Trowbridge-Reitz Sample, near normal, slope_x

Percentage Accurate: 98.9% → 98.9%

Time: 10.3s

Alternatives: 10

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = (((single(1.0) / u1) + single(-1.0)) ^ single(-0.5)) * cos((u2 * single(6.28318530718)));
end

Derivation

1. Initial program 99.1%

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

   1. add-sqr-sqrt 95.3%

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

   2. sqrt-unprod 96.5%

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

   3. swap-sqr 96.5%

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

   4. add-sqr-sqrt 96.6%

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

   5. pow2 96.6%

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

4. Applied egg-rr 96.6%

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

   1. *-lft-identity 96.6%

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

   2. associate-*l/ 96.4%

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

   3. associate-/r/ 96.5%

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

   4. associate-*l/ 96.4%

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

   5. *-lft-identity 96.4%

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

   6. div-sub 96.4%

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

   7. sub-neg 96.4%

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

   8. *-inverses 96.4%

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

   9. metadata-eval 96.4%

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

6. Simplified 96.4%

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

7. Taylor expanded in u2 around inf 99.0%

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

   1. *-commutative 99.0%

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

   2. *-commutative 99.0%

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

   3. sub-neg 99.0%

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

   4. metadata-eval 99.0%

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

9. Simplified 99.0%

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

    1. inv-pow 99.0%

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

    2. sqrt-pow1 99.1%

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

    3. metadata-eval 99.1%

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

11. Applied egg-rr 99.1%

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

Alternative 2: 93.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.0012000000569969416:\\ \;\;\;\;{\left(\frac{1}{u1} + -1\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1 \cdot \left(1 + u1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 6.28318530718) 0.0012000000569969416)
   (pow (+ (/ 1.0 u1) -1.0) -0.5)
   (* (cos (* u2 6.28318530718)) (sqrt (* u1 (+ 1.0 u1))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * 6.28318530718f) <= 0.0012000000569969416f) {
		tmp = powf(((1.0f / u1) + -1.0f), -0.5f);
	} else {
		tmp = cosf((u2 * 6.28318530718f)) * sqrtf((u1 * (1.0f + 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) <= 0.0012000000569969416e0) then
        tmp = ((1.0e0 / u1) + (-1.0e0)) ** (-0.5e0)
    else
        tmp = cos((u2 * 6.28318530718e0)) * sqrt((u1 * (1.0e0 + 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(0.0012000000569969416))
		tmp = Float32(Float32(Float32(1.0) / u1) + Float32(-1.0)) ^ Float32(-0.5);
	else
		tmp = Float32(cos(Float32(u2 * Float32(6.28318530718))) * sqrt(Float32(u1 * Float32(Float32(1.0) + u1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((u2 * single(6.28318530718)) <= single(0.0012000000569969416))
		tmp = ((single(1.0) / u1) + single(-1.0)) ^ single(-0.5);
	else
		tmp = cos((u2 * single(6.28318530718))) * sqrt((u1 * (single(1.0) + u1)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.00120000006

