Trowbridge-Reitz Sample, near normal, slope_y

Percentage Accurate: 98.3% → 98.3%

Time: 14.4s

Alternatives: 16

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.3% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

function code(cosTheta_i, u1, u2)
	return expm1(log1p(Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * sin(Float32(Float32(6.28318530718) * 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. Step-by-step derivation

   1. expm1-log1p-u 98.5%

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

4. Applied egg-rr 98.5%

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

5. Final simplification 98.5%

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

Alternative 2: 98.3% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

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((6.28318530718e0 * u2)) * sqrt(((1.0e0 + (1.0e0 / u1)) / ((-1.0e0) + (u1 ** (-2.0e0)))))
end function

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sin((single(6.28318530718) * u2)) * sqrt(((single(1.0) + (single(1.0) / u1)) / (single(-1.0) + (u1 ^ single(-2.0)))));
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. clear-num 98.4%

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

   2. inv-pow 98.4%

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

4. Applied egg-rr 98.4%

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

   1. sqr-pow 98.3%

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

   2. div-sub 98.2%

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

   3. *-inverses 98.2%

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

   4. sub-neg 98.2%

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

   5. metadata-eval 98.2%

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

   6. metadata-eval 98.2%

      \[\leadsto \sqrt{{\left(\frac{1}{u1} + -1\right)}^{\color{blue}{-0.5}} \cdot {\left(\frac{1 - u1}{u1}\right)}^{\left(\frac{-1}{2}\right)}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   7. div-sub 98.2%

      \[\leadsto \sqrt{{\left(\frac{1}{u1} + -1\right)}^{-0.5} \cdot {\color{blue}{\left(\frac{1}{u1} - \frac{u1}{u1}\right)}}^{\left(\frac{-1}{2}\right)}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   8. *-inverses 98.2%

      \[\leadsto \sqrt{{\left(\frac{1}{u1} + -1\right)}^{-0.5} \cdot {\left(\frac{1}{u1} - \color{blue}{1}\right)}^{\left(\frac{-1}{2}\right)}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   9. sub-neg 98.2%

      \[\leadsto \sqrt{{\left(\frac{1}{u1} + -1\right)}^{-0.5} \cdot {\color{blue}{\left(\frac{1}{u1} + \left(-1\right)\right)}}^{\left(\frac{-1}{2}\right)}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   10. metadata-eval 98.2%

       \[\leadsto \sqrt{{\left(\frac{1}{u1} + -1\right)}^{-0.5} \cdot {\left(\frac{1}{u1} + \color{blue}{-1}\right)}^{\left(\frac{-1}{2}\right)}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   11. metadata-eval 98.2%

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

6. Applied egg-rr 98.2%

   \[\leadsto \sqrt{\color{blue}{{\left(\frac{1}{u1} + -1\right)}^{-0.5} \cdot {\left(\frac{1}{u1} + -1\right)}^{-0.5}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
7. Step-by-step derivation

   1. pow-sqr 98.3%

      \[\leadsto \sqrt{\color{blue}{{\left(\frac{1}{u1} + -1\right)}^{\left(2 \cdot -0.5\right)}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   2. metadata-eval 98.3%

