Trowbridge-Reitz Sample, near normal, slope_x

Percentage Accurate: 99.0% → 98.9%

Time: 10.8s

Alternatives: 8

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = (((single(1.0) / u1) + single(-1.0)) ^ single(-0.5)) * 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. Step-by-step derivation

   1. add-cbrt-cube 99.0%

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

   2. add-cbrt-cube 99.0%

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

   3. cbrt-unprod 99.0%

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

   4. add-sqr-sqrt 99.0%

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

   5. pow1 99.0%

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

   6. pow1/2 99.0%

      \[\leadsto \sqrt[3]{\left({\left(\frac{u1}{1 - u1}\right)}^{1} \cdot \color{blue}{{\left(\frac{u1}{1 - u1}\right)}^{0.5}}\right) \cdot \left(\left(\cos \left(6.28318530718 \cdot u2\right) \cdot \cos \left(6.28318530718 \cdot u2\right)\right) \cdot \cos \left(6.28318530718 \cdot u2\right)\right)} \]

   7. pow-prod-up 99.0%

      \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{u1}{1 - u1}\right)}^{\left(1 + 0.5\right)}} \cdot \left(\left(\cos \left(6.28318530718 \cdot u2\right) \cdot \cos \left(6.28318530718 \cdot u2\right)\right) \cdot \cos \left(6.28318530718 \cdot u2\right)\right)} \]

   8. metadata-eval 99.0%

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

   9. pow3 99.0%

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

3. Applied egg-rr 99.0%

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

   1. add-sqr-sqrt 95.2%

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

   2. sqrt-unprod 96.2%

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

   3. cbrt-prod 96.2%

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

   4. rem-cbrt-cube 96.2%

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

   5. cbrt-prod 96.1%

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

   6. rem-cbrt-cube 96.1%

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

   7. swap-sqr 96.0%

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

5. Applied egg-rr 98.6%

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

   1. associate-/r/ 98.7%

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

   2. pow1/2 98.7%

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

   3. pow-flip 99.1%

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

   4. div-sub 99.1%

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

   5. *-inverses 99.1%

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

   6. sub-neg 99.1%

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

   7. metadata-eval 99.1%

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

   8. metadata-eval 99.1%

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

7. Applied egg-rr 99.1%

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

8. Final simplification 99.1%

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

Alternative 2: 89.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.004600000102072954:\\ \;\;\;\;{\left(\frac{1}{u1} + -1\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.004600000102072954)
   (pow (+ (/ 1.0 u1) -1.0) -0.5)
   (* (cos (* 6.28318530718 u2)) (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.004600000102072954f) {
		tmp = powf(((1.0f / u1) + -1.0f), -0.5f);
	} 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) <= 0.004600000102072954e0) then
        tmp = ((1.0e0 / u1) + (-1.0e0)) ** (-0.5e0)
    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(0.004600000102072954))
		tmp = Float32(Float32(Float32(1.0) / u1) + Float32(-1.0)) ^ Float32(-0.5);
	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(0.004600000102072954))
		tmp = ((single(1.0) / u1) + single(-1.0)) ^ single(-0.5);
	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) < 0.0046000001

   1. Initial program 99.4%

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

   2. Taylor expanded in u2 around 0 97.9%

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

      1. add-cbrt-cube 97.9%

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

      2. add-sqr-sqrt 97.9%

         \[\leadsto \sqrt[3]{\color{blue}{\frac{u1}{1 - u1}} \cdot \sqrt{\frac{u1}{1 - u1}}} \]

      3. pow1 97.9%

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

      4. pow1/2 97.9%

         \[\leadsto \sqrt[3]{{\left(\frac{u1}{1 - u1}\right)}^{1} \cdot \color{blue}{{\left(\frac{u1}{1 - u1}\right)}^{0.5}}} \]

      5. pow-prod-up 97.8%

         \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{u1}{1 - u1}\right)}^{\left(1 + 0.5\right)}}} \]

      6. metadata-eval 97.8%

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

   4. Applied egg-rr 97.8%

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

      1. pow1/3 95.0%

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

      2. pow-pow 97.9%

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

      3. metadata-eval 97.9%

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

      4. pow1/2 97.9%

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

      5. clear-num 97.8%

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

      6. sqrt-div 97.4%

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

      7. metadata-eval 97.4%

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

      8. pow1/2 97.4%

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

      9. pow-flip 97.8%

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

      10. div-sub 97.9%

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

      11. *-inverses 97.9%

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

      12. sub-neg 97.9%

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

      13. metadata-eval 97.9%

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

      14. metadata-eval 97.9%

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

   6. Applied egg-rr 97.9%

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

   if 0.0046000001 < (*.f32 314159265359/50000000000 u2)

