Trowbridge-Reitz Sample, near normal, slope_y

Percentage Accurate: 98.3% → 98.4%

Time: 13.9s

Alternatives: 11

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

   1. add-sqr-sqrt 97.7%

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

   2. sqrt-unprod 98.5%

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

   3. *-commutative 98.5%

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

   4. *-commutative 98.5%

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

   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. Final simplification 98.6%

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

Alternative 2: 98.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{log1p}\left(\mathsf{expm1}\left(\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(u2 \cdot 6.28318530718\right)\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (log1p (expm1 (* (sqrt (/ u1 (- 1.0 u1))) (sin (* u2 6.28318530718))))))

C

float code(float cosTheta_i, float u1, float u2) {
	return log1pf(expm1f((sqrtf((u1 / (1.0f - u1))) * sinf((u2 * 6.28318530718f)))));
}

Julia

function code(cosTheta_i, u1, u2)
	return log1p(expm1(Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * sin(Float32(u2 * Float32(6.28318530718))))))
end

Derivation

1. Initial program 98.5%

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

   1. log1p-expm1-u 98.5%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right)\right)\right)} \]

4. Applied egg-rr 98.5%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right)\right)\right)} \]

5. Final simplification 98.5%

   \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(u2 \cdot 6.28318530718\right)\right)\right) \]
6. Add Preprocessing

Alternative 3: 98.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \sin \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{\frac{u1}{1 - {u1}^{2}} \cdot \left(u1 + 1\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sin (* u2 6.28318530718))
  (sqrt (* (/ u1 (- 1.0 (pow u1 2.0))) (+ u1 1.0)))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sinf((u2 * 6.28318530718f)) * sqrtf(((u1 / (1.0f - powf(u1, 2.0f))) * (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(((u1 / (1.0e0 - (u1 ** 2.0e0))) * (u1 + 1.0e0)))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sin(Float32(u2 * Float32(6.28318530718))) * sqrt(Float32(Float32(u1 / Float32(Float32(1.0) - (u1 ^ Float32(2.0)))) * Float32(u1 + Float32(1.0)))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sin((u2 * single(6.28318530718))) * sqrt(((u1 / (single(1.0) - (u1 ^ single(2.0)))) * (u1 + single(1.0))));
end

Derivation

1. Initial program 98.5%

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

   1. flip-- 98.5%

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

   2. associate-/r/ 98.5%

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

   3. metadata-eval 98.5%

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

   4. pow2 98.5%

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

   5. +-commutative 98.5%

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

4. Applied egg-rr 98.5%

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

5. Final simplification 98.5%

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

Alternative 4: 90.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.0210999995470047:\\ \;\;\;\;\frac{6.28318530718}{\frac{\sqrt{\frac{1 - u1}{u1}}}{u2}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 6.28318530718) 0.0210999995470047)
   (/ 6.28318530718 (/ (sqrt (/ (- 1.0 u1) u1)) u2))
   (* (sin (* u2 6.28318530718)) (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * 6.28318530718f) <= 0.0210999995470047f) {
		tmp = 6.28318530718f / (sqrtf(((1.0f - u1) / u1)) / u2);
	} else {
		tmp = sinf((u2 * 6.28318530718f)) * sqrtf(u1);
	}
	return tmp;
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    real(4) :: tmp
    if ((u2 * 6.28318530718e0) <= 0.0210999995470047e0) then
        tmp = 6.28318530718e0 / (sqrt(((1.0e0 - u1) / u1)) / u2)
    else
        tmp = sin((u2 * 6.28318530718e0)) * sqrt(u1)
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(u2 * Float32(6.28318530718)) <= Float32(0.0210999995470047))
		tmp = Float32(Float32(6.28318530718) / Float32(sqrt(Float32(Float32(Float32(1.0) - u1) / u1)) / u2));
	else
		tmp = Float32(sin(Float32(u2 * Float32(6.28318530718))) * sqrt(u1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((u2 * single(6.28318530718)) <= single(0.0210999995470047))
		tmp = single(6.28318530718) / (sqrt(((single(1.0) - u1) / u1)) / u2);
	else
		tmp = sin((u2 * single(6.28318530718))) * sqrt(u1);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 314159265359/50000000000 u2) < 0.0210999995

