Trowbridge-Reitz Sample, near normal, slope_x

Percentage Accurate: 99.0% → 99.0%

Time: 10.7s

Alternatives: 7

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

2. Final simplification 99.0%

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

Alternative 2: 89.6% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.004000000189989805)
   (sqrt (/ 1.0 (+ (/ 1.0 u1) -1.0)))
   (* (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((1.0f / ((1.0f / u1) + -1.0f)));
	} 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((1.0e0 / ((1.0e0 / u1) + (-1.0e0))))
    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(Float32(1.0) / Float32(Float32(Float32(1.0) / u1) + Float32(-1.0))));
	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((single(1.0) / ((single(1.0) / u1) + single(-1.0))));
	else
		tmp = cos((single(6.28318530718) * u2)) * sqrt(u1);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 314159265359/50000000000 u2) < 0.00400000019

   1. Initial program 99.5%

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

      1. *-commutative 99.5%

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

      2. sqrt-div 99.1%

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

      3. associate-*r/ 99.1%

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

   3. Applied egg-rr 99.1%

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

      1. associate-/l* 98.9%

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

   5. Simplified 98.9%

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

      1. add-exp-log 96.6%

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

      2. sqrt-undiv 96.7%

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

   7. Applied egg-rr 96.7%

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

      1. clear-num 96.6%

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

      2. associate-/r/ 96.7%

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

      3. rem-exp-log 99.0%

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

      4. pow1/2 99.0%

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

      5. pow-flip 99.4%

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

      6. div-sub 99.5%

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

      7. *-inverses 99.5%

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

      8. sub-neg 99.5%

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

      9. metadata-eval 99.5%

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

      10. metadata-eval 99.5%

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

   9. Applied egg-rr 99.5%

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

   10. Taylor expanded in u2 around 0 97.9%

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

   if 0.00400000019 < (*.f32 314159265359/50000000000 u2)

   1. Initial program 98.1%

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

   2. Taylor expanded in u1 around 0 72.9%

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

4. Final simplification 89.8%

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

Alternative 3: 79.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

   1. *-commutative 99.0%

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

   2. sqrt-div 98.7%

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

   3. associate-*r/ 98.7%

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

3. Applied egg-rr 98.7%

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

   1. associate-/l* 98.6%

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

5. Simplified 98.6%

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

   1. add-exp-log 96.3%

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

   2. sqrt-undiv 96.3%

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

7. Applied egg-rr 96.3%

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

   1. clear-num 96.3%

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

   2. associate-/r/ 96.4%

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

   3. rem-exp-log 98.5%

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

   4. pow1/2 98.5%

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

   5. pow-flip 98.9%

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

   6. div-sub 99.0%

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

   7. *-inverses 99.0%

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

   8. sub-neg 99.0%

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

   9. metadata-eval 99.0%

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

   10. metadata-eval 99.0%

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

9. Applied egg-rr 99.0%

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

10. Taylor expanded in u2 around 0 79.9%

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

11. Final simplification 79.9%

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

Alternative 4: 79.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

2. Taylor expanded in u2 around 0 79.8%

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

   1. pow1/2 79.8%

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

   2. div-inv 79.7%

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

   3. unpow-prod-down 79.5%

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

   4. pow1/2 79.5%

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

4. Applied egg-rr 79.5%

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

   1. unpow1/2 79.5%

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

6. Simplified 79.5%

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

   1. *-commutative 79.5%

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

   2. sqrt-div 79.3%

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

   3. metadata-eval 79.3%

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

   4. associate-/r/ 79.4%

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

   5. sqrt-div 79.6%

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

   6. pow1/2 79.6%

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

   7. pow-flip 79.8%

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

   8. div-sub 79.8%

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

   9. *-inverses 79.8%

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

   10. sub-neg 79.8%

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

   11. metadata-eval 79.8%

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

   12. metadata-eval 79.8%

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

8. Applied egg-rr 79.8%

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

9. Final simplification 79.8%

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

Alternative 5: 79.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{u1}{1 - u1}} \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

2. Taylor expanded in u2 around 0 79.8%

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

3. Final simplification 79.8%

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

Alternative 6: 63.2% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

2. Taylor expanded in u2 around 0 79.8%

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

3. Taylor expanded in u1 around 0 62.5%

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

4. Final simplification 62.5%

   \[\leadsto \sqrt{u1} \]

Alternative 7: 19.1% 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 99.0%

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

2. Taylor expanded in u1 around 0 85.4%

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

   1. +-commutative 85.4%

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

   2. unpow2 85.4%

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

   3. fma-def 85.4%

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

4. Simplified 85.4%

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

5. Taylor expanded in u1 around inf 20.0%

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

6. Taylor expanded in u2 around 0 19.5%

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

7. Final simplification 19.5%

   \[\leadsto u1 \]

Reproduce

herbie shell --seed 2023306 
(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))))