   1. Initial program 98.2%

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

   2. Taylor expanded in u1 around 0 77.4%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.004600000102072954:\\ \;\;\;\;{\left(\frac{1}{u1} + -1\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \]

Alternative 3: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = cos((single(6.28318530718) * u2)) * 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. Final simplification 99.0%

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

Alternative 4: 79.4% 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 82.9%

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

   1. add-cbrt-cube 82.8%

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

   2. add-sqr-sqrt 82.8%

      \[\leadsto \sqrt[3]{\color{blue}{\frac{u1}{1 - u1}} \cdot \sqrt{\frac{u1}{1 - u1}}} \]

   3. pow1 82.8%

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

   4. pow1/2 82.8%

      \[\leadsto \sqrt[3]{{\left(\frac{u1}{1 - u1}\right)}^{1} \cdot \color{blue}{{\left(\frac{u1}{1 - u1}\right)}^{0.5}}} \]

   5. pow-prod-up 82.8%

      \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{u1}{1 - u1}\right)}^{\left(1 + 0.5\right)}}} \]

   6. metadata-eval 82.8%

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

4. Applied egg-rr 82.8%

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

   1. pow1/3 80.8%

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

   2. pow-pow 82.9%

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

   3. metadata-eval 82.9%

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

   4. pow1/2 82.9%

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

   5. clear-num 82.8%

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

   6. sqrt-div 82.5%

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

   7. metadata-eval 82.5%

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

   8. pow1/2 82.5%

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

   9. pow-flip 82.8%

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

   10. div-sub 82.9%

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

   11. *-inverses 82.9%

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

   12. sub-neg 82.9%

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

   13. metadata-eval 82.9%

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

   14. metadata-eval 82.9%

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

6. Applied egg-rr 82.9%

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

7. Final simplification 82.9%

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

Alternative 5: 70.9% accurate, 2.0× speedup

Math

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

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 82.9%

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

3. Taylor expanded in u1 around 0 76.4%

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

   1. +-commutative 76.4%

      \[\leadsto \sqrt{\color{blue}{\left({u1}^{2} + {u1}^{3}\right) + u1}} \]

   2. unpow2 76.4%

      \[\leadsto \sqrt{\left(\color{blue}{u1 \cdot u1} + {u1}^{3}\right) + u1} \]

   3. cube-mult 76.4%

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

   4. unpow2 76.4%

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

   5. distribute-lft-out 76.4%

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

   6. unpow2 76.4%

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

   7. distribute-rgt1-in 76.4%

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

   8. *-commutative 76.4%

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

   9. *-rgt-identity 76.4%

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

   10. distribute-lft-in 76.3%

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

   11. fma-udef 76.3%

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

5. Simplified 76.3%

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

6. Taylor expanded in u1 around 0 73.1%

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

7. Final simplification 73.1%

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

Alternative 6: 79.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

2. Taylor expanded in u2 around 0 82.9%

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

3. Final simplification 82.9%

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

Alternative 7: 62.6% 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 82.9%

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

3. Taylor expanded in u1 around 0 64.0%

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

4. Final simplification 64.0%

   \[\leadsto \sqrt{u1} \]

Alternative 8: 20.1% accurate, 69.7× speedup

Math

\[\begin{array}{l} \\ u1 + 0.5 \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2) :precision binary32 (+ u1 0.5))

C

float code(float cosTheta_i, float u1, float u2) {
	return u1 + 0.5f;
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = u1 + 0.5e0
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(u1 + Float32(0.5))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = u1 + single(0.5);
end

Derivation

1. Initial program 99.0%

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

2. Taylor expanded in u2 around 0 82.9%

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

3. Taylor expanded in u1 around 0 76.4%

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

   1. +-commutative 76.4%

      \[\leadsto \sqrt{\color{blue}{\left({u1}^{2} + {u1}^{3}\right) + u1}} \]

   2. unpow2 76.4%

      \[\leadsto \sqrt{\left(\color{blue}{u1 \cdot u1} + {u1}^{3}\right) + u1} \]

   3. cube-mult 76.4%

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

   4. unpow2 76.4%

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

   5. distribute-lft-out 76.4%

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

   6. unpow2 76.4%

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

   7. distribute-rgt1-in 76.4%

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

   8. *-commutative 76.4%

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

   9. *-rgt-identity 76.4%

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

   10. distribute-lft-in 76.3%

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

   11. fma-udef 76.3%

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

5. Simplified 76.3%

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

6. Taylor expanded in u1 around 0 73.1%

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

7. Taylor expanded in u1 around inf 20.7%

   \[\leadsto \color{blue}{0.5 + u1} \]
8. Step-by-step derivation

   1. +-commutative 20.7%

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

9. Simplified 20.7%

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

10. Final simplification 20.7%

    \[\leadsto u1 + 0.5 \]

Reproduce

herbie shell --seed 2023319 
(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))))