   1. Initial program 98.7%

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

   3. Taylor expanded in u2 around 0 95.3%

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

      1. *-commutative 95.3%

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

      2. sqrt-div 95.1%

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

      3. associate-*r/ 95.0%

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

   5. Applied egg-rr 95.0%

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

      1. associate-/l* 95.2%

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

   7. Simplified 95.2%

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

      1. expm1-log1p-u 95.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(6.28318530718 \cdot \frac{u2}{\frac{\sqrt{1 - u1}}{\sqrt{u1}}}\right)\right)} \]

      2. expm1-udef 25.4%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(6.28318530718 \cdot \frac{u2}{\frac{\sqrt{1 - u1}}{\sqrt{u1}}}\right)} - 1} \]

      3. clear-num 25.4%

         \[\leadsto e^{\mathsf{log1p}\left(6.28318530718 \cdot \color{blue}{\frac{1}{\frac{\frac{\sqrt{1 - u1}}{\sqrt{u1}}}{u2}}}\right)} - 1 \]

      4. un-div-inv 25.4%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{6.28318530718}{\frac{\frac{\sqrt{1 - u1}}{\sqrt{u1}}}{u2}}}\right)} - 1 \]

      5. sqrt-undiv 25.4%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{6.28318530718}{\frac{\color{blue}{\sqrt{\frac{1 - u1}{u1}}}}{u2}}\right)} - 1 \]

   9. Applied egg-rr 25.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{6.28318530718}{\frac{\sqrt{\frac{1 - u1}{u1}}}{u2}}\right)} - 1} \]
   10. Step-by-step derivation

       1. expm1-def 95.4%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{6.28318530718}{\frac{\sqrt{\frac{1 - u1}{u1}}}{u2}}\right)\right)} \]

       2. expm1-log1p 95.4%

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

   11. Simplified 95.4%

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

   if 0.0210999995 < (*.f32 314159265359/50000000000 u2)

   1. Initial program 98.1%

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

   3. Taylor expanded in u1 around 0 70.7%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.0210999995470047:\\ \;\;\;\;\frac{6.28318530718}{\frac{\sqrt{\frac{1 - u1}{u1}}}{u2}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.3% accurate, 1.0× speedup

Math

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

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 98.5%

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

3. Final simplification 98.5%

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

Alternative 6: 81.1% 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 98.5%

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

3. Taylor expanded in u2 around 0 82.4%

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

4. Final simplification 82.4%

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

Alternative 7: 81.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in u2 around 0 82.4%

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

4. Final simplification 82.4%

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

Alternative 8: 81.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in u2 around 0 82.4%

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

   1. *-commutative 82.4%

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

   2. sqrt-div 82.3%

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

   3. associate-*r/ 82.1%

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

5. Applied egg-rr 82.1%

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

   1. associate-/l* 82.3%

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

7. Simplified 82.3%

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

   1. expm1-log1p-u 82.3%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(6.28318530718 \cdot \frac{u2}{\frac{\sqrt{1 - u1}}{\sqrt{u1}}}\right)\right)} \]

   2. expm1-udef 29.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(6.28318530718 \cdot \frac{u2}{\frac{\sqrt{1 - u1}}{\sqrt{u1}}}\right)} - 1} \]

   3. clear-num 29.1%

      \[\leadsto e^{\mathsf{log1p}\left(6.28318530718 \cdot \color{blue}{\frac{1}{\frac{\frac{\sqrt{1 - u1}}{\sqrt{u1}}}{u2}}}\right)} - 1 \]

   4. un-div-inv 29.1%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{6.28318530718}{\frac{\frac{\sqrt{1 - u1}}{\sqrt{u1}}}{u2}}}\right)} - 1 \]

   5. sqrt-undiv 29.1%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{6.28318530718}{\frac{\color{blue}{\sqrt{\frac{1 - u1}{u1}}}}{u2}}\right)} - 1 \]

9. Applied egg-rr 29.1%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{6.28318530718}{\frac{\sqrt{\frac{1 - u1}{u1}}}{u2}}\right)} - 1} \]
10. Step-by-step derivation

