Trowbridge-Reitz Sample, near normal, slope_y

Percentage Accurate: 98.3% → 98.4%

Time: 10.7s

Alternatives: 11

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - u1))) * sinf(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 = sqrt((u1 / (1.0e0 - u1))) * sin(sqrt(((u2 ** 2.0e0) * 39.47841760436263e0)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.2%

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

   1. add-sqr-sqrt 97.5%

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

   2. sqrt-unprod 98.2%

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

   3. *-commutative 98.2%

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

   4. *-commutative 98.2%

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

   5. swap-sqr 97.9%

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

   6. pow2 97.9%

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

   7. metadata-eval 98.4%

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

4. Applied egg-rr 98.4%

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

Alternative 2: 93.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.009999999776482582:\\ \;\;\;\;6.28318530718 \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1 \cdot \left(u1 + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 6.28318530718) 0.009999999776482582)
   (* 6.28318530718 (* (sqrt (/ u1 (- 1.0 u1))) u2))
   (* (sin (* u2 6.28318530718)) (sqrt (* u1 (+ u1 1.0))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * 6.28318530718f) <= 0.009999999776482582f) {
		tmp = 6.28318530718f * (sqrtf((u1 / (1.0f - u1))) * u2);
	} else {
		tmp = sinf((u2 * 6.28318530718f)) * 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 ((u2 * 6.28318530718e0) <= 0.009999999776482582e0) then
        tmp = 6.28318530718e0 * (sqrt((u1 / (1.0e0 - u1))) * u2)
    else
        tmp = sin((u2 * 6.28318530718e0)) * sqrt((u1 * (u1 + 1.0e0)))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(u2 * Float32(6.28318530718)) <= Float32(0.009999999776482582))
		tmp = Float32(Float32(6.28318530718) * Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * u2));
	else
		tmp = Float32(sin(Float32(u2 * Float32(6.28318530718))) * 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 ((u2 * single(6.28318530718)) <= single(0.009999999776482582))
		tmp = single(6.28318530718) * (sqrt((u1 / (single(1.0) - u1))) * u2);
	else
		tmp = sin((u2 * single(6.28318530718))) * 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) < 0.00999999978

   1. Initial program 98.4%

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

   3. Taylor expanded in u2 around 0 96.7%

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

   if 0.00999999978 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 97.9%

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

   3. Taylor expanded in u1 around 0 85.1%

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

      1. +-commutative 85.1%

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

   5. Simplified 85.1%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.009999999776482582:\\ \;\;\;\;6.28318530718 \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1 \cdot \left(u1 + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.009999999776482582:\\ \;\;\;\;6.28318530718 \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot 6.28318530718\right) \cdot {\left(\frac{1}{u1}\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 6.28318530718) 0.009999999776482582)
   (* 6.28318530718 (* (sqrt (/ u1 (- 1.0 u1))) u2))
   (* (sin (* u2 6.28318530718)) (pow (/ 1.0 u1) -0.5))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * 6.28318530718f) <= 0.009999999776482582f) {
		tmp = 6.28318530718f * (sqrtf((u1 / (1.0f - u1))) * u2);
	} else {
		tmp = sinf((u2 * 6.28318530718f)) * powf((1.0f / u1), -0.5f);
	}
	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) <= 0.009999999776482582e0) then
        tmp = 6.28318530718e0 * (sqrt((u1 / (1.0e0 - u1))) * u2)
    else
        tmp = sin((u2 * 6.28318530718e0)) * ((1.0e0 / u1) ** (-0.5e0))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(u2 * Float32(6.28318530718)) <= Float32(0.009999999776482582))
		tmp = Float32(Float32(6.28318530718) * Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * u2));
	else
		tmp = Float32(sin(Float32(u2 * Float32(6.28318530718))) * (Float32(Float32(1.0) / u1) ^ Float32(-0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((u2 * single(6.28318530718)) <= single(0.009999999776482582))
		tmp = single(6.28318530718) * (sqrt((u1 / (single(1.0) - u1))) * u2);
	else
		tmp = sin((u2 * single(6.28318530718))) * ((single(1.0) / u1) ^ single(-0.5));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.00999999978