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

   3. unpow-1 98.3%

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

8. Simplified 98.3%

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

   1. flip-+ 98.3%

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

   2. associate-/r/ 98.3%

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

   3. metadata-eval 98.3%

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

   4. sub-neg 98.3%

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

   5. inv-pow 98.3%

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

   6. inv-pow 98.3%

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

   7. pow-prod-up 98.4%

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

   8. metadata-eval 98.4%

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

   9. metadata-eval 98.4%

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

   10. sub-neg 98.4%

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

   11. metadata-eval 98.4%

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

10. Applied egg-rr 98.4%

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

    1. associate-*l/ 98.5%

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

    2. +-commutative 98.5%

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

    3. *-lft-identity 98.5%

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

    4. +-commutative 98.5%

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

12. Simplified 98.5%

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

13. Final simplification 98.5%

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

Alternative 3: 93.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.0020000000949949026:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \left(6.28318530718 \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1 \cdot \left(u1 + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.0020000000949949026)
   (* (sqrt (/ u1 (- 1.0 u1))) (* 6.28318530718 u2))
   (* (sin (* 6.28318530718 u2)) (sqrt (* u1 (+ u1 1.0))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.0020000000949949026f) {
		tmp = sqrtf((u1 / (1.0f - u1))) * (6.28318530718f * u2);
	} else {
		tmp = sinf((6.28318530718f * u2)) * sqrtf((u1 * (u1 + 1.0f)));
	}
	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) <= 0.0020000000949949026e0) then
        tmp = sqrt((u1 / (1.0e0 - u1))) * (6.28318530718e0 * u2)
    else
        tmp = sin((6.28318530718e0 * u2)) * sqrt((u1 * (u1 + 1.0e0)))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(6.28318530718) * u2) <= Float32(0.0020000000949949026))
		tmp = Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * Float32(Float32(6.28318530718) * u2));
	else
		tmp = Float32(sin(Float32(Float32(6.28318530718) * u2)) * sqrt(Float32(u1 * Float32(u1 + Float32(1.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((single(6.28318530718) * u2) <= single(0.0020000000949949026))
		tmp = sqrt((u1 / (single(1.0) - u1))) * (single(6.28318530718) * u2);
	else
		tmp = sin((single(6.28318530718) * u2)) * sqrt((u1 * (u1 + single(1.0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.00200000009

   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 98.6%

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

   if 0.00200000009 < (*.f32 #s(literal 314159265359/50000000000 binary32) 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 87.5%

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

      1. +-commutative 50.9%

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

   5. Simplified 87.5%

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(u1 + 1\right)}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.0020000000949949026:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \left(6.28318530718 \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1 \cdot \left(u1 + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.017500000074505806:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \left(6.28318530718 \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.017500000074505806)
   (* (sqrt (/ u1 (- 1.0 u1))) (* 6.28318530718 u2))
   (* (sin (* 6.28318530718 u2)) (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.017500000074505806f) {
		tmp = sqrtf((u1 / (1.0f - u1))) * (6.28318530718f * u2);
	} else {
		tmp = sinf((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) <= 0.017500000074505806e0) then
        tmp = sqrt((u1 / (1.0e0 - u1))) * (6.28318530718e0 * u2)
    else
        tmp = sin((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(0.017500000074505806))
		tmp = Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * Float32(Float32(6.28318530718) * u2));
	else
		tmp = Float32(sin(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(0.017500000074505806))
		tmp = sqrt((u1 / (single(1.0) - u1))) * (single(6.28318530718) * u2);
	else
		tmp = sin((single(6.28318530718) * u2)) * sqrt(u1);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.0175000001

   1. Initial program 98.8%

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

   3. Taylor expanded in u2 around 0 96.1%

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

   if 0.0175000001 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 97.8%

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

   3. Taylor expanded in u1 around 0 73.5%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.017500000074505806:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \left(6.28318530718 \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.3% accurate, 1.0× speedup

Math

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

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

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%

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

Alternative 6: 73.3% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = single(6.28318530718) * (u2 * sqrt((u1 * (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. Taylor expanded in u2 around 0 81.4%

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

4. Taylor expanded in u1 around 0 74.1%

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

   1. +-commutative 74.1%

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

6. Simplified 74.1%

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

7. Final simplification 74.1%

   \[\leadsto 6.28318530718 \cdot \left(u2 \cdot \sqrt{u1 \cdot \left(u1 + 1\right)}\right) \]
8. Add Preprocessing

Alternative 7: 81.7% 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 81.4%

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

4. Final simplification 81.4%

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

Alternative 8: 81.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((u1 / (single(1.0) - u1))) * (single(6.28318530718) * 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 81.6%

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

4. Final simplification 81.6%

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

Alternative 9: 64.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in u2 around 0 81.4%

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

4. Taylor expanded in u1 around 0 66.3%

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

5. Final simplification 66.3%

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

Alternative 10: 64.8% 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 81.4%

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

4. Taylor expanded in u1 around 0 66.3%

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

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

   \[\leadsto \color{blue}{-6.28318530718 \cdot \left(\sqrt{u1} \cdot \left(u2 \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)} \]
6. Step-by-step derivation

   1. *-commutative -0.0%

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

   2. associate-*r* -0.0%

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

   3. *-commutative -0.0%

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

   4. unpow2 -0.0%

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

   5. rem-square-sqrt 66.3%

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

   6. associate-*l* 66.3%

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

   7. metadata-eval 66.3%

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

   8. associate-*r* 66.3%

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

7. Simplified 66.3%

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

8. Final simplification 66.3%

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

Alternative 11: 64.8% accurate, 2.0× speedup

Math

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

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(Float32(6.28318530718) * 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 81.4%

