdot \frac{1}{u1}\right)} \]
12. Step-by-step derivation

    1. distribute-rgt-in 20.7%

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

    2. *-lft-identity 20.7%

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

    3. associate-*l* 20.7%

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

    4. lft-mult-inverse 20.7%

       \[\leadsto u1 + 0.5 \cdot \color{blue}{1} \]

    5. metadata-eval 20.7%

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

13. Simplified 20.7%

    \[\leadsto \color{blue}{u1 + 0.5} \]
14. Add Preprocessing

Alternative 10: 19.2% accurate, 209.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.2%

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

3. Taylor expanded in u2 around 0 83.1%

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

4. Taylor expanded in u1 around 0 74.8%

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

   1. +-commutative 74.8%

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

   2. *-commutative 74.8%

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

   3. distribute-lft1-in 74.9%

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

   4. fma-define 74.9%

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

6. Simplified 74.9%

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

7. Taylor expanded in u1 around inf 19.5%

   \[\leadsto \color{blue}{u1} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024117 
(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))))