Trowbridge-Reitz Sample, near normal, slope_x

Percentage Accurate: 98.9% → 98.9%

Time: 9.7s

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: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.9% accurate, 1.0× speedup

Math

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

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

Derivation

1. Initial program 98.9%

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

2. Final simplification 98.9%

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

Alternative 2: 96.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(6.28318530718 \cdot u2\right)\\ \mathbf{if}\;t_0 \leq 0.9983000159263611:\\ \;\;\;\;t_0 \cdot \sqrt{u1 + u1 \cdot u1}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + -19.739208802181317 \cdot \left(u2 \cdot u2\right)\right) \cdot {\left(-1 + \frac{1}{u1}\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* 6.28318530718 u2))))
   (if (<= t_0 0.9983000159263611)
     (* t_0 (sqrt (+ u1 (* u1 u1))))
     (*
      (+ 1.0 (* -19.739208802181317 (* u2 u2)))
      (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 <= 0.9983000159263611f) {
		tmp = t_0 * sqrtf((u1 + (u1 * u1)));
	} else {
		tmp = (1.0f + (-19.739208802181317f * (u2 * u2))) * 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 <= 0.9983000159263611e0) then
        tmp = t_0 * sqrt((u1 + (u1 * u1)))
    else
        tmp = (1.0e0 + ((-19.739208802181317e0) * (u2 * u2))) * (((-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(0.9983000159263611))
		tmp = Float32(t_0 * sqrt(Float32(u1 + Float32(u1 * u1))));
	else
		tmp = Float32(Float32(Float32(1.0) + Float32(Float32(-19.739208802181317) * Float32(u2 * u2))) * (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(0.9983000159263611))
		tmp = t_0 * sqrt((u1 + (u1 * u1)));
	else
		tmp = (single(1.0) + (single(-19.739208802181317) * (u2 * u2))) * ((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)) < 0.998300016

   1. Initial program 97.3%

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

      1. clear-num 97.4%

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

      2. inv-pow 97.4%

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

      3. div-sub 97.3%

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

      4. pow1 97.3%

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

      5. pow1 97.3%

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

      6. pow-div 97.3%

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

      7. metadata-eval 97.3%

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

      8. metadata-eval 97.3%

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

   3. Applied egg-rr 97.3%

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

      1. unpow-1 97.3%

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

      2. sub-neg 97.3%

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

      3. metadata-eval 97.3%

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

   5. Simplified 97.3%

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

   6. Taylor expanded in u1 around 0 89.9%

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

      1. +-commutative 89.9%

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

      2. unpow2 89.9%

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

   8. Simplified 89.9%

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

   if 0.998300016 < (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. Step-by-step derivation

      1. clear-num 99.2%

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

      2. inv-pow 99.2%

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

      3. div-sub 99.3%

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

      4. pow1 99.3%

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

      5. pow1 99.3%

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

      6. pow-div 99.3%

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

      7. metadata-eval 99.3%

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

      8. metadata-eval 99.3%

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

   3. Applied egg-rr 99.3%

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

      1. unpow-1 99.3%

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

      2. sub-neg 99.3%

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

      3. metadata-eval 99.3%

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

   5. Simplified 99.3%

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

   6. Taylor expanded in u2 around 0 99.3%

      \[\leadsto \color{blue}{-19.739208802181317 \cdot \left({u2}^{2} \cdot \sqrt{\frac{1}{\frac{1}{u1} - 1}}\right) + \sqrt{\frac{1}{\frac{1}{u1} - 1}}} \]
   7. Step-by-step derivation

      1. associate-*r* 99.3%

         \[\leadsto \color{blue}{\left(-19.739208802181317 \cdot {u2}^{2}\right) \cdot \sqrt{\frac{1}{\frac{1}{u1} - 1}}} + \sqrt{\frac{1}{\frac{1}{u1} - 1}} \]

      2. distribute-lft1-in 99.2%

         \[\leadsto \color{blue}{\left(-19.739208802181317 \cdot {u2}^{2} + 1\right) \cdot \sqrt{\frac{1}{\frac{1}{u1} - 1}}} \]

      3. unpow2 99.2%

         \[\leadsto \left(-19.739208802181317 \cdot \color{blue}{\left(u2 \cdot u2\right)} + 1\right) \cdot \sqrt{\frac{1}{\frac{1}{u1} - 1}} \]

