Trowbridge-Reitz Sample, near normal, slope_x

Percentage Accurate: 99.0% → 99.0%

Time: 9.4s

Alternatives: 9

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

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

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. Add Preprocessing
3. Add Preprocessing

Alternative 2: 94.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.0015999999595806003:\\ \;\;\;\;{\left(\frac{1}{u1} + -1\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1 \cdot \left(u1 + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.0015999999595806003)
   (pow (+ (/ 1.0 u1) -1.0) -0.5)
   (* (cos (* 6.28318530718 u2)) (sqrt (* u1 (+ u1 1.0))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.0015999999595806003f) {
		tmp = powf(((1.0f / u1) + -1.0f), -0.5f);
	} else {
		tmp = cosf((6.28318530718f * u2)) * sqrtf((u1 * (u1 + 1.0f)));
	}
	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) <= 0.0015999999595806003e0) then
        tmp = ((1.0e0 / u1) + (-1.0e0)) ** (-0.5e0)
    else
        tmp = cos((6.28318530718e0 * u2)) * sqrt((u1 * (u1 + 1.0e0)))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(6.28318530718) * u2) <= Float32(0.0015999999595806003))
		tmp = Float32(Float32(Float32(1.0) / u1) + Float32(-1.0)) ^ Float32(-0.5);
	else
		tmp = Float32(cos(Float32(Float32(6.28318530718) * u2)) * sqrt(Float32(u1 * Float32(u1 + Float32(1.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((single(6.28318530718) * u2) <= single(0.0015999999595806003))
		tmp = ((single(1.0) / u1) + single(-1.0)) ^ single(-0.5);
	else
		tmp = cos((single(6.28318530718) * u2)) * sqrt((u1 * (u1 + single(1.0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.00159999996

   1. Initial program 99.5%

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

   3. Taylor expanded in u2 around 0 98.7%

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

      1. clear-num 98.7%

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

      2. sqrt-div 98.3%

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

      3. metadata-eval 98.3%

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

   5. Applied egg-rr 98.3%

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

      1. expm1-log1p-u 98.2%

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

      2. expm1-undefine 81.4%

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

      3. pow1/2 81.4%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{\color{blue}{{\left(\frac{1 - u1}{u1}\right)}^{0.5}}}\right)} - 1 \]

      4. pow-flip 81.4%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{{\left(\frac{1 - u1}{u1}\right)}^{\left(-0.5\right)}}\right)} - 1 \]

      5. div-sub 81.3%

         \[\leadsto e^{\mathsf{log1p}\left({\color{blue}{\left(\frac{1}{u1} - \frac{u1}{u1}\right)}}^{\left(-0.5\right)}\right)} - 1 \]

      6. *-inverses 81.3%

         \[\leadsto e^{\mathsf{log1p}\left({\left(\frac{1}{u1} - \color{blue}{1}\right)}^{\left(-0.5\right)}\right)} - 1 \]

      7. sub-neg 81.3%

         \[\leadsto e^{\mathsf{log1p}\left({\color{blue}{\left(\frac{1}{u1} + \left(-1\right)\right)}}^{\left(-0.5\right)}\right)} - 1 \]

      8. metadata-eval 81.3%

         \[\leadsto e^{\mathsf{log1p}\left({\left(\frac{1}{u1} + \color{blue}{-1}\right)}^{\left(-0.5\right)}\right)} - 1 \]

      9. metadata-eval 81.3%

         \[\leadsto e^{\mathsf{log1p}\left({\left(\frac{1}{u1} + -1\right)}^{\color{blue}{-0.5}}\right)} - 1 \]

   7. Applied egg-rr 81.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(\frac{1}{u1} + -1\right)}^{-0.5}\right)} - 1} \]
   8. Step-by-step derivation

      1. log1p-undefine 81.3%

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

      2. rem-exp-log 81.3%

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

      3. associate-+r- 83.7%

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

      4. +-commutative 83.7%

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

      5. associate-+l- 98.8%

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

      6. metadata-eval 98.8%

         \[\leadsto {\left(\frac{1}{u1} + -1\right)}^{-0.5} - \color{blue}{0} \]