   1. Initial program 99.5%

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

      1. add-sqr-sqrt 98.1%

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

      2. sqrt-unprod 99.5%

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

      3. swap-sqr 99.4%

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

      4. add-sqr-sqrt 99.5%

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

      5. pow2 99.5%

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

   4. Applied egg-rr 99.5%

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

      1. *-lft-identity 99.5%

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

      2. associate-*l/ 99.3%

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

      3. associate-/r/ 99.4%

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

      4. associate-*l/ 99.4%

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

      5. *-lft-identity 99.4%

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

      6. div-sub 99.3%

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

      7. sub-neg 99.3%

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

      8. *-inverses 99.3%

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

      9. metadata-eval 99.3%

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

   6. Simplified 99.3%

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

   7. Taylor expanded in u2 around inf 99.4%

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

      1. *-commutative 99.4%

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

      2. *-commutative 99.4%

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

      3. sub-neg 99.4%

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

      4. metadata-eval 99.4%

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

   9. Simplified 99.4%

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

   10. Taylor expanded in u2 around 0 98.9%

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

       1. sub-neg 98.9%

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

       2. metadata-eval 98.9%

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

       3. unpow-1 98.9%

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

       4. metadata-eval 98.9%

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

       5. pow-sqr 99.1%

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

       6. rem-sqrt-square 99.1%

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

       7. rem-square-sqrt 97.7%

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

       8. fabs-sqr 97.7%

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

       9. rem-square-sqrt 99.1%

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

       10. +-commutative 99.1%

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

   12. Simplified 99.1%

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

   if 0.00120000006 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 98.4%

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

   3. Taylor expanded in u1 around 0 85.3%

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

      1. +-commutative 85.3%

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

   5. Simplified 85.3%

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

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.0012000000569969416:\\ \;\;\;\;{\left(\frac{1}{u1} + -1\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1 \cdot \left(1 + u1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.019999999552965164:\\ \;\;\;\;{\left(\frac{1}{u1} + -1\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 6.28318530718) 0.019999999552965164)
   (pow (+ (/ 1.0 u1) -1.0) -0.5)
   (* (cos (* u2 6.28318530718)) (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * 6.28318530718f) <= 0.019999999552965164f) {
		tmp = powf(((1.0f / u1) + -1.0f), -0.5f);
	} else {
		tmp = cosf((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) <= 0.019999999552965164e0) then
        tmp = ((1.0e0 / u1) + (-1.0e0)) ** (-0.5e0)
    else
        tmp = cos((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(0.019999999552965164))
		tmp = Float32(Float32(Float32(1.0) / u1) + Float32(-1.0)) ^ Float32(-0.5);
	else
		tmp = Float32(cos(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(0.019999999552965164))
		tmp = ((single(1.0) / u1) + single(-1.0)) ^ single(-0.5);
	else
		tmp = cos((u2 * single(6.28318530718))) * sqrt(u1);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.0199999996

   1. Initial program 99.4%

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

      1. add-sqr-sqrt 98.1%

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

      2. sqrt-unprod 99.4%

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

      3. swap-sqr 99.4%

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

      4. add-sqr-sqrt 99.4%

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

      5. pow2 99.4%

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

   4. Applied egg-rr 99.4%

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

      1. *-lft-identity 99.4%

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

      2. associate-*l/ 99.3%

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

      3. associate-/r/ 99.3%

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

      4. associate-*l/ 99.2%

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

      5. *-lft-identity 99.2%

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

      6. div-sub 99.2%

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

      7. sub-neg 99.2%

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

      8. *-inverses 99.2%

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

      9. metadata-eval 99.2%

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

   6. Simplified 99.2%

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

   7. Taylor expanded in u2 around inf 99.3%

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

      1. *-commutative 99.3%

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

      2. *-commutative 99.3%

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

      3. sub-neg 99.3%

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

      4. metadata-eval 99.3%

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

   9. Simplified 99.3%

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

   10. Taylor expanded in u2 around 0 95.8%

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

       1. sub-neg 95.8%

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

       2. metadata-eval 95.8%

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

       3. unpow-1 95.8%

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

       4. metadata-eval 95.8%

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

       5. pow-sqr 95.9%

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

       6. rem-sqrt-square 95.9%

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

       7. rem-square-sqrt 94.8%

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

       8. fabs-sqr 94.8%

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

       9. rem-square-sqrt 95.9%

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

       10. +-commutative 95.9%

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

   12. Simplified 95.9%

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

   if 0.0199999996 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 97.9%

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

   3. Taylor expanded in u1 around 0 73.8%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.019999999552965164:\\ \;\;\;\;{\left(\frac{1}{u1} + -1\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