      4. sub-neg 99.2%

         \[\leadsto \left(-19.739208802181317 \cdot \left(u2 \cdot u2\right) + 1\right) \cdot \sqrt{\frac{1}{\color{blue}{\frac{1}{u1} + \left(-1\right)}}} \]

      5. metadata-eval 99.2%

         \[\leadsto \left(-19.739208802181317 \cdot \left(u2 \cdot u2\right) + 1\right) \cdot \sqrt{\frac{1}{\frac{1}{u1} + \color{blue}{-1}}} \]

      6. unpow-1 99.2%

         \[\leadsto \left(-19.739208802181317 \cdot \left(u2 \cdot u2\right) + 1\right) \cdot \sqrt{\color{blue}{{\left(\frac{1}{u1} + -1\right)}^{-1}}} \]

      7. metadata-eval 99.2%

         \[\leadsto \left(-19.739208802181317 \cdot \left(u2 \cdot u2\right) + 1\right) \cdot \sqrt{{\left(\frac{1}{u1} + -1\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]

      8. pow-sqr 99.4%

         \[\leadsto \left(-19.739208802181317 \cdot \left(u2 \cdot u2\right) + 1\right) \cdot \sqrt{\color{blue}{{\left(\frac{1}{u1} + -1\right)}^{-0.5} \cdot {\left(\frac{1}{u1} + -1\right)}^{-0.5}}} \]

      9. rem-sqrt-square 99.4%

         \[\leadsto \left(-19.739208802181317 \cdot \left(u2 \cdot u2\right) + 1\right) \cdot \color{blue}{\left|{\left(\frac{1}{u1} + -1\right)}^{-0.5}\right|} \]

      10. sqr-pow 98.3%

          \[\leadsto \left(-19.739208802181317 \cdot \left(u2 \cdot u2\right) + 1\right) \cdot \left|\color{blue}{{\left(\frac{1}{u1} + -1\right)}^{\left(\frac{-0.5}{2}\right)} \cdot {\left(\frac{1}{u1} + -1\right)}^{\left(\frac{-0.5}{2}\right)}}\right| \]

      11. fabs-sqr 98.3%

          \[\leadsto \left(-19.739208802181317 \cdot \left(u2 \cdot u2\right) + 1\right) \cdot \color{blue}{\left({\left(\frac{1}{u1} + -1\right)}^{\left(\frac{-0.5}{2}\right)} \cdot {\left(\frac{1}{u1} + -1\right)}^{\left(\frac{-0.5}{2}\right)}\right)} \]

      12. sqr-pow 99.4%

          \[\leadsto \left(-19.739208802181317 \cdot \left(u2 \cdot u2\right) + 1\right) \cdot \color{blue}{{\left(\frac{1}{u1} + -1\right)}^{-0.5}} \]

      13. +-commutative 99.4%

          \[\leadsto \left(-19.739208802181317 \cdot \left(u2 \cdot u2\right) + 1\right) \cdot {\color{blue}{\left(-1 + \frac{1}{u1}\right)}}^{-0.5} \]

   8. Simplified 99.4%

      \[\leadsto \color{blue}{\left(-19.739208802181317 \cdot \left(u2 \cdot u2\right) + 1\right) \cdot {\left(-1 + \frac{1}{u1}\right)}^{-0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(6.28318530718 \cdot u2\right) \leq 0.9983000159263611:\\ \;\;\;\;\cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1 + u1 \cdot u1}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + -19.739208802181317 \cdot \left(u2 \cdot u2\right)\right) \cdot {\left(-1 + \frac{1}{u1}\right)}^{-0.5}\\ \end{array} \]

Alternative 3: 93.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.05999999865889549:\\ \;\;\;\;\left(1 + -19.739208802181317 \cdot \left(u2 \cdot u2\right)\right) \cdot {\left(-1 + \frac{1}{u1}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.05999999865889549)
   (* (+ 1.0 (* -19.739208802181317 (* u2 u2))) (pow (+ -1.0 (/ 1.0 u1)) -0.5))
   (* (cos (* 6.28318530718 u2)) (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.05999999865889549f) {
		tmp = (1.0f + (-19.739208802181317f * (u2 * u2))) * powf((-1.0f + (1.0f / u1)), -0.5f);
	} else {
		tmp = cosf((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) <= 0.05999999865889549e0) then
        tmp = (1.0e0 + ((-19.739208802181317e0) * (u2 * u2))) * (((-1.0e0) + (1.0e0 / u1)) ** (-0.5e0))
    else
        tmp = cos((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(0.05999999865889549))
		tmp = Float32(Float32(Float32(1.0) + Float32(Float32(-19.739208802181317) * Float32(u2 * u2))) * (Float32(Float32(-1.0) + Float32(Float32(1.0) / u1)) ^ Float32(-0.5)));
	else
		tmp = Float32(cos(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(0.05999999865889549))
		tmp = (single(1.0) + (single(-19.739208802181317) * (u2 * u2))) * ((single(-1.0) + (single(1.0) / u1)) ^ single(-0.5));
	else
		tmp = cos((single(6.28318530718) * u2)) * sqrt(u1);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 314159265359/50000000000 u2) < 0.0599999987