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

4. Taylor expanded in u1 around 0 66.3%

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

   1. *-commutative 66.3%

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

   2. associate-*l* 66.3%

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

   3. *-commutative 66.3%

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

6. Simplified 66.3%

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

7. Final simplification 66.3%

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

Alternative 12: 20.6% accurate, 19.0× speedup

Math

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

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* 6.28318530718 (* u1 (+ u2 (* 0.5 (/ u2 u1))))))

C

float code(float cosTheta_i, float u1, float u2) {
	return 6.28318530718f * (u1 * (u2 + (0.5f * (u2 / u1))));
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = 6.28318530718e0 * (u1 * (u2 + (0.5e0 * (u2 / u1))))
end function

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = single(6.28318530718) * (u1 * (u2 + (single(0.5) * (u2 / 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 81.4%

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

4. Taylor expanded in u1 around 0 74.1%

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

   1. distribute-rgt-in 74.1%

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

   2. unpow2 74.1%

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

   3. *-lft-identity 74.1%

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

6. Simplified 74.1%

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

7. Taylor expanded in u1 around inf 20.1%

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

8. Final simplification 20.1%

   \[\leadsto 6.28318530718 \cdot \left(u1 \cdot \left(u2 + 0.5 \cdot \frac{u2}{u1}\right)\right) \]
9. Add Preprocessing

Alternative 13: 20.6% accurate, 19.0× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return 6.28318530718f * (u2 * (u1 * (1.0f + (0.5f / u1))));
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = 6.28318530718e0 * (u2 * (u1 * (1.0e0 + (0.5e0 / u1))))
end function

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = single(6.28318530718) * (u2 * (u1 * (single(1.0) + (single(0.5) / 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 81.4%

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

4. Taylor expanded in u1 around 0 74.1%

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

   1. distribute-rgt-in 74.1%

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

   2. unpow2 74.1%

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

   3. *-lft-identity 74.1%

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

6. Simplified 74.1%

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

7. Taylor expanded in u1 around inf 20.1%

   \[\leadsto 6.28318530718 \cdot \left(\color{blue}{\left(u1 \cdot \left(1 + 0.5 \cdot \frac{1}{u1}\right)\right)} \cdot u2\right) \]
8. Step-by-step derivation

   1. associate-*r/ 20.1%

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

   2. metadata-eval 20.1%

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

9. Simplified 20.1%

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

10. Final simplification 20.1%

    \[\leadsto 6.28318530718 \cdot \left(u2 \cdot \left(u1 \cdot \left(1 + \frac{0.5}{u1}\right)\right)\right) \]
11. Add Preprocessing

Alternative 14: 4.8% accurate, 41.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = (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 81.4%

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

4. Taylor expanded in u1 around 0 74.1%

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

   1. distribute-rgt-in 74.1%

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

   2. unpow2 74.1%

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

   3. *-lft-identity 74.1%

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

6. Simplified 74.1%

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

7. Taylor expanded in u1 around -inf 5.0%

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

8. Final simplification 5.0%

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

Alternative 15: 19.4% accurate, 41.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = single(6.28318530718) * (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 81.4%

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

4. Taylor expanded in u1 around 0 74.1%

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

   1. distribute-rgt-in 74.1%

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

   2. unpow2 74.1%

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

   3. *-lft-identity 74.1%

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

6. Simplified 74.1%

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

7. Taylor expanded in u1 around inf 19.1%

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

8. Final simplification 19.1%

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

Alternative 16: 19.4% accurate, 41.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = u2 * (u1 * 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 81.4%

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

4. Taylor expanded in u1 around 0 74.1%

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

   1. distribute-rgt-in 74.1%

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

   2. unpow2 74.1%

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

   3. *-lft-identity 74.1%

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

6. Simplified 74.1%

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

7. Taylor expanded in u1 around inf 19.1%

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

   1. *-commutative 19.1%

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

   2. associate-*r* 19.1%

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

   3. *-commutative 19.1%

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

   4. associate-*l* 19.1%

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

9. Simplified 19.1%

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

10. Final simplification 19.1%

    \[\leadsto u2 \cdot \left(u1 \cdot 6.28318530718\right) \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024076 
(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))))