3. Final simplification 99.1%

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

Alternative 5: 79.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

   1. add-sqr-sqrt 95.3%

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

   2. sqrt-unprod 96.5%

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

   3. swap-sqr 96.5%

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

   4. add-sqr-sqrt 96.6%

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

   5. pow2 96.6%

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

4. Applied egg-rr 96.6%

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

   1. *-lft-identity 96.6%

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

   2. associate-*l/ 96.4%

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

   3. associate-/r/ 96.5%

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

   4. associate-*l/ 96.4%

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

   5. *-lft-identity 96.4%

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

   6. div-sub 96.4%

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

   7. sub-neg 96.4%

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

   8. *-inverses 96.4%

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

   9. metadata-eval 96.4%

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

6. Simplified 96.4%

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

7. Taylor expanded in u2 around inf 99.0%

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

   1. *-commutative 99.0%

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

   2. *-commutative 99.0%

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

   3. sub-neg 99.0%

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

   4. metadata-eval 99.0%

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

9. Simplified 99.0%

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

10. Taylor expanded in u2 around 0 82.7%

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

    1. sub-neg 82.7%

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

    2. metadata-eval 82.7%

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

    3. unpow-1 82.7%

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

    4. metadata-eval 82.7%

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

    5. pow-sqr 82.8%

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

    6. rem-sqrt-square 82.8%

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

    7. rem-square-sqrt 81.9%

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

    8. fabs-sqr 81.9%

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

    9. rem-square-sqrt 82.8%

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

    10. +-commutative 82.8%

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

12. Simplified 82.8%

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

13. Final simplification 82.8%

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

Alternative 6: 79.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in u2 around 0 82.7%

   \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]
4. Add Preprocessing

Alternative 7: 71.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in u2 around 0 82.7%

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

4. Taylor expanded in u1 around 0 72.3%

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

   1. +-commutative 84.7%

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

6. Simplified 72.3%

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

7. Final simplification 72.3%

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

Alternative 8: 63.6% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in u2 around 0 82.7%

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

4. Taylor expanded in u1 around 0 63.6%

   \[\leadsto \sqrt{\color{blue}{u1}} \]
5. Add Preprocessing

Alternative 9: 20.1% accurate, 29.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in u2 around 0 82.7%

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

4. Taylor expanded in u1 around 0 72.3%

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

   1. distribute-lft-in 72.4%

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

   2. *-rgt-identity 72.4%

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

   3. unpow2 72.4%

      \[\leadsto \sqrt{u1 + \color{blue}{{u1}^{2}}} \]

6. Simplified 72.4%

   \[\leadsto \sqrt{\color{blue}{u1 + {u1}^{2}}} \]

7. Taylor expanded in u1 around inf 21.1%

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

   1. associate-*r/ 21.1%

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

   2. metadata-eval 21.1%

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

9. Simplified 21.1%

   \[\leadsto \color{blue}{u1 \cdot \left(1 + \frac{0.5}{u1}\right)} \]
10. Add Preprocessing

Alternative 10: 19.0% accurate, 209.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in u2 around 0 82.7%

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

4. Taylor expanded in u1 around 0 72.3%

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

   1. distribute-lft-in 72.4%

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

   2. *-rgt-identity 72.4%

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

   3. unpow2 72.4%

      \[\leadsto \sqrt{u1 + \color{blue}{{u1}^{2}}} \]

6. Simplified 72.4%

   \[\leadsto \sqrt{\color{blue}{u1 + {u1}^{2}}} \]

7. Taylor expanded in u1 around inf 5.4%

   \[\leadsto \color{blue}{u1 \cdot \left(\left(1 + 0.5 \cdot \frac{1}{u1}\right) - \frac{0.125}{{u1}^{2}}\right)} \]
8. Step-by-step derivation

   1. associate-*r/ 5.4%

      \[\leadsto u1 \cdot \left(\left(1 + \color{blue}{\frac{0.5 \cdot 1}{u1}}\right) - \frac{0.125}{{u1}^{2}}\right) \]

   2. metadata-eval 5.4%

      \[\leadsto u1 \cdot \left(\left(1 + \frac{\color{blue}{0.5}}{u1}\right) - \frac{0.125}{{u1}^{2}}\right) \]

9. Simplified 5.4%

   \[\leadsto \color{blue}{u1 \cdot \left(\left(1 + \frac{0.5}{u1}\right) - \frac{0.125}{{u1}^{2}}\right)} \]

10. Taylor expanded in u1 around inf 19.8%

    \[\leadsto \color{blue}{u1} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024106 
(FPCore (cosTheta_i u1 u2)
  :name "Trowbridge-Reitz Sample, near normal, slope_x"
  :precision binary32
  :pre (and (and (and (> cosTheta_i 0.9999) (<= cosTheta_i 1.0)) (and (<= 2.328306437e-10 u1) (<= u1 1.0))) (and (<= 2.328306437e-10 u2) (<= u2 1.0)))
  (* (sqrt (/ u1 (- 1.0 u1))) (cos (* 6.28318530718 u2))))