Trowbridge-Reitz Sample, near normal, slope_x

Percentage Accurate: 99.0% → 99.0%

Time: 14.4s

Alternatives: 7

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: 99.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.8%

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

   1. add-sqr-sqrt 98.7%

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

   2. sqrt-unprod 98.8%

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

   3. swap-sqr 98.9%

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

   4. *-commutative 98.9%

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

   5. pow2 98.9%

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

   6. metadata-eval 99.0%

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

3. Applied egg-rr 99.0%

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

4. Final simplification 99.0%

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

Alternative 2: 89.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(u2 \cdot 6.28318530718\right)\\ \mathbf{if}\;t_0 \leq 0.9999099969863892:\\ \;\;\;\;t_0 \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* u2 6.28318530718))))
   (if (<= t_0 0.9999099969863892)
     (* t_0 (sqrt u1))
     (sqrt (/ u1 (- 1.0 u1))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = cosf((u2 * 6.28318530718f));
	float tmp;
	if (t_0 <= 0.9999099969863892f) {
		tmp = t_0 * sqrtf(u1);
	} else {
		tmp = 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) :: t_0
    real(4) :: tmp
    t_0 = cos((u2 * 6.28318530718e0))
    if (t_0 <= 0.9999099969863892e0) then
        tmp = t_0 * sqrt(u1)
    else
        tmp = sqrt((u1 / (1.0e0 - u1)))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	t_0 = cos(Float32(u2 * Float32(6.28318530718)))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.9999099969863892))
		tmp = Float32(t_0 * sqrt(u1));
	else
		tmp = sqrt(Float32(u1 / Float32(Float32(1.0) - u1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	t_0 = cos((u2 * single(6.28318530718)));
	tmp = single(0.0);
	if (t_0 <= single(0.9999099969863892))
		tmp = t_0 * sqrt(u1);
	else
		tmp = sqrt((u1 / (single(1.0) - u1)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 314159265359/50000000000 u2)) < 0.999909997

   1. Initial program 97.6%

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

   2. Taylor expanded in u1 around 0 72.7%

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

   if 0.999909997 < (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. Taylor expanded in u2 around 0 96.0%

      \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(u2 \cdot 6.28318530718\right) \leq 0.9999099969863892:\\ \;\;\;\;\cos \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}}\\ \end{array} \]

Alternative 3: 99.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 + u2 \cdot 6.28318530718\\ \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(\frac{t_0}{\frac{t_0}{u2 \cdot 6.28318530718}}\right) \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (+ 2.0 (* u2 6.28318530718))))
   (* (sqrt (/ u1 (- 1.0 u1))) (cos (/ t_0 (/ t_0 (* u2 6.28318530718)))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = 2.0f + (u2 * 6.28318530718f);
	return sqrtf((u1 / (1.0f - u1))) * cosf((t_0 / (t_0 / (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
    real(4) :: t_0
    t_0 = 2.0e0 + (u2 * 6.28318530718e0)
    code = sqrt((u1 / (1.0e0 - u1))) * cos((t_0 / (t_0 / (u2 * 6.28318530718e0))))
end function

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(Float32(2.0) + Float32(u2 * Float32(6.28318530718)))
	return Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * cos(Float32(t_0 / Float32(t_0 / Float32(u2 * Float32(6.28318530718))))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	t_0 = single(2.0) + (u2 * single(6.28318530718));
	tmp = sqrt((u1 / (single(1.0) - u1))) * cos((t_0 / (t_0 / (u2 * single(6.28318530718)))));
end

Derivation

1. Initial program 98.8%

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

   1. expm1-log1p-u 98.8%

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

3. Applied egg-rr 98.8%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(6.28318530718 \cdot u2\right)\right)\right)} \]
4. Step-by-step derivation

   1. expm1-udef 98.8%

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

   2. flip-- 98.8%

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

   3. log1p-udef 98.8%

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

   4. add-exp-log 98.9%

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

   5. *-commutative 98.9%

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

   6. log1p-udef 98.8%

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

   7. add-exp-log 98.8%

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

   8. *-commutative 98.8%

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

   9. metadata-eval 98.8%

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

   10. +-commutative 98.8%

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

   11. log1p-udef 98.8%

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

   12. add-exp-log 98.8%

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

   13. associate-+r+ 98.8%

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

   14. metadata-eval 98.8%

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

   15. *-commutative 98.8%

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

5. Applied egg-rr 98.8%

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

   1. difference-of-sqr-1 98.8%

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

   2. +-commutative 98.8%

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

   3. associate-+r+ 98.8%

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

   4. metadata-eval 98.8%

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

   5. +-commutative 98.8%

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

   6. associate--l+ 98.8%

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

   7. metadata-eval 98.8%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(\frac{\left(2 + u2 \cdot 6.28318530718\right) \cdot \left(u2 \cdot 6.28318530718 + \color{blue}{0}\right)}{2 + u2 \cdot 6.28318530718}\right) \]

   8. +-rgt-identity 98.8%

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

   9. associate-/l* 98.9%

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

7. Simplified 98.9%

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

8. Final simplification 98.9%

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

Alternative 4: 99.0% 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 98.8%

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

2. Final simplification 98.8%

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

Alternative 5: 79.9% 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 98.8%

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

2. Taylor expanded in u2 around 0 78.9%

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

3. Final simplification 78.9%

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

Alternative 6: 62.9% 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 98.8%

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

2. Taylor expanded in u2 around 0 78.9%

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

3. Taylor expanded in u1 around 0 62.3%

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

4. Final simplification 62.3%

   \[\leadsto \sqrt{u1} \]

Alternative 7: 19.2% 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 98.8%

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

2. Taylor expanded in u1 around 0 86.0%

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

   1. +-commutative 86.0%

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

   2. unpow2 86.0%

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

   3. fma-udef 86.0%

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

4. Simplified 86.0%

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

5. Taylor expanded in u1 around inf 19.9%

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

   1. *-commutative 19.9%

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

7. Simplified 19.9%

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

8. Taylor expanded in u2 around 0 19.1%

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

9. Final simplification 19.1%

   \[\leadsto u1 \]

Reproduce

herbie shell --seed 2023301 
(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))))