Trowbridge-Reitz Sample, near normal, slope_x

Percentage Accurate: 99.0% → 99.0%

Time: 9.8s

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: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in u1 around inf 98.8%

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

   1. pow1/2 98.8%

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

   2. pow-to-exp 96.2%

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

   3. associate-/r* 96.2%

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

   4. log-div 96.2%

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

   5. *-inverses 96.2%

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

   6. *-inverses 96.2%

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

   7. div-sub 96.3%

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

   8. log-div 96.3%

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

   9. sub-neg 96.3%

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

   10. rem-log-exp 95.9%

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

   11. clear-num 96.0%

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

   12. pow-to-exp 98.1%

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

   13. pow1/2 98.1%

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

   14. rem-log-exp 98.9%

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

   15. sub-neg 98.9%

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

5. Applied egg-rr 98.9%

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

   1. *-lft-identity 98.9%

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

7. Simplified 98.9%

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

8. Taylor expanded in u1 around 0 99.0%

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

   1. neg-mul-1 99.0%

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

   2. sub-neg 99.0%

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

10. Simplified 99.0%

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

Alternative 2: 89.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(6.28318530718 \cdot u2\right)\\ \mathbf{if}\;t\_0 \leq 0.9999499917030334:\\ \;\;\;\;t\_0 \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* 6.28318530718 u2))))
   (if (<= t_0 0.9999499917030334)
     (* t_0 (sqrt u1))
     (sqrt (/ u1 (- 1.0 u1))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = cosf((6.28318530718f * u2));
	float tmp;
	if (t_0 <= 0.9999499917030334f) {
		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((6.28318530718e0 * u2))
    if (t_0 <= 0.9999499917030334e0) 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(Float32(6.28318530718) * u2))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.9999499917030334))
		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((single(6.28318530718) * u2));
	tmp = single(0.0);
	if (t_0 <= single(0.9999499917030334))
		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 #s(literal 314159265359/50000000000 binary32) u2)) < 0.999949992

   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 70.7%

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

   if 0.999949992 < (cos.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2))

   1. Initial program 99.3%

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

   3. Taylor expanded in u2 around 0 96.1%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(6.28318530718 \cdot u2\right) \leq 0.9999499917030334:\\ \;\;\;\;\cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.00139999995008111:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1 \cdot \left(1 + u1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.00139999995008111)
   (sqrt (/ u1 (- 1.0 u1)))
   (* (cos (* 6.28318530718 u2)) (sqrt (* u1 (+ 1.0 u1))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.00139999995008111f) {
		tmp = sqrtf((u1 / (1.0f - u1)));
	} else {
		tmp = cosf((6.28318530718f * u2)) * 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 ((6.28318530718e0 * u2) <= 0.00139999995008111e0) then
        tmp = sqrt((u1 / (1.0e0 - u1)))
    else
        tmp = cos((6.28318530718e0 * u2)) * sqrt((u1 * (1.0e0 + 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(0.00139999995008111))
		tmp = sqrt(Float32(u1 / Float32(Float32(1.0) - u1)));
	else
		tmp = Float32(cos(Float32(Float32(6.28318530718) * u2)) * 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 ((single(6.28318530718) * u2) <= single(0.00139999995008111))
		tmp = sqrt((u1 / (single(1.0) - u1)));
	else
		tmp = cos((single(6.28318530718) * u2)) * 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) < 0.00139999995

   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%

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

   if 0.00139999995 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 98.1%

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

   3. Taylor expanded in u1 around 0 83.0%

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

      1. +-commutative 83.0%

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

   5. Simplified 83.0%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.00139999995008111:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1 \cdot \left(1 + u1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.009999999776482582:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(6.28318530718 \cdot u2\right)}{\sqrt{\frac{1}{u1}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.009999999776482582)
   (sqrt (/ u1 (- 1.0 u1)))
   (/ (cos (* 6.28318530718 u2)) (sqrt (/ 1.0 u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.009999999776482582f) {
		tmp = sqrtf((u1 / (1.0f - u1)));
	} else {
		tmp = cosf((6.28318530718f * u2)) / sqrtf((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 ((6.28318530718e0 * u2) <= 0.009999999776482582e0) then
        tmp = sqrt((u1 / (1.0e0 - u1)))
    else
        tmp = cos((6.28318530718e0 * u2)) / sqrt((1.0e0 / 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(0.009999999776482582))
		tmp = sqrt(Float32(u1 / Float32(Float32(1.0) - u1)));
	else
		tmp = Float32(cos(Float32(Float32(6.28318530718) * u2)) / sqrt(Float32(Float32(1.0) / u1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((single(6.28318530718) * u2) <= single(0.009999999776482582))
		tmp = sqrt((u1 / (single(1.0) - u1)));
	else
		tmp = cos((single(6.28318530718) * u2)) / sqrt((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) < 0.00999999978

   1. Initial program 99.3%

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

   3. Taylor expanded in u2 around 0 96.1%

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

   if 0.00999999978 < (*.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 inf 97.9%

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

      1. *-un-lft-identity 97.9%

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

      2. times-frac 98.0%

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

      3. sub-neg 98.0%

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

      4. metadata-eval 98.0%

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

   5. Applied egg-rr 98.0%

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

      1. add-sqr-sqrt 73.3%

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

      2. sqrt-unprod 74.8%

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

      3. swap-sqr 74.7%

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

      4. add-sqr-sqrt 74.9%

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

      5. frac-times 74.9%

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

      6. *-un-lft-identity 74.9%

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

      7. distribute-rgt-in 74.9%

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

      8. lft-mult-inverse 75.0%

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

      9. neg-mul-1 75.0%

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

      10. sub-neg 75.0%

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

      11. add-sqr-sqrt 74.8%

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

      12. rem-log-exp 64.4%

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

      13. rem-log-exp 62.7%

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

   7. Applied egg-rr 97.8%

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

   8. Taylor expanded in u1 around 0 70.8%

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

Alternative 5: 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 98.9%

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

3. Final simplification 98.9%

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

Alternative 6: 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 98.9%

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

3. Taylor expanded in u2 around 0 78.3%

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

Alternative 7: 71.6% 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 98.9%

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

3. Taylor expanded in u2 around 0 78.3%

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

4. Taylor expanded in u1 around 0 69.9%

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

   1. +-commutative 85.1%

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

6. Simplified 69.9%

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

7. Final simplification 69.9%

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

Alternative 8: 63.3% 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 98.9%

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

3. Taylor expanded in u2 around 0 78.3%

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

4. Taylor expanded in u1 around 0 62.5%

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

Reproduce

herbie shell --seed 2024167 
(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))))