      7. --rgt-identity 98.8%

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

   9. Simplified 98.8%

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

   if 0.00159999996 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 98.0%

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

   3. Taylor expanded in u1 around 0 84.0%

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

      1. +-commutative 84.0%

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

   5. Simplified 84.0%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.0015999999595806003:\\ \;\;\;\;{\left(\frac{1}{u1} + -1\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1 \cdot \left(u1 + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.00800000037997961:\\ \;\;\;\;{\left(\frac{1}{u1} + -1\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.00800000037997961)
   (pow (+ (/ 1.0 u1) -1.0) -0.5)
   (* (cos (* 6.28318530718 u2)) (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.00800000037997961f) {
		tmp = powf(((1.0f / u1) + -1.0f), -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) <= 0.00800000037997961e0) then
        tmp = ((1.0e0 / u1) + (-1.0e0)) ** (-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(0.00800000037997961))
		tmp = Float32(Float32(Float32(1.0) / u1) + Float32(-1.0)) ^ 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(0.00800000037997961))
		tmp = ((single(1.0) / u1) + single(-1.0)) ^ 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 #s(literal 314159265359/50000000000 binary32) u2) < 0.00800000038

   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 96.8%

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

      1. clear-num 96.7%

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

      2. sqrt-div 96.4%

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

      3. metadata-eval 96.4%

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

   5. Applied egg-rr 96.4%

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

      1. expm1-log1p-u 96.3%

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

      2. expm1-undefine 80.5%

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

      3. pow1/2 80.5%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{\color{blue}{{\left(\frac{1 - u1}{u1}\right)}^{0.5}}}\right)} - 1 \]

      4. pow-flip 80.5%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{{\left(\frac{1 - u1}{u1}\right)}^{\left(-0.5\right)}}\right)} - 1 \]

      5. div-sub 80.5%

         \[\leadsto e^{\mathsf{log1p}\left({\color{blue}{\left(\frac{1}{u1} - \frac{u1}{u1}\right)}}^{\left(-0.5\right)}\right)} - 1 \]

      6. *-inverses 80.5%

         \[\leadsto e^{\mathsf{log1p}\left({\left(\frac{1}{u1} - \color{blue}{1}\right)}^{\left(-0.5\right)}\right)} - 1 \]

      7. sub-neg 80.5%

         \[\leadsto e^{\mathsf{log1p}\left({\color{blue}{\left(\frac{1}{u1} + \left(-1\right)\right)}}^{\left(-0.5\right)}\right)} - 1 \]

      8. metadata-eval 80.5%

         \[\leadsto e^{\mathsf{log1p}\left({\left(\frac{1}{u1} + \color{blue}{-1}\right)}^{\left(-0.5\right)}\right)} - 1 \]

      9. metadata-eval 80.5%

         \[\leadsto e^{\mathsf{log1p}\left({\left(\frac{1}{u1} + -1\right)}^{\color{blue}{-0.5}}\right)} - 1 \]

   7. Applied egg-rr 80.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(\frac{1}{u1} + -1\right)}^{-0.5}\right)} - 1} \]
   8. Step-by-step derivation

      1. log1p-undefine 80.4%

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

      2. rem-exp-log 80.4%

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

      3. associate-+r- 82.9%

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

      4. +-commutative 82.9%

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

      5. associate-+l- 96.8%

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

      6. metadata-eval 96.8%

         \[\leadsto {\left(\frac{1}{u1} + -1\right)}^{-0.5} - \color{blue}{0} \]

      7. --rgt-identity 96.8%

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

   9. Simplified 96.8%

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

   if 0.00800000038 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 97.8%

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

   3. Taylor expanded in u1 around 0 70.7%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.00800000037997961:\\ \;\;\;\;{\left(\frac{1}{u1} + -1\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = ((single(1.0) / u1) + single(-1.0)) ^ single(-0.5);
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 78.1%