   1. Initial program 99.4%

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

      1. clear-num 99.2%

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

      2. inv-pow 99.2%

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

      3. div-sub 99.2%

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

      4. pow1 99.2%

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

      5. pow1 99.2%

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

      6. pow-div 99.2%

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

      7. metadata-eval 99.2%

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

      8. metadata-eval 99.2%

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

   3. Applied egg-rr 99.2%

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

      1. unpow-1 99.2%

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

      2. sub-neg 99.2%

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

      3. metadata-eval 99.2%

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

   5. Simplified 99.2%

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

   6. Taylor expanded in u2 around 0 99.2%

      \[\leadsto \color{blue}{-19.739208802181317 \cdot \left({u2}^{2} \cdot \sqrt{\frac{1}{\frac{1}{u1} - 1}}\right) + \sqrt{\frac{1}{\frac{1}{u1} - 1}}} \]
   7. Step-by-step derivation

      1. associate-*r* 99.2%

         \[\leadsto \color{blue}{\left(-19.739208802181317 \cdot {u2}^{2}\right) \cdot \sqrt{\frac{1}{\frac{1}{u1} - 1}}} + \sqrt{\frac{1}{\frac{1}{u1} - 1}} \]

      2. distribute-lft1-in 99.2%

         \[\leadsto \color{blue}{\left(-19.739208802181317 \cdot {u2}^{2} + 1\right) \cdot \sqrt{\frac{1}{\frac{1}{u1} - 1}}} \]

      3. unpow2 99.2%

         \[\leadsto \left(-19.739208802181317 \cdot \color{blue}{\left(u2 \cdot u2\right)} + 1\right) \cdot \sqrt{\frac{1}{\frac{1}{u1} - 1}} \]

      4. sub-neg 99.2%

         \[\leadsto \left(-19.739208802181317 \cdot \left(u2 \cdot u2\right) + 1\right) \cdot \sqrt{\frac{1}{\color{blue}{\frac{1}{u1} + \left(-1\right)}}} \]

      5. metadata-eval 99.2%

         \[\leadsto \left(-19.739208802181317 \cdot \left(u2 \cdot u2\right) + 1\right) \cdot \sqrt{\frac{1}{\frac{1}{u1} + \color{blue}{-1}}} \]

      6. unpow-1 99.2%

         \[\leadsto \left(-19.739208802181317 \cdot \left(u2 \cdot u2\right) + 1\right) \cdot \sqrt{\color{blue}{{\left(\frac{1}{u1} + -1\right)}^{-1}}} \]

      7. metadata-eval 99.2%

         \[\leadsto \left(-19.739208802181317 \cdot \left(u2 \cdot u2\right) + 1\right) \cdot \sqrt{{\left(\frac{1}{u1} + -1\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]

      8. pow-sqr 99.3%

         \[\leadsto \left(-19.739208802181317 \cdot \left(u2 \cdot u2\right) + 1\right) \cdot \sqrt{\color{blue}{{\left(\frac{1}{u1} + -1\right)}^{-0.5} \cdot {\left(\frac{1}{u1} + -1\right)}^{-0.5}}} \]

      9. rem-sqrt-square 99.3%

         \[\leadsto \left(-19.739208802181317 \cdot \left(u2 \cdot u2\right) + 1\right) \cdot \color{blue}{\left|{\left(\frac{1}{u1} + -1\right)}^{-0.5}\right|} \]

      10. sqr-pow 98.2%

          \[\leadsto \left(-19.739208802181317 \cdot \left(u2 \cdot u2\right) + 1\right) \cdot \left|\color{blue}{{\left(\frac{1}{u1} + -1\right)}^{\left(\frac{-0.5}{2}\right)} \cdot {\left(\frac{1}{u1} + -1\right)}^{\left(\frac{-0.5}{2}\right)}}\right| \]

