Trowbridge-Reitz Sample, near normal, slope_y

Percentage Accurate: 98.3% → 98.3%

Time: 11.5s

Alternatives: 11

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.2%

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

   1. clear-num 98.1%

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

   2. inv-pow 98.1%

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

3. Applied egg-rr 98.1%

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

   1. unpow-1 98.1%

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

   2. clear-num 98.2%

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

   3. sqrt-div 97.8%

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

   4. associate-*l/ 98.0%

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

   5. *-commutative 98.0%

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

   6. associate-/l* 97.9%

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

   7. sqrt-div 98.2%

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

   8. div-sub 98.2%

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

   9. *-inverses 98.2%

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

   10. sub-neg 98.2%

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

   11. metadata-eval 98.2%

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

5. Applied egg-rr 98.2%

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

6. Final simplification 98.2%

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

Alternative 2: 90.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.01600000075995922:\\ \;\;\;\;6.28318530718 \cdot \left(u2 \cdot {\left(\frac{1}{u1} + -1\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.01600000075995922)
   (* 6.28318530718 (* u2 (pow (+ (/ 1.0 u1) -1.0) -0.5)))
   (* (sin (* 6.28318530718 u2)) (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.01600000075995922f) {
		tmp = 6.28318530718f * (u2 * powf(((1.0f / u1) + -1.0f), -0.5f));
	} else {
		tmp = sinf((6.28318530718f * u2)) * sqrtf(u1);
	}
	return tmp;
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    real(4) :: tmp
    if ((6.28318530718e0 * u2) <= 0.01600000075995922e0) then
        tmp = 6.28318530718e0 * (u2 * (((1.0e0 / u1) + (-1.0e0)) ** (-0.5e0)))
    else
        tmp = sin((6.28318530718e0 * u2)) * sqrt(u1)
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(6.28318530718) * u2) <= Float32(0.01600000075995922))
		tmp = Float32(Float32(6.28318530718) * Float32(u2 * (Float32(Float32(Float32(1.0) / u1) + Float32(-1.0)) ^ Float32(-0.5))));
	else
		tmp = Float32(sin(Float32(Float32(6.28318530718) * u2)) * sqrt(u1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((single(6.28318530718) * u2) <= single(0.01600000075995922))
		tmp = single(6.28318530718) * (u2 * (((single(1.0) / u1) + single(-1.0)) ^ single(-0.5)));
	else
		tmp = sin((single(6.28318530718) * u2)) * sqrt(u1);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 314159265359/50000000000 u2) < 0.0160000008

   1. Initial program 98.4%

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

   2. Taylor expanded in u2 around 0 96.1%

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

      1. clear-num 96.0%

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

      2. unpow-1 96.0%

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

      3. sqrt-pow1 96.2%

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

      4. div-sub 96.1%

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

      5. *-inverses 96.1%

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

      6. sub-neg 96.1%

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

      7. metadata-eval 96.1%

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

      8. metadata-eval 96.1%

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

   4. Applied egg-rr 96.1%

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

   if 0.0160000008 < (*.f32 314159265359/50000000000 u2)

   1. Initial program 97.8%

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

   2. Taylor expanded in u1 around 0 74.8%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.01600000075995922:\\ \;\;\;\;6.28318530718 \cdot \left(u2 \cdot {\left(\frac{1}{u1} + -1\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \]

Alternative 3: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return sinf((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 = sin((6.28318530718e0 * u2)) * sqrt((u1 / (1.0e0 - u1)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.2%

