Trowbridge-Reitz Sample, near normal, slope_y

Percentage Accurate: 98.3% → 98.3%

Time: 12.4s

Alternatives: 13

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: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sin((u2 * single(6.28318530718))) / sqrt(((single(1.0) / 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. Step-by-step derivation

   1. add-sqr-sqrt 97.8%

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

   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. swap-sqr 98.0%

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

   4. metadata-eval 98.4%

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

3. Applied egg-rr 98.4%

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

   1. clear-num 98.3%

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

   2. sqrt-div 98.3%

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

   3. metadata-eval 98.3%

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

5. Applied egg-rr 98.3%

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

   1. associate-*l/ 98.4%

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

   2. *-un-lft-identity 98.4%

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

   3. add-sqr-sqrt 95.8%

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

   4. add-sqr-sqrt 98.4%

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

   5. *-commutative 98.4%

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

   6. metadata-eval 98.2%

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

   7. swap-sqr 98.4%

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

   8. sqrt-unprod 97.6%

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

   9. add-sqr-sqrt 98.4%

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

   10. div-sub 98.4%

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

   11. *-inverses 98.4%

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

   12. sub-neg 98.4%

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

   13. metadata-eval 98.4%

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

7. Applied egg-rr 98.4%

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

8. Final simplification 98.4%

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

Alternative 2: 93.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.00812000036239624:\\ \;\;\;\;\sqrt{39.47841760436263 \cdot \frac{u2}{\frac{\frac{1}{u1} + -1}{u2}}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1 + u1 \cdot u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 6.28318530718) 0.00812000036239624)
   (sqrt (* 39.47841760436263 (/ u2 (/ (+ (/ 1.0 u1) -1.0) u2))))
   (* (sin (* u2 6.28318530718)) (sqrt (+ u1 (* u1 u1))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * 6.28318530718f) <= 0.00812000036239624f) {
		tmp = sqrtf((39.47841760436263f * (u2 / (((1.0f / u1) + -1.0f) / u2))));
	} else {
		tmp = sinf((u2 * 6.28318530718f)) * sqrtf((u1 + (u1 * 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) <= 0.00812000036239624e0) then
        tmp = sqrt((39.47841760436263e0 * (u2 / (((1.0e0 / u1) + (-1.0e0)) / u2))))
    else
        tmp = sin((u2 * 6.28318530718e0)) * sqrt((u1 + (u1 * 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(0.00812000036239624))
		tmp = sqrt(Float32(Float32(39.47841760436263) * Float32(u2 / Float32(Float32(Float32(Float32(1.0) / u1) + Float32(-1.0)) / u2))));
	else
		tmp = Float32(sin(Float32(u2 * Float32(6.28318530718))) * sqrt(Float32(u1 + Float32(u1 * u1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((u2 * single(6.28318530718)) <= single(0.00812000036239624))
		tmp = sqrt((single(39.47841760436263) * (u2 / (((single(1.0) / u1) + single(-1.0)) / u2))));
	else
		tmp = sin((u2 * single(6.28318530718))) * sqrt((u1 + (u1 * u1)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 314159265359/50000000000 u2) < 0.0081200004

   1. Initial program 98.4%

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

   2. Taylor expanded in u2 around 0 97.1%

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

      1. clear-num 98.5%

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

      2. sqrt-div 98.4%

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

      3. metadata-eval 98.4%

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

   4. Applied egg-rr 97.0%

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

      1. add-sqr-sqrt 96.6%

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

      2. sqrt-unprod 97.0%

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

      3. swap-sqr 97.0%

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

      4. metadata-eval 97.1%

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

      5. un-div-inv 97.4%

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

      6. un-div-inv 97.4%

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

      7. frac-times 97.3%

         \[\leadsto \sqrt{39.47841760436263 \cdot \color{blue}{\frac{u2 \cdot u2}{\sqrt{\frac{1 - u1}{u1}} \cdot \sqrt{\frac{1 - u1}{u1}}}}} \]

      8. add-sqr-sqrt 97.7%

         \[\leadsto \sqrt{39.47841760436263 \cdot \frac{u2 \cdot u2}{\color{blue}{\frac{1 - u1}{u1}}}} \]

      9. div-sub 97.7%

         \[\leadsto \sqrt{39.47841760436263 \cdot \frac{u2 \cdot u2}{\color{blue}{\frac{1}{u1} - \frac{u1}{u1}}}} \]

      10. *-inverses 97.7%

          \[\leadsto \sqrt{39.47841760436263 \cdot \frac{u2 \cdot u2}{\frac{1}{u1} - \color{blue}{1}}} \]

      11. sub-neg 97.7%

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

      12. metadata-eval 97.7%

          \[\leadsto \sqrt{39.47841760436263 \cdot \frac{u2 \cdot u2}{\frac{1}{u1} + \color{blue}{-1}}} \]