    1. expm1-def 82.5%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{6.28318530718}{\frac{\sqrt{\frac{1 - u1}{u1}}}{u2}}\right)\right)} \]

    2. expm1-log1p 82.5%

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

11. Simplified 82.5%

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

12. Final simplification 82.5%

    \[\leadsto \frac{6.28318530718}{\frac{\sqrt{\frac{1 - u1}{u1}}}{u2}} \]
13. Add Preprocessing

Alternative 9: 64.2% 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 98.5%

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

3. Taylor expanded in u2 around 0 82.4%

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

4. Taylor expanded in u1 around 0 66.5%

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

5. Final simplification 66.5%

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

Alternative 10: 64.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in u2 around 0 82.4%

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

4. Taylor expanded in u1 around 0 66.5%

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

   1. expm1-log1p-u 66.5%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(6.28318530718 \cdot \left(\sqrt{u1} \cdot u2\right)\right)\right)} \]

   2. expm1-udef 26.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(6.28318530718 \cdot \left(\sqrt{u1} \cdot u2\right)\right)} - 1} \]

   3. associate-*r* 26.8%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(6.28318530718 \cdot \sqrt{u1}\right) \cdot u2}\right)} - 1 \]

   4. *-commutative 26.8%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{u2 \cdot \left(6.28318530718 \cdot \sqrt{u1}\right)}\right)} - 1 \]

6. Applied egg-rr 26.8%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(u2 \cdot \left(6.28318530718 \cdot \sqrt{u1}\right)\right)} - 1} \]
7. Step-by-step derivation

   1. expm1-def 66.5%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(u2 \cdot \left(6.28318530718 \cdot \sqrt{u1}\right)\right)\right)} \]

   2. expm1-log1p 66.5%

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

8. Simplified 66.5%

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

9. Final simplification 66.5%

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

Alternative 11: -0.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ u2 \cdot \sqrt{-39.47841760436263} \end{array} \]

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return u2 * sqrtf(-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 = u2 * sqrt((-39.47841760436263e0))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in u2 around 0 82.4%

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

   1. *-commutative 82.4%

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

   2. sqrt-div 82.3%

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

   3. associate-*r/ 82.1%

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

5. Applied egg-rr 82.1%

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

   1. associate-/l* 82.3%

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

7. Simplified 82.3%

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

   1. add-sqr-sqrt 81.9%

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

   2. sqrt-unprod 82.3%

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

   3. associate-*r/ 82.3%

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

   4. associate-*r/ 82.3%

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

   5. frac-times 82.1%

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

   6. *-commutative 82.1%

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

   7. *-commutative 82.1%

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

   8. swap-sqr 82.1%

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

   9. pow2 82.1%

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

   10. metadata-eval 82.4%

       \[\leadsto \sqrt{\frac{{u2}^{2} \cdot \color{blue}{39.47841760436263}}{\frac{\sqrt{1 - u1}}{\sqrt{u1}} \cdot \frac{\sqrt{1 - u1}}{\sqrt{u1}}}} \]

   11. sqrt-undiv 82.5%

       \[\leadsto \sqrt{\frac{{u2}^{2} \cdot 39.47841760436263}{\color{blue}{\sqrt{\frac{1 - u1}{u1}}} \cdot \frac{\sqrt{1 - u1}}{\sqrt{u1}}}} \]

   12. sqrt-undiv 82.5%

       \[\leadsto \sqrt{\frac{{u2}^{2} \cdot 39.47841760436263}{\sqrt{\frac{1 - u1}{u1}} \cdot \color{blue}{\sqrt{\frac{1 - u1}{u1}}}}} \]

   13. add-sqr-sqrt 82.7%

       \[\leadsto \sqrt{\frac{{u2}^{2} \cdot 39.47841760436263}{\color{blue}{\frac{1 - u1}{u1}}}} \]

9. Applied egg-rr 82.7%

   \[\leadsto \color{blue}{\sqrt{\frac{{u2}^{2} \cdot 39.47841760436263}{\frac{1 - u1}{u1}}}} \]

10. Taylor expanded in u1 around -inf -0.0%

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

11. Final simplification -0.0%

    \[\leadsto u2 \cdot \sqrt{-39.47841760436263} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024024 
(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))))