   1. Initial program 98.4%

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

   3. Taylor expanded in u2 around 0 96.7%

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

   if 0.00999999978 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 97.9%

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

      1. clear-num 97.8%

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

      2. sqrt-div 97.8%

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

      3. metadata-eval 97.8%

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

   4. Applied egg-rr 97.8%

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

      1. *-un-lft-identity 97.8%

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

      2. inv-pow 97.8%

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

      3. sqrt-pow2 97.9%

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

      4. div-sub 97.8%

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

      5. *-inverses 97.8%

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

      6. sub-neg 97.8%

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

      7. metadata-eval 97.8%

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

      8. metadata-eval 97.8%

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

   6. Applied egg-rr 97.8%

      \[\leadsto \color{blue}{\left(1 \cdot {\left(\frac{1}{u1} + -1\right)}^{-0.5}\right)} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
   7. Step-by-step derivation

      1. *-lft-identity 97.8%

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

   8. Simplified 97.8%

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

   9. Taylor expanded in u1 around 0 73.2%

      \[\leadsto {\color{blue}{\left(\frac{1}{u1}\right)}}^{-0.5} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.009999999776482582:\\ \;\;\;\;6.28318530718 \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot 6.28318530718\right) \cdot {\left(\frac{1}{u1}\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.04399999976158142:\\ \;\;\;\;6.28318530718 \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 6.28318530718) 0.04399999976158142)
   (* 6.28318530718 (* (sqrt (/ u1 (- 1.0 u1))) u2))
   (* (sin (* u2 6.28318530718)) (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * 6.28318530718f) <= 0.04399999976158142f) {
		tmp = 6.28318530718f * (sqrtf((u1 / (1.0f - u1))) * u2);
	} else {
		tmp = sinf((u2 * 6.28318530718f)) * sqrtf(u1);
	}
	return tmp;
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    real(4) :: tmp
    if ((u2 * 6.28318530718e0) <= 0.04399999976158142e0) then
        tmp = 6.28318530718e0 * (sqrt((u1 / (1.0e0 - u1))) * u2)
    else
        tmp = sin((u2 * 6.28318530718e0)) * sqrt(u1)
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(u2 * Float32(6.28318530718)) <= Float32(0.04399999976158142))
		tmp = Float32(Float32(6.28318530718) * Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * u2));
	else
		tmp = Float32(sin(Float32(u2 * Float32(6.28318530718))) * sqrt(u1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((u2 * single(6.28318530718)) <= single(0.04399999976158142))
		tmp = single(6.28318530718) * (sqrt((u1 / (single(1.0) - u1))) * u2);
	else
		tmp = sin((u2 * single(6.28318530718))) * sqrt(u1);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.0439999998

   1. Initial program 98.4%

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

   3. Taylor expanded in u2 around 0 94.0%

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

   if 0.0439999998 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 97.6%

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

   3. Taylor expanded in u1 around 0 74.3%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.04399999976158142:\\ \;\;\;\;6.28318530718 \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.2%

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

   1. clear-num 98.2%

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

   2. sqrt-div 98.1%

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

   3. metadata-eval 98.1%

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

4. Applied egg-rr 98.1%

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

   1. *-un-lft-identity 98.1%

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

   2. inv-pow 98.1%

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

   3. sqrt-pow2 98.2%

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

   4. div-sub 98.1%

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

   5. *-inverses 98.1%

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

   6. sub-neg 98.1%

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

   7. metadata-eval 98.1%

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

   8. metadata-eval 98.1%

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

6. Applied egg-rr 98.1%

   \[\leadsto \color{blue}{\left(1 \cdot {\left(\frac{1}{u1} + -1\right)}^{-0.5}\right)} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
7. Step-by-step derivation