      11. fabs-sqr 98.2%

          \[\leadsto \left(-19.739208802181317 \cdot \left(u2 \cdot u2\right) + 1\right) \cdot \color{blue}{\left({\left(\frac{1}{u1} + -1\right)}^{\left(\frac{-0.5}{2}\right)} \cdot {\left(\frac{1}{u1} + -1\right)}^{\left(\frac{-0.5}{2}\right)}\right)} \]

      12. sqr-pow 99.3%

          \[\leadsto \left(-19.739208802181317 \cdot \left(u2 \cdot u2\right) + 1\right) \cdot \color{blue}{{\left(\frac{1}{u1} + -1\right)}^{-0.5}} \]

      13. +-commutative 99.3%

          \[\leadsto \left(-19.739208802181317 \cdot \left(u2 \cdot u2\right) + 1\right) \cdot {\color{blue}{\left(-1 + \frac{1}{u1}\right)}}^{-0.5} \]

   8. Simplified 99.3%

      \[\leadsto \color{blue}{\left(-19.739208802181317 \cdot \left(u2 \cdot u2\right) + 1\right) \cdot {\left(-1 + \frac{1}{u1}\right)}^{-0.5}} \]

   if 0.0599999987 < (*.f32 314159265359/50000000000 u2)

   1. Initial program 97.3%

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

   2. Taylor expanded in u1 around 0 75.9%

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

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.05999999865889549:\\ \;\;\;\;\left(1 + -19.739208802181317 \cdot \left(u2 \cdot u2\right)\right) \cdot {\left(-1 + \frac{1}{u1}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \]

Alternative 4: 87.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \left(1 + -19.739208802181317 \cdot \left(u2 \cdot u2\right)\right) \cdot {\left(-1 + \frac{1}{u1}\right)}^{-0.5} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (+ 1.0 (* -19.739208802181317 (* u2 u2))) (pow (+ -1.0 (/ 1.0 u1)) -0.5)))

C

float code(float cosTheta_i, float u1, float u2) {
	return (1.0f + (-19.739208802181317f * (u2 * 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 = (1.0e0 + ((-19.739208802181317e0) * (u2 * u2))) * (((-1.0e0) + (1.0e0 / u1)) ** (-0.5e0))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(Float32(Float32(1.0) + Float32(Float32(-19.739208802181317) * Float32(u2 * u2))) * (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(-19.739208802181317) * (u2 * u2))) * ((single(-1.0) + (single(1.0) / u1)) ^ single(-0.5));
end

Derivation

1. Initial program 98.9%

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

   1. clear-num 98.8%

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

   2. inv-pow 98.8%

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

   3. div-sub 98.8%

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

   4. pow1 98.8%

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

   5. pow1 98.8%

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

   6. pow-div 98.8%

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

   7. metadata-eval 98.8%

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

   8. metadata-eval 98.8%

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

3. Applied egg-rr 98.8%

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

   1. unpow-1 98.8%

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

   2. sub-neg 98.8%

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

   3. metadata-eval 98.8%

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

5. Simplified 98.8%

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

6. Taylor expanded in u2 around 0 86.1%

   \[\leadsto \color{blue}{-19.739208802181317 \cdot \left({u2}^{2} \cdot \sqrt{\frac{1}{\frac{1}{u1} - 1}}\right) + \sqrt{\frac{1}{\frac{1}{u1} - 1}}} \]
7. Step-by-step derivation

   1. associate-*r* 86.1%

      \[\leadsto \color{blue}{\left(-19.739208802181317 \cdot {u2}^{2}\right) \cdot \sqrt{\frac{1}{\frac{1}{u1} - 1}}} + \sqrt{\frac{1}{\frac{1}{u1} - 1}} \]

   2. distribute-lft1-in 86.1%

      \[\leadsto \color{blue}{\left(-19.739208802181317 \cdot {u2}^{2} + 1\right) \cdot \sqrt{\frac{1}{\frac{1}{u1} - 1}}} \]

   3. unpow2 86.1%

      \[\leadsto \left(-19.739208802181317 \cdot \color{blue}{\left(u2 \cdot u2\right)} + 1\right) \cdot \sqrt{\frac{1}{\frac{1}{u1} - 1}} \]