   6. Applied egg-rr 97.7%

      \[\leadsto \color{blue}{\sqrt{39.47841760436263 \cdot \frac{u2 \cdot u2}{\frac{1}{u1} + -1}}} \]
   7. Step-by-step derivation

      1. associate-/l* 97.8%

         \[\leadsto \sqrt{39.47841760436263 \cdot \color{blue}{\frac{u2}{\frac{\frac{1}{u1} + -1}{u2}}}} \]

      2. +-commutative 97.8%

         \[\leadsto \sqrt{39.47841760436263 \cdot \frac{u2}{\frac{\color{blue}{-1 + \frac{1}{u1}}}{u2}}} \]

   8. Simplified 97.8%

      \[\leadsto \color{blue}{\sqrt{39.47841760436263 \cdot \frac{u2}{\frac{-1 + \frac{1}{u1}}{u2}}}} \]

   if 0.0081200004 < (*.f32 314159265359/50000000000 u2)

   1. Initial program 97.9%

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

      1. flip-- 48.9%

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

      2. associate-/r/ 48.9%

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

      3. metadata-eval 48.9%

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

      4. +-commutative 48.9%

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

   3. Applied egg-rr 98.0%

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

   4. Taylor expanded in u1 around 0 82.9%

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

      1. +-commutative 46.1%

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

      2. unpow2 46.1%

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

   6. Simplified 82.9%

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

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.00812000036239624:\\ \;\;\;\;\sqrt{39.47841760436263 \cdot \frac{u2}{\frac{\frac{1}{u1} + -1}{u2}}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1 + u1 \cdot u1}\\ \end{array} \]

Alternative 3: 90.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.008500000461935997:\\ \;\;\;\;\sqrt{39.47841760436263 \cdot \frac{u2}{\frac{\frac{1}{u1} + -1}{u2}}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 6.28318530718) 0.008500000461935997)
   (sqrt (* 39.47841760436263 (/ u2 (/ (+ (/ 1.0 u1) -1.0) u2))))
   (* (sin (* u2 6.28318530718)) (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * 6.28318530718f) <= 0.008500000461935997f) {
		tmp = sqrtf((39.47841760436263f * (u2 / (((1.0f / u1) + -1.0f) / 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) <= 0.008500000461935997e0) then
        tmp = sqrt((39.47841760436263e0 * (u2 / (((1.0e0 / u1) + (-1.0e0)) / 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(0.008500000461935997))
		tmp = sqrt(Float32(Float32(39.47841760436263) * Float32(u2 / Float32(Float32(Float32(Float32(1.0) / u1) + Float32(-1.0)) / 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(0.008500000461935997))
		tmp = sqrt((single(39.47841760436263) * (u2 / (((single(1.0) / u1) + single(-1.0)) / 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 314159265359/50000000000 u2) < 0.00850000046

   1. Initial program 98.4%

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

   2. Taylor expanded in u2 around 0 97.0%

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

      1. clear-num 98.5%

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

      2. sqrt-div 98.4%

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

      3. metadata-eval 98.4%

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

   4. Applied egg-rr 96.9%

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

      1. add-sqr-sqrt 96.5%

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

      2. sqrt-unprod 96.9%

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

      3. swap-sqr 96.9%

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

      4. metadata-eval 97.0%

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

      5. un-div-inv 97.3%

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

      6. un-div-inv 97.3%

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

      7. frac-times 97.2%

         \[\leadsto \sqrt{39.47841760436263 \cdot \color{blue}{\frac{u2 \cdot u2}{\sqrt{\frac{1 - u1}{u1}} \cdot \sqrt{\frac{1 - u1}{u1}}}}} \]

      8. add-sqr-sqrt 97.6%

         \[\leadsto \sqrt{39.47841760436263 \cdot \frac{u2 \cdot u2}{\color{blue}{\frac{1 - u1}{u1}}}} \]

      9. div-sub 97.6%

         \[\leadsto \sqrt{39.47841760436263 \cdot \frac{u2 \cdot u2}{\color{blue}{\frac{1}{u1} - \frac{u1}{u1}}}} \]

      10. *-inverses 97.6%

          \[\leadsto \sqrt{39.47841760436263 \cdot \frac{u2 \cdot u2}{\frac{1}{u1} - \color{blue}{1}}} \]

      11. sub-neg 97.6%

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

      12. metadata-eval 97.6%

          \[\leadsto \sqrt{39.47841760436263 \cdot \frac{u2 \cdot u2}{\frac{1}{u1} + \color{blue}{-1}}} \]

   6. Applied egg-rr 97.6%

      \[\leadsto \color{blue}{\sqrt{39.47841760436263 \cdot \frac{u2 \cdot u2}{\frac{1}{u1} + -1}}} \]
   7. Step-by-step derivation