   1. *-lft-identity 98.1%

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

8. Simplified 98.1%

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

9. Taylor expanded in u1 around 0 98.2%

   \[\leadsto {\color{blue}{\left(\frac{1 + -1 \cdot u1}{u1}\right)}}^{-0.5} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
10. Step-by-step derivation

    1. mul-1-neg 98.2%

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

    2. sub-neg 98.2%

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

11. Simplified 98.2%

    \[\leadsto {\color{blue}{\left(\frac{1 - u1}{u1}\right)}}^{-0.5} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
12. Step-by-step derivation

    1. clear-num 98.2%

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

    2. inv-pow 98.2%

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

13. Applied egg-rr 98.2%

    \[\leadsto {\color{blue}{\left({\left(\frac{u1}{1 - u1}\right)}^{-1}\right)}}^{-0.5} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
14. Step-by-step derivation

    1. unpow-1 98.2%

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

15. Simplified 98.2%

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

16. Final simplification 98.2%

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

Alternative 6: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.2%

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

3. Final simplification 98.2%

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

Alternative 7: 81.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.2%

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

3. Taylor expanded in u2 around 0 80.5%

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

Alternative 8: 73.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.2%

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

3. Taylor expanded in u2 around 0 80.5%

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

4. Taylor expanded in u1 around 0 71.5%

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

   1. +-commutative 85.2%

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

6. Simplified 71.5%

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

7. Final simplification 71.5%

   \[\leadsto 6.28318530718 \cdot \left(u2 \cdot \sqrt{u1 \cdot \left(u1 + 1\right)}\right) \]
8. Add Preprocessing

Alternative 9: 64.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.2%

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

3. Taylor expanded in u2 around 0 80.5%

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

4. Taylor expanded in u1 around 0 63.7%

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

5. Final simplification 63.7%

   \[\leadsto 6.28318530718 \cdot \left(u2 \cdot \sqrt{u1}\right) \]
6. Add Preprocessing

Alternative 10: 20.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.2%

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

   1. add-cbrt-cube 98.2%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\sqrt{\frac{u1}{1 - u1}} \cdot \sqrt{\frac{u1}{1 - u1}}\right) \cdot \sqrt{\frac{u1}{1 - u1}}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   2. pow1/3 95.9%

      \[\leadsto \color{blue}{{\left(\left(\sqrt{\frac{u1}{1 - u1}} \cdot \sqrt{\frac{u1}{1 - u1}}\right) \cdot \sqrt{\frac{u1}{1 - u1}}\right)}^{0.3333333333333333}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   3. add-sqr-sqrt 95.9%

      \[\leadsto {\left(\color{blue}{\frac{u1}{1 - u1}} \cdot \sqrt{\frac{u1}{1 - u1}}\right)}^{0.3333333333333333} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   4. pow1 95.9%

      \[\leadsto {\left(\color{blue}{{\left(\frac{u1}{1 - u1}\right)}^{1}} \cdot \sqrt{\frac{u1}{1 - u1}}\right)}^{0.3333333333333333} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   5. pow1/2 95.9%

      \[\leadsto {\left({\left(\frac{u1}{1 - u1}\right)}^{1} \cdot \color{blue}{{\left(\frac{u1}{1 - u1}\right)}^{0.5}}\right)}^{0.3333333333333333} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   6. pow-prod-up 95.9%

      \[\leadsto {\color{blue}{\left({\left(\frac{u1}{1 - u1}\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   7. metadata-eval 95.9%

      \[\leadsto {\left({\left(\frac{u1}{1 - u1}\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

4. Applied egg-rr 95.9%

   \[\leadsto \color{blue}{{\left({\left(\frac{u1}{1 - u1}\right)}^{1.5}\right)}^{0.3333333333333333}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
5. Step-by-step derivation

   1. unpow1/3 98.2%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(\frac{u1}{1 - u1}\right)}^{1.5}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

