Trowbridge-Reitz Sample, near normal, slope_x

Percentage Accurate: 99.0% → 98.9%

Time: 11.3s

Alternatives: 8

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = cos((single(6.28318530718) * u2)) * ((single(-1.0) + (single(1.0) / u1)) ^ 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. clear-num 99.0%

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

   2. sqrt-div 98.7%

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

   3. metadata-eval 98.7%

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

4. Applied egg-rr 98.7%

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

   1. div-sub 98.8%

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

   2. sub-neg 98.8%

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

   3. *-inverses 98.8%

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

   4. metadata-eval 98.8%

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

6. Simplified 98.8%

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

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. sub-neg 99.0%

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

   2. metadata-eval 99.0%

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

   3. unpow-1 99.0%

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

   4. metadata-eval 99.0%

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

   5. pow-sqr 99.2%

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

   6. rem-sqrt-square 99.2%

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

   7. sqr-pow 98.0%

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

   8. fabs-sqr 98.0%

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

   9. sqr-pow 99.2%

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

   10. +-commutative 99.2%

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

9. Simplified 99.2%

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

10. Final simplification 99.2%

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

Alternative 2: 94.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(6.28318530718 \cdot u2\right)\\ \mathbf{if}\;t_0 \leq 0.9999979734420776:\\ \;\;\;\;t_0 \cdot \sqrt{u1 \cdot \left(1 + u1\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(-1 + \frac{1}{u1}\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* 6.28318530718 u2))))
   (if (<= t_0 0.9999979734420776)
     (* t_0 (sqrt (* u1 (+ 1.0 u1))))
     (pow (+ -1.0 (/ 1.0 u1)) -0.5))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = cosf((6.28318530718f * u2));
	float tmp;
	if (t_0 <= 0.9999979734420776f) {
		tmp = t_0 * sqrtf((u1 * (1.0f + u1)));
	} else {
		tmp = powf((-1.0f + (1.0f / u1)), -0.5f);
	}
	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) :: t_0
    real(4) :: tmp
    t_0 = cos((6.28318530718e0 * u2))
    if (t_0 <= 0.9999979734420776e0) then
        tmp = t_0 * sqrt((u1 * (1.0e0 + u1)))
    else
        tmp = ((-1.0e0) + (1.0e0 / u1)) ** (-0.5e0)
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	t_0 = cos(Float32(Float32(6.28318530718) * u2))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.9999979734420776))
		tmp = Float32(t_0 * sqrt(Float32(u1 * Float32(Float32(1.0) + u1))));
	else
		tmp = Float32(Float32(-1.0) + Float32(Float32(1.0) / u1)) ^ Float32(-0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	t_0 = cos((single(6.28318530718) * u2));
	tmp = single(0.0);
	if (t_0 <= single(0.9999979734420776))
		tmp = t_0 * sqrt((u1 * (single(1.0) + u1)));
	else
		tmp = (single(-1.0) + (single(1.0) / u1)) ^ single(-0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 314159265359/50000000000 u2)) < 0.999997973

   1. Initial program 98.5%

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

      1. clear-num 98.5%

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

      2. associate-/r/ 98.4%

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

   4. Applied egg-rr 98.4%

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

   5. Taylor expanded in u1 around 0 85.7%

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

      1. +-commutative 85.7%

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

   7. Simplified 85.7%

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

   if 0.999997973 < (cos.f32 (*.f32 314159265359/50000000000 u2))