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

2. Final simplification 98.2%

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

Alternative 4: 81.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.2%

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

2. Taylor expanded in u2 around 0 81.3%

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

   1. clear-num 81.3%

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

   2. unpow-1 81.3%

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

   3. sqrt-pow1 81.4%

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

   4. div-sub 81.4%

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

   5. *-inverses 81.4%

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

   6. sub-neg 81.4%

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

   7. metadata-eval 81.4%

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

   8. metadata-eval 81.4%

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

4. Applied egg-rr 81.4%

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

5. Final simplification 81.4%

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

Alternative 5: 81.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.2%

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

2. Taylor expanded in u2 around 0 81.3%

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

3. Final simplification 81.3%

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

Alternative 6: 64.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.2%

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

2. Taylor expanded in u2 around 0 81.3%

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

3. Taylor expanded in u1 around 0 65.2%

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

4. Final simplification 65.2%

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

Alternative 7: 64.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.2%

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

2. Taylor expanded in u2 around 0 81.3%

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

3. Taylor expanded in u1 around 0 65.2%

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

   1. expm1-log1p-u 65.2%

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

   2. expm1-udef 27.1%

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

   3. *-commutative 27.1%

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

   4. associate-*r* 27.1%

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

5. Applied egg-rr 27.1%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\right)} - 1} \]
6. Step-by-step derivation

   1. expm1-def 65.2%

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

   2. expm1-log1p 65.2%

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

7. Simplified 65.2%

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

8. Final simplification 65.2%

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

Alternative 8: 20.6% accurate, 29.9× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return 6.28318530718f * (u2 * (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 = 6.28318530718e0 * (u2 * (u1 + 0.5e0))
end function

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = single(6.28318530718) * (u2 * (u1 + single(0.5)));
end

Derivation

1. Initial program 98.2%

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

2. Taylor expanded in u2 around 0 81.3%

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

3. Taylor expanded in u1 around 0 73.1%

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

   1. +-commutative 73.1%

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

   2. unpow2 73.1%

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

   3. fma-def 73.1%

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

5. Simplified 73.1%

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

6. Taylor expanded in u1 around inf 20.7%

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

   1. +-commutative 20.7%

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

8. Simplified 20.7%

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

9. Final simplification 20.7%

   \[\leadsto 6.28318530718 \cdot \left(u2 \cdot \left(u1 + 0.5\right)\right) \]

Alternative 9: 4.8% accurate, 41.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.2%

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

2. Taylor expanded in u2 around 0 81.3%

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

3. Taylor expanded in u1 around 0 73.1%

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

   1. +-commutative 73.1%

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

   2. unpow2 73.1%

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

   3. fma-def 73.1%

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

5. Simplified 73.1%

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

6. Taylor expanded in u1 around -inf 4.7%

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

7. Final simplification 4.7%

   \[\leadsto \left(u2 \cdot u1\right) \cdot -6.28318530718 \]

Alternative 10: 19.4% accurate, 41.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.2%

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

2. Taylor expanded in u2 around 0 81.3%

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

3. Taylor expanded in u1 around 0 73.1%

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

   1. +-commutative 73.1%

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

   2. unpow2 73.1%

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

   3. fma-def 73.1%

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

5. Simplified 73.1%

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

6. Taylor expanded in u1 around inf 19.4%

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

7. Final simplification 19.4%

   \[\leadsto 6.28318530718 \cdot \left(u2 \cdot u1\right) \]

Alternative 11: 19.4% accurate, 41.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.2%

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

2. Taylor expanded in u2 around 0 81.3%

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

3. Taylor expanded in u1 around 0 73.1%

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

   1. +-commutative 73.1%

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

   2. unpow2 73.1%

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

   3. fma-def 73.1%

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

5. Simplified 73.1%

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

6. Taylor expanded in u1 around inf 19.4%

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

   1. associate-*r* 19.4%

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

8. Simplified 19.4%

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

9. Final simplification 19.4%

   \[\leadsto u2 \cdot \left(6.28318530718 \cdot u1\right) \]

Reproduce

herbie shell --seed 2023318 
(FPCore (cosTheta_i u1 u2)
  :name "Trowbridge-Reitz Sample, near normal, slope_y"
  :precision binary32
  :pre (and (and (and (> cosTheta_i 0.9999) (<= cosTheta_i 1.0)) (and (<= 2.328306437e-10 u1) (<= u1 1.0))) (and (<= 2.328306437e-10 u2) (<= u2 1.0)))
  (* (sqrt (/ u1 (- 1.0 u1))) (sin (* 6.28318530718 u2))))