6. Simplified 98.2%

   \[\leadsto \color{blue}{\sqrt[3]{{\left(\frac{u1}{1 - u1}\right)}^{1.5}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
7. Step-by-step derivation

   1. pow1/3 95.9%

      \[\leadsto \color{blue}{{\left({\left(\frac{u1}{1 - u1}\right)}^{1.5}\right)}^{0.3333333333333333}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   2. pow-pow 98.2%

      \[\leadsto \color{blue}{{\left(\frac{u1}{1 - u1}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   3. metadata-eval 98.2%

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

   4. pow1/2 98.2%

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

   5. clear-num 98.2%

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

   6. sqrt-div 98.1%

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

   7. metadata-eval 98.1%

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

   8. add-sqr-sqrt 97.3%

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

   9. associate-/r* 97.5%

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

   10. metadata-eval 97.5%

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

   11. sqrt-div 97.8%

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

   12. sqrt-div 97.8%

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

   13. clear-num 97.8%

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

   14. sqrt-div 97.9%

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

   15. pow1/2 97.9%

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

   16. sqrt-pow1 97.9%

       \[\leadsto \frac{\color{blue}{{\left(\frac{u1}{1 - u1}\right)}^{\left(\frac{0.5}{2}\right)}}}{\sqrt{\sqrt{\frac{1 - u1}{u1}}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   17. metadata-eval 97.9%

       \[\leadsto \frac{{\left(\frac{u1}{1 - u1}\right)}^{\color{blue}{0.25}}}{\sqrt{\sqrt{\frac{1 - u1}{u1}}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   18. pow1/2 97.9%

       \[\leadsto \frac{{\left(\frac{u1}{1 - u1}\right)}^{0.25}}{\sqrt{\color{blue}{{\left(\frac{1 - u1}{u1}\right)}^{0.5}}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   19. sqrt-pow1 97.9%

       \[\leadsto \frac{{\left(\frac{u1}{1 - u1}\right)}^{0.25}}{\color{blue}{{\left(\frac{1 - u1}{u1}\right)}^{\left(\frac{0.5}{2}\right)}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

8. Applied egg-rr 98.0%

   \[\leadsto \color{blue}{\frac{{\left(\frac{u1}{1 - u1}\right)}^{0.25}}{{\left(\frac{1}{u1} + -1\right)}^{0.25}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

9. Taylor expanded in u1 around inf 20.2%

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

10. Final simplification 20.2%

    \[\leadsto \sin \left(u2 \cdot 6.28318530718\right) \]
11. Add Preprocessing

Alternative 11: 19.5% accurate, 69.7× speedup

Math

\[\begin{array}{l} \\ u2 \cdot 6.28318530718 \end{array} \]

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return 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 = u2 * 6.28318530718e0
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.2%

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

   1. add-cbrt-cube 98.2%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\sqrt{\frac{u1}{1 - u1}} \cdot \sqrt{\frac{u1}{1 - u1}}\right) \cdot \sqrt{\frac{u1}{1 - u1}}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   2. pow1/3 95.9%

      \[\leadsto \color{blue}{{\left(\left(\sqrt{\frac{u1}{1 - u1}} \cdot \sqrt{\frac{u1}{1 - u1}}\right) \cdot \sqrt{\frac{u1}{1 - u1}}\right)}^{0.3333333333333333}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   3. add-sqr-sqrt 95.9%

      \[\leadsto {\left(\color{blue}{\frac{u1}{1 - u1}} \cdot \sqrt{\frac{u1}{1 - u1}}\right)}^{0.3333333333333333} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   4. pow1 95.9%

      \[\leadsto {\left(\color{blue}{{\left(\frac{u1}{1 - u1}\right)}^{1}} \cdot \sqrt{\frac{u1}{1 - u1}}\right)}^{0.3333333333333333} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   5. pow1/2 95.9%

      \[\leadsto {\left({\left(\frac{u1}{1 - u1}\right)}^{1} \cdot \color{blue}{{\left(\frac{u1}{1 - u1}\right)}^{0.5}}\right)}^{0.3333333333333333} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   6. pow-prod-up 95.9%

      \[\leadsto {\color{blue}{\left({\left(\frac{u1}{1 - u1}\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   7. metadata-eval 95.9%

      \[\leadsto {\left({\left(\frac{u1}{1 - u1}\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

4. Applied egg-rr 95.9%

   \[\leadsto \color{blue}{{\left({\left(\frac{u1}{1 - u1}\right)}^{1.5}\right)}^{0.3333333333333333}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
5. Step-by-step derivation

   1. unpow1/3 98.2%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(\frac{u1}{1 - u1}\right)}^{1.5}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

6. Simplified 98.2%

   \[\leadsto \color{blue}{\sqrt[3]{{\left(\frac{u1}{1 - u1}\right)}^{1.5}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
7. Step-by-step derivation

   1. pow1/3 95.9%

      \[\leadsto \color{blue}{{\left({\left(\frac{u1}{1 - u1}\right)}^{1.5}\right)}^{0.3333333333333333}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   2. pow-pow 98.2%

      \[\leadsto \color{blue}{{\left(\frac{u1}{1 - u1}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   3. metadata-eval 98.2%

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

   4. pow1/2 98.2%

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

   5. clear-num 98.2%

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

   6. sqrt-div 98.1%

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

   7. metadata-eval 98.1%

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

   8. add-sqr-sqrt 97.3%

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

   9. associate-/r* 97.5%

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

   10. metadata-eval 97.5%

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

   11. sqrt-div 97.8%

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

   12. sqrt-div 97.8%

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

   13. clear-num 97.8%

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

   14. sqrt-div 97.9%

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

   15. pow1/2 97.9%

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

   16. sqrt-pow1 97.9%

       \[\leadsto \frac{\color{blue}{{\left(\frac{u1}{1 - u1}\right)}^{\left(\frac{0.5}{2}\right)}}}{\sqrt{\sqrt{\frac{1 - u1}{u1}}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   17. metadata-eval 97.9%

       \[\leadsto \frac{{\left(\frac{u1}{1 - u1}\right)}^{\color{blue}{0.25}}}{\sqrt{\sqrt{\frac{1 - u1}{u1}}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   18. pow1/2 97.9%

       \[\leadsto \frac{{\left(\frac{u1}{1 - u1}\right)}^{0.25}}{\sqrt{\color{blue}{{\left(\frac{1 - u1}{u1}\right)}^{0.5}}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   19. sqrt-pow1 97.9%

       \[\leadsto \frac{{\left(\frac{u1}{1 - u1}\right)}^{0.25}}{\color{blue}{{\left(\frac{1 - u1}{u1}\right)}^{\left(\frac{0.5}{2}\right)}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

8. Applied egg-rr 98.0%

   \[\leadsto \color{blue}{\frac{{\left(\frac{u1}{1 - u1}\right)}^{0.25}}{{\left(\frac{1}{u1} + -1\right)}^{0.25}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

9. Taylor expanded in u1 around inf 20.2%

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

10. Taylor expanded in u2 around 0 19.5%

    \[\leadsto \color{blue}{6.28318530718 \cdot u2} \]

11. Final simplification 19.5%

    \[\leadsto u2 \cdot 6.28318530718 \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024163 
(FPCore (cosTheta_i u1 u2)
  :name "Trowbridge-Reitz Sample, near normal, slope_y"
  :precision binary32
  :pre (and (and (and (> cosTheta_i 0.9999) (<= cosTheta_i 1.0)) (and (<= 2.328306437e-10 u1) (<= u1 1.0))) (and (<= 2.328306437e-10 u2) (<= u2 1.0)))
  (* (sqrt (/ u1 (- 1.0 u1))) (sin (* 6.28318530718 u2))))