   4. sub-neg 86.1%

      \[\leadsto \left(-19.739208802181317 \cdot \left(u2 \cdot u2\right) + 1\right) \cdot \sqrt{\frac{1}{\color{blue}{\frac{1}{u1} + \left(-1\right)}}} \]

   5. metadata-eval 86.1%

      \[\leadsto \left(-19.739208802181317 \cdot \left(u2 \cdot u2\right) + 1\right) \cdot \sqrt{\frac{1}{\frac{1}{u1} + \color{blue}{-1}}} \]

   6. unpow-1 86.1%

      \[\leadsto \left(-19.739208802181317 \cdot \left(u2 \cdot u2\right) + 1\right) \cdot \sqrt{\color{blue}{{\left(\frac{1}{u1} + -1\right)}^{-1}}} \]

   7. metadata-eval 86.1%

      \[\leadsto \left(-19.739208802181317 \cdot \left(u2 \cdot u2\right) + 1\right) \cdot \sqrt{{\left(\frac{1}{u1} + -1\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]

   8. pow-sqr 86.2%

      \[\leadsto \left(-19.739208802181317 \cdot \left(u2 \cdot u2\right) + 1\right) \cdot \sqrt{\color{blue}{{\left(\frac{1}{u1} + -1\right)}^{-0.5} \cdot {\left(\frac{1}{u1} + -1\right)}^{-0.5}}} \]

   9. rem-sqrt-square 86.2%

      \[\leadsto \left(-19.739208802181317 \cdot \left(u2 \cdot u2\right) + 1\right) \cdot \color{blue}{\left|{\left(\frac{1}{u1} + -1\right)}^{-0.5}\right|} \]

   10. sqr-pow 85.3%

       \[\leadsto \left(-19.739208802181317 \cdot \left(u2 \cdot u2\right) + 1\right) \cdot \left|\color{blue}{{\left(\frac{1}{u1} + -1\right)}^{\left(\frac{-0.5}{2}\right)} \cdot {\left(\frac{1}{u1} + -1\right)}^{\left(\frac{-0.5}{2}\right)}}\right| \]

   11. fabs-sqr 85.3%

       \[\leadsto \left(-19.739208802181317 \cdot \left(u2 \cdot u2\right) + 1\right) \cdot \color{blue}{\left({\left(\frac{1}{u1} + -1\right)}^{\left(\frac{-0.5}{2}\right)} \cdot {\left(\frac{1}{u1} + -1\right)}^{\left(\frac{-0.5}{2}\right)}\right)} \]

   12. sqr-pow 86.2%

       \[\leadsto \left(-19.739208802181317 \cdot \left(u2 \cdot u2\right) + 1\right) \cdot \color{blue}{{\left(\frac{1}{u1} + -1\right)}^{-0.5}} \]

   13. +-commutative 86.2%

       \[\leadsto \left(-19.739208802181317 \cdot \left(u2 \cdot u2\right) + 1\right) \cdot {\color{blue}{\left(-1 + \frac{1}{u1}\right)}}^{-0.5} \]

8. Simplified 86.2%

   \[\leadsto \color{blue}{\left(-19.739208802181317 \cdot \left(u2 \cdot u2\right) + 1\right) \cdot {\left(-1 + \frac{1}{u1}\right)}^{-0.5}} \]

9. Final simplification 86.2%

   \[\leadsto \left(1 + -19.739208802181317 \cdot \left(u2 \cdot u2\right)\right) \cdot {\left(-1 + \frac{1}{u1}\right)}^{-0.5} \]

Alternative 5: 79.2% 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. Taylor expanded in u2 around 0 79.0%

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

3. Final simplification 79.0%

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

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 98.9%

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

2. Taylor expanded in u2 around 0 79.0%

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

3. Taylor expanded in u1 around 0 61.6%

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

4. Final simplification 61.6%

   \[\leadsto \sqrt{u1} \]

Alternative 7: 19.0% accurate, 209.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

2. Taylor expanded in u1 around 0 86.4%

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

   1. unpow2 86.4%

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

   2. fma-udef 86.4%

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

4. Simplified 86.4%

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

5. Taylor expanded in u1 around inf 19.9%

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

   1. *-commutative 19.9%

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

7. Simplified 19.9%

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

8. Taylor expanded in u2 around 0 18.9%

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

9. Final simplification 18.9%

   \[\leadsto u1 \]

Reproduce

herbie shell --seed 2023238 
(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))))