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

   1. clear-num 78.1%

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

   2. sqrt-div 77.8%

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

   3. metadata-eval 77.8%

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

5. Applied egg-rr 77.8%

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

   1. expm1-log1p-u 77.8%

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

   2. expm1-undefine 66.9%

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

   3. pow1/2 66.9%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{\color{blue}{{\left(\frac{1 - u1}{u1}\right)}^{0.5}}}\right)} - 1 \]

   4. pow-flip 66.9%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{{\left(\frac{1 - u1}{u1}\right)}^{\left(-0.5\right)}}\right)} - 1 \]

   5. div-sub 66.9%

      \[\leadsto e^{\mathsf{log1p}\left({\color{blue}{\left(\frac{1}{u1} - \frac{u1}{u1}\right)}}^{\left(-0.5\right)}\right)} - 1 \]

   6. *-inverses 66.9%

      \[\leadsto e^{\mathsf{log1p}\left({\left(\frac{1}{u1} - \color{blue}{1}\right)}^{\left(-0.5\right)}\right)} - 1 \]

   7. sub-neg 66.9%

      \[\leadsto e^{\mathsf{log1p}\left({\color{blue}{\left(\frac{1}{u1} + \left(-1\right)\right)}}^{\left(-0.5\right)}\right)} - 1 \]

   8. metadata-eval 66.9%

      \[\leadsto e^{\mathsf{log1p}\left({\left(\frac{1}{u1} + \color{blue}{-1}\right)}^{\left(-0.5\right)}\right)} - 1 \]

   9. metadata-eval 66.9%

      \[\leadsto e^{\mathsf{log1p}\left({\left(\frac{1}{u1} + -1\right)}^{\color{blue}{-0.5}}\right)} - 1 \]

7. Applied egg-rr 66.9%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(\frac{1}{u1} + -1\right)}^{-0.5}\right)} - 1} \]
8. Step-by-step derivation

   1. log1p-undefine 66.9%

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

   2. rem-exp-log 66.9%

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

   3. associate-+r- 68.6%

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

   4. +-commutative 68.6%

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

   5. associate-+l- 78.1%

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

   6. metadata-eval 78.1%

      \[\leadsto {\left(\frac{1}{u1} + -1\right)}^{-0.5} - \color{blue}{0} \]

   7. --rgt-identity 78.1%

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

9. Simplified 78.1%

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

Alternative 5: 80.0% 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 78.1%

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

Alternative 6: 71.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{u1 \cdot \left(u1 + 1\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2) :precision binary32 (sqrt (* u1 (+ u1 1.0))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 * (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 = sqrt((u1 * (u1 + 1.0e0)))
end function

Julia

function code(cosTheta_i, u1, u2)
	return sqrt(Float32(u1 * Float32(u1 + Float32(1.0))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((u1 * (u1 + single(1.0))));
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 78.1%

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

4. Taylor expanded in u1 around 0 68.9%

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

   1. +-commutative 84.4%

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

6. Simplified 68.9%

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

Alternative 7: 63.1% 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 78.1%

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

4. Taylor expanded in u1 around 0 60.8%

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

Alternative 8: 20.2% accurate, 69.7× speedup

Math

\[\begin{array}{l} \\ u1 + 0.5 \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = 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. Add Preprocessing

3. Taylor expanded in u2 around 0 78.1%

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

4. Taylor expanded in u1 around 0 68.9%

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

   1. +-commutative 68.9%

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

   2. distribute-lft-in 68.9%

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

   3. *-rgt-identity 68.9%

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

   4. fma-define 68.9%

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

6. Simplified 68.9%

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

7. Taylor expanded in u1 around inf 20.2%

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

   1. distribute-lft-in 20.2%

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

   2. *-rgt-identity 20.2%

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

   3. *-commutative 20.2%

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

   4. associate-*r* 20.2%

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

   5. rgt-mult-inverse 20.2%

      \[\leadsto u1 + \color{blue}{1} \cdot 0.5 \]

   6. metadata-eval 20.2%

      \[\leadsto u1 + \color{blue}{0.5} \]

9. Simplified 20.2%

   \[\leadsto \color{blue}{u1 + 0.5} \]
10. Add Preprocessