Trowbridge-Reitz Sample, near normal, slope_y

Percentage Accurate: 98.3% → 98.4%

Time: 12.7s

Alternatives: 9

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - u1))) * sinf(sqrtf((powf(u2, 2.0f) * 39.47841760436263f)));
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = sqrt((u1 / (1.0e0 - u1))) * sin(sqrt(((u2 ** 2.0e0) * 39.47841760436263e0)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.2%

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

   1. add-sqr-sqrt 97.5%

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

   2. sqrt-unprod 98.2%

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

   3. *-commutative 98.2%

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

   4. *-commutative 98.2%

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

   5. swap-sqr 98.0%

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

   6. pow2 98.0%

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

   7. metadata-eval 98.4%

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

4. Applied egg-rr 98.4%

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

5. Final simplification 98.4%

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

Alternative 2: 98.2% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.2%

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

   1. clear-num 98.3%

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

   2. inv-pow 98.3%

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

4. Applied egg-rr 98.3%

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

5. Final simplification 98.3%

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

Alternative 3: 93.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.30000001192092896:\\ \;\;\;\;\frac{\sqrt{\frac{u1}{1 - u1}}}{u2 \cdot 1.0471975511966667 + 0.15915494309188485 \cdot \frac{1}{u2}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 6.28318530718) 0.30000001192092896)
   (/
    (sqrt (/ u1 (- 1.0 u1)))
    (+ (* u2 1.0471975511966667) (* 0.15915494309188485 (/ 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) <= 0.30000001192092896f) {
		tmp = sqrtf((u1 / (1.0f - u1))) / ((u2 * 1.0471975511966667f) + (0.15915494309188485f * (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) <= 0.30000001192092896e0) then
        tmp = sqrt((u1 / (1.0e0 - u1))) / ((u2 * 1.0471975511966667e0) + (0.15915494309188485e0 * (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(0.30000001192092896))
		tmp = Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) / Float32(Float32(u2 * Float32(1.0471975511966667)) + Float32(Float32(0.15915494309188485) * Float32(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(0.30000001192092896))
		tmp = sqrt((u1 / (single(1.0) - u1))) / ((u2 * single(1.0471975511966667)) + (single(0.15915494309188485) * (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) < 0.300000012

   1. Initial program 98.4%

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

      1. add-sqr-sqrt 97.8%

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

      2. sqrt-unprod 98.4%

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

      3. *-commutative 98.4%

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

      4. *-commutative 98.4%

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

      5. swap-sqr 98.3%

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

      6. pow2 98.3%

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

      7. metadata-eval 98.6%

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

   4. Applied egg-rr 98.6%

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

      1. add-sqr-sqrt 98.0%

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

      2. sqrt-div 97.8%

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

      3. add-sqr-sqrt 98.4%

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

      4. *-commutative 98.4%

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

      5. sqrt-prod 98.2%

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

      6. metadata-eval 98.1%

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

      7. unpow2 98.1%

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

      8. sqrt-prod 97.5%

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

      9. add-sqr-sqrt 98.1%

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

      10. associate-/r/ 98.3%

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

      11. div-inv 98.2%

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

      12. associate-/r* 98.0%

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

      13. sqrt-div 98.4%

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

      14. *-commutative 98.4%

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

   6. Applied egg-rr 98.4%

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

   7. Taylor expanded in u2 around 0 96.9%

      \[\leadsto \frac{\sqrt{\frac{u1}{1 - u1}}}{\color{blue}{1.0471975511966667 \cdot u2 + 0.15915494309188485 \cdot \frac{1}{u2}}} \]

   if 0.300000012 < (*.f32 314159265359/50000000000 u2)

   1. Initial program 97.3%

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

   3. Taylor expanded in u1 around 0 76.5%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.30000001192092896:\\ \;\;\;\;\frac{\sqrt{\frac{u1}{1 - u1}}}{u2 \cdot 1.0471975511966667 + 0.15915494309188485 \cdot \frac{1}{u2}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1}\\ \end{array} \]
5. Add Preprocessing