      1. associate-/l* 97.7%

         \[\leadsto \sqrt{39.47841760436263 \cdot \color{blue}{\frac{u2}{\frac{\frac{1}{u1} + -1}{u2}}}} \]

      2. +-commutative 97.7%

         \[\leadsto \sqrt{39.47841760436263 \cdot \frac{u2}{\frac{\color{blue}{-1 + \frac{1}{u1}}}{u2}}} \]

   8. Simplified 97.7%

      \[\leadsto \color{blue}{\sqrt{39.47841760436263 \cdot \frac{u2}{\frac{-1 + \frac{1}{u1}}{u2}}}} \]

   if 0.00850000046 < (*.f32 314159265359/50000000000 u2)

   1. Initial program 97.9%

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

   2. Taylor expanded in u1 around 0 70.5%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.008500000461935997:\\ \;\;\;\;\sqrt{39.47841760436263 \cdot \frac{u2}{\frac{\frac{1}{u1} + -1}{u2}}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1}\\ \end{array} \]

Alternative 4: 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. Final simplification 98.2%

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

Alternative 5: 81.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \sqrt{39.47841760436263 \cdot \frac{u2}{\frac{\frac{1}{u1} + -1}{u2}}} \end{array} \]

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((39.47841760436263f * (u2 / (((1.0f / u1) + -1.0f) / 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((39.47841760436263e0 * (u2 / (((1.0e0 / u1) + (-1.0e0)) / u2))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.2%

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

2. Taylor expanded in u2 around 0 80.9%

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

   1. clear-num 98.3%

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

   2. sqrt-div 98.3%

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

   3. metadata-eval 98.3%

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

4. Applied egg-rr 80.9%

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

   1. add-sqr-sqrt 80.6%

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

   2. sqrt-unprod 80.9%

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

   3. swap-sqr 80.8%

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

   4. metadata-eval 80.9%

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

   5. un-div-inv 81.1%

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

   6. un-div-inv 81.1%

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

   7. frac-times 81.1%

      \[\leadsto \sqrt{39.47841760436263 \cdot \color{blue}{\frac{u2 \cdot u2}{\sqrt{\frac{1 - u1}{u1}} \cdot \sqrt{\frac{1 - u1}{u1}}}}} \]

   8. add-sqr-sqrt 81.3%

      \[\leadsto \sqrt{39.47841760436263 \cdot \frac{u2 \cdot u2}{\color{blue}{\frac{1 - u1}{u1}}}} \]

   9. div-sub 81.3%

      \[\leadsto \sqrt{39.47841760436263 \cdot \frac{u2 \cdot u2}{\color{blue}{\frac{1}{u1} - \frac{u1}{u1}}}} \]

   10. *-inverses 81.3%

       \[\leadsto \sqrt{39.47841760436263 \cdot \frac{u2 \cdot u2}{\frac{1}{u1} - \color{blue}{1}}} \]

   11. sub-neg 81.3%

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

   12. metadata-eval 81.3%

       \[\leadsto \sqrt{39.47841760436263 \cdot \frac{u2 \cdot u2}{\frac{1}{u1} + \color{blue}{-1}}} \]

6. Applied egg-rr 81.3%

   \[\leadsto \color{blue}{\sqrt{39.47841760436263 \cdot \frac{u2 \cdot u2}{\frac{1}{u1} + -1}}} \]
7. Step-by-step derivation

   1. associate-/l* 81.4%

      \[\leadsto \sqrt{39.47841760436263 \cdot \color{blue}{\frac{u2}{\frac{\frac{1}{u1} + -1}{u2}}}} \]

   2. +-commutative 81.4%

      \[\leadsto \sqrt{39.47841760436263 \cdot \frac{u2}{\frac{\color{blue}{-1 + \frac{1}{u1}}}{u2}}} \]

8. Simplified 81.4%

   \[\leadsto \color{blue}{\sqrt{39.47841760436263 \cdot \frac{u2}{\frac{-1 + \frac{1}{u1}}{u2}}}} \]

9. Final simplification 81.4%

   \[\leadsto \sqrt{39.47841760436263 \cdot \frac{u2}{\frac{\frac{1}{u1} + -1}{u2}}} \]

Alternative 6: 73.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.2%

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

2. Taylor expanded in u2 around 0 80.9%

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

   1. flip-- 80.9%

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

   2. associate-/r/ 80.9%

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

   3. metadata-eval 80.9%

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

   4. +-commutative 80.9%

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

4. Applied egg-rr 80.9%

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

5. Taylor expanded in u1 around 0 71.6%

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

   1. +-commutative 71.6%

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

   2. unpow2 71.6%

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

7. Simplified 71.6%

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

8. Final simplification 71.6%

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

Alternative 7: 81.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.2%

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

2. Taylor expanded in u2 around 0 80.9%

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

3. Final simplification 80.9%

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

Alternative 8: 81.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.2%

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

2. Taylor expanded in u2 around 0 80.9%

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

   1. *-commutative 80.9%

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

   2. associate-*l* 81.0%

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

4. Simplified 81.0%

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

5. Final simplification 81.0%

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

Alternative 9: 81.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ 6.28318530718 \cdot \frac{u2}{\sqrt{\frac{1}{u1} + -1}} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = single(6.28318530718) * (u2 / sqrt(((single(1.0) / 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. Taylor expanded in u2 around 0 80.9%