   1. Initial program 99.4%

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

   3. Taylor expanded in u2 around 0 98.8%

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

      1. sqrt-div 98.4%

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

      2. div-inv 98.0%

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

   5. Applied egg-rr 98.0%

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

      1. associate-*r/ 98.4%

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

      2. *-rgt-identity 98.4%

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

   7. Simplified 98.4%

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

      1. sqrt-undiv 98.8%

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

      2. clear-num 98.6%

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

      3. inv-pow 98.6%

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

      4. sqrt-pow1 98.8%

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

      5. div-sub 98.8%

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

      6. *-inverses 98.8%

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

      7. sub-neg 98.8%

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

      8. metadata-eval 98.8%

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

      9. metadata-eval 98.8%

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

   9. Applied egg-rr 98.8%

      \[\leadsto \color{blue}{{\left(\frac{1}{u1} + -1\right)}^{-0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(6.28318530718 \cdot u2\right) \leq 0.9999979734420776:\\ \;\;\;\;\cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1 \cdot \left(1 + u1\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(-1 + \frac{1}{u1}\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(6.28318530718 \cdot u2\right)\\ \mathbf{if}\;t_0 \leq 0.9999880194664001:\\ \;\;\;\;t_0 \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;{\left(-1 + \frac{1}{u1}\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* 6.28318530718 u2))))
   (if (<= t_0 0.9999880194664001)
     (* t_0 (sqrt u1))
     (pow (+ -1.0 (/ 1.0 u1)) -0.5))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = cosf((6.28318530718f * u2));
	float tmp;
	if (t_0 <= 0.9999880194664001f) {
		tmp = t_0 * sqrtf(u1);
	} else {
		tmp = powf((-1.0f + (1.0f / u1)), -0.5f);
	}
	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) :: t_0
    real(4) :: tmp
    t_0 = cos((6.28318530718e0 * u2))
    if (t_0 <= 0.9999880194664001e0) then
        tmp = t_0 * sqrt(u1)
    else
        tmp = ((-1.0e0) + (1.0e0 / u1)) ** (-0.5e0)
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	t_0 = cos(Float32(Float32(6.28318530718) * u2))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.9999880194664001))
		tmp = Float32(t_0 * sqrt(u1));
	else
		tmp = Float32(Float32(-1.0) + Float32(Float32(1.0) / u1)) ^ Float32(-0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	t_0 = cos((single(6.28318530718) * u2));
	tmp = single(0.0);
	if (t_0 <= single(0.9999880194664001))
		tmp = t_0 * sqrt(u1);
	else
		tmp = (single(-1.0) + (single(1.0) / u1)) ^ single(-0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 314159265359/50000000000 u2)) < 0.999988019

   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 71.3%

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

   if 0.999988019 < (cos.f32 (*.f32 314159265359/50000000000 u2))

   1. Initial program 99.4%

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

   3. Taylor expanded in u2 around 0 97.8%

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

      1. sqrt-div 97.5%

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

      2. div-inv 97.1%

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

   5. Applied egg-rr 97.1%

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

      1. associate-*r/ 97.5%

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

      2. *-rgt-identity 97.5%

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

   7. Simplified 97.5%

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

      1. sqrt-undiv 97.8%

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

      2. clear-num 97.7%

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

      3. inv-pow 97.7%

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

      4. sqrt-pow1 97.9%

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

      5. div-sub 97.8%

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

      6. *-inverses 97.8%

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

      7. sub-neg 97.8%

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

      8. metadata-eval 97.8%

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

      9. metadata-eval 97.8%

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

   9. Applied egg-rr 97.8%

      \[\leadsto \color{blue}{{\left(\frac{1}{u1} + -1\right)}^{-0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(6.28318530718 \cdot u2\right) \leq 0.9999880194664001:\\ \;\;\;\;\cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;{\left(-1 + \frac{1}{u1}\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

3. Final simplification 99.1%

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

Alternative 5: 79.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = (single(-1.0) + (single(1.0) / u1)) ^ 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. Taylor expanded in u2 around 0 80.6%

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

   1. sqrt-div 80.4%

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

   2. div-inv 80.1%

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

5. Applied egg-rr 80.1%

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

   1. associate-*r/ 80.4%

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

   2. *-rgt-identity 80.4%

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

7. Simplified 80.4%

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

   1. sqrt-undiv 80.6%

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

   2. clear-num 80.5%

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

   3. inv-pow 80.5%

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

   4. sqrt-pow1 80.6%

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

   5. div-sub 80.6%

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

   6. *-inverses 80.6%

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

   7. sub-neg 80.6%

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

   8. metadata-eval 80.6%

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

   9. metadata-eval 80.6%

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

9. Applied egg-rr 80.6%

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

10. Final simplification 80.6%

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

Alternative 6: 79.7% 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 80.6%

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

4. Final simplification 80.6%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \]
5. Add Preprocessing

Alternative 7: 62.8% 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 80.6%

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

4. Taylor expanded in u1 around 0 64.8%

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

5. Final simplification 64.8%

   \[\leadsto \sqrt{u1} \]
6. Add Preprocessing

Alternative 8: 19.1% 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 u1 around 0 87.9%

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

   1. +-commutative 87.9%

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

   2. unpow2 87.9%

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

   3. fma-def 87.9%

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

5. Simplified 87.9%

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

6. Taylor expanded in u1 around inf 19.7%

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

7. Taylor expanded in u2 around 0 19.0%

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

8. Final simplification 19.0%

   \[\leadsto u1 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024024 
(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))))