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. Add Preprocessing

3. Final simplification 98.2%

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

Alternative 5: 88.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \frac{\sqrt{\frac{u1}{1 - u1}}}{u2 \cdot 1.0471975511966667 + 0.15915494309188485 \cdot \frac{1}{u2}} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (/
  (sqrt (/ u1 (- 1.0 u1)))
  (+ (* u2 1.0471975511966667) (* 0.15915494309188485 (/ 1.0 u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - u1))) / ((u2 * 1.0471975511966667f) + (0.15915494309188485f * (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((u1 / (1.0e0 - u1))) / ((u2 * 1.0471975511966667e0) + (0.15915494309188485e0 * (1.0e0 / u2)))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) / Float32(Float32(u2 * Float32(1.0471975511966667)) + Float32(Float32(0.15915494309188485) * Float32(Float32(1.0) / u2))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((u1 / (single(1.0) - u1))) / ((u2 * single(1.0471975511966667)) + (single(0.15915494309188485) * (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. Add Preprocessing
3. Step-by-step derivation

   1. add-sqr-sqrt 97.5%

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

   2. sqrt-unprod 98.2%

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

   3. *-commutative 98.2%

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

   4. *-commutative 98.2%

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

   5. swap-sqr 98.0%

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

   6. pow2 98.0%

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

   7. metadata-eval 98.4%

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

4. Applied egg-rr 98.4%

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

   1. add-sqr-sqrt 94.5%

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

   2. sqrt-div 94.3%

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

   3. add-sqr-sqrt 98.2%

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

   4. *-commutative 98.2%

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

   5. sqrt-prod 97.7%

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

   6. metadata-eval 97.9%

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

   7. unpow2 97.9%

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

   8. sqrt-prod 97.3%

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

   9. add-sqr-sqrt 97.9%

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

   10. associate-/r/ 98.0%

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

   11. div-inv 97.9%

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

   12. associate-/r* 97.8%

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

   13. sqrt-div 98.1%

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

   14. *-commutative 98.1%

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

6. Applied egg-rr 98.1%

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

7. Taylor expanded in u2 around 0 85.7%

   \[\leadsto \frac{\sqrt{\frac{u1}{1 - u1}}}{\color{blue}{1.0471975511966667 \cdot u2 + 0.15915494309188485 \cdot \frac{1}{u2}}} \]

8. Final simplification 85.7%

   \[\leadsto \frac{\sqrt{\frac{u1}{1 - u1}}}{u2 \cdot 1.0471975511966667 + 0.15915494309188485 \cdot \frac{1}{u2}} \]
9. Add Preprocessing

Alternative 6: 81.3% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.2%

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

3. Taylor expanded in u2 around 0 79.0%

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

4. Final simplification 79.0%

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

Alternative 7: 81.3% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.2%

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

3. Taylor expanded in u2 around 0 79.0%

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

   1. associate-*r* 79.1%

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

5. Simplified 79.1%

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

6. Final simplification 79.1%

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

Alternative 8: 4.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.2%

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

3. Taylor expanded in u2 around 0 79.0%

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

4. Taylor expanded in u1 around 0 61.8%

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

   1. add-sqr-sqrt 61.7%

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

   2. sqrt-unprod 61.8%

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

   3. swap-sqr 61.8%

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

   4. add-sqr-sqrt 61.9%

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

   5. pow2 61.9%

      \[\leadsto 6.28318530718 \cdot \sqrt{u1 \cdot \color{blue}{{u2}^{2}}} \]

6. Applied egg-rr 61.9%

   \[\leadsto 6.28318530718 \cdot \color{blue}{\sqrt{u1 \cdot {u2}^{2}}} \]

7. Taylor expanded in u2 around -inf 4.7%

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

8. Final simplification 4.7%

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

Alternative 9: 64.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.2%

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

3. Taylor expanded in u2 around 0 79.0%

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

4. Taylor expanded in u1 around 0 61.8%

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

5. Final simplification 61.8%

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

Reproduce

herbie shell --seed 2024011 
(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))))