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

   1. clear-num 98.3%

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

   2. sqrt-div 98.3%

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

   3. metadata-eval 98.3%

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

4. Applied egg-rr 80.9%

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

   1. associate-*r* 80.9%

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

   2. *-commutative 80.9%

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

   3. un-div-inv 81.0%

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

   4. div-sub 81.0%

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

   5. *-inverses 81.0%

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

   6. sub-neg 81.0%

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

   7. metadata-eval 81.0%

      \[\leadsto \frac{u2 \cdot 6.28318530718}{\sqrt{\frac{1}{u1} + \color{blue}{-1}}} \]

6. Applied egg-rr 81.0%

   \[\leadsto \color{blue}{\frac{u2 \cdot 6.28318530718}{\sqrt{\frac{1}{u1} + -1}}} \]
7. Step-by-step derivation

   1. associate-/l* 81.0%

      \[\leadsto \color{blue}{\frac{u2}{\frac{\sqrt{\frac{1}{u1} + -1}}{6.28318530718}}} \]

   2. associate-/r/ 81.0%

      \[\leadsto \color{blue}{\frac{u2}{\sqrt{\frac{1}{u1} + -1}} \cdot 6.28318530718} \]

   3. +-commutative 81.0%

      \[\leadsto \frac{u2}{\sqrt{\color{blue}{-1 + \frac{1}{u1}}}} \cdot 6.28318530718 \]

8. Simplified 81.0%

   \[\leadsto \color{blue}{\frac{u2}{\sqrt{-1 + \frac{1}{u1}}} \cdot 6.28318530718} \]

9. Final simplification 81.0%

   \[\leadsto 6.28318530718 \cdot \frac{u2}{\sqrt{\frac{1}{u1} + -1}} \]

Alternative 10: 81.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{u2 \cdot 6.28318530718}{\sqrt{\frac{1}{u1} + -1}} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = (u2 * single(6.28318530718)) / sqrt(((single(1.0) / 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. Taylor expanded in u2 around 0 80.9%

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

   1. clear-num 98.3%

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

   2. sqrt-div 98.3%

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

   3. metadata-eval 98.3%

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

4. Applied egg-rr 80.9%

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

   1. associate-*r* 80.9%

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

   2. *-commutative 80.9%

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

   3. un-div-inv 81.0%

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

   4. div-sub 81.0%

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

   5. *-inverses 81.0%

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

   6. sub-neg 81.0%

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

   7. metadata-eval 81.0%

      \[\leadsto \frac{u2 \cdot 6.28318530718}{\sqrt{\frac{1}{u1} + \color{blue}{-1}}} \]

6. Applied egg-rr 81.0%

   \[\leadsto \color{blue}{\frac{u2 \cdot 6.28318530718}{\sqrt{\frac{1}{u1} + -1}}} \]

7. Final simplification 81.0%

   \[\leadsto \frac{u2 \cdot 6.28318530718}{\sqrt{\frac{1}{u1} + -1}} \]

Alternative 11: 64.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.2%

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

2. Taylor expanded in u2 around 0 80.9%

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

3. Taylor expanded in u1 around 0 63.5%

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

   1. add-sqr-sqrt 63.4%

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

   2. sqrt-unprod 63.5%

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

   3. *-commutative 63.5%

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

   4. *-commutative 63.5%

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

   5. swap-sqr 63.5%

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

   6. add-sqr-sqrt 63.6%

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

5. Applied egg-rr 63.6%

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

6. Final simplification 63.6%

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

Alternative 12: 64.8% 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. Taylor expanded in u2 around 0 80.9%

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

3. Taylor expanded in u1 around 0 63.5%

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

4. Final simplification 63.5%

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

Alternative 13: 19.3% accurate, 41.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.2%

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

2. Taylor expanded in u1 around 0 84.4%

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

   1. unpow2 84.4%

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

   2. fma-udef 84.4%

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

4. Simplified 84.4%

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

5. Taylor expanded in u1 around inf 20.2%

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

   1. *-commutative 20.2%

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

7. Simplified 20.2%

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

8. Taylor expanded in u2 around 0 20.0%

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

9. Final simplification 20.0%

   \[\leadsto 6.28318530718 \cdot \left(u2 \cdot u1\right) \]

Reproduce

herbie shell --seed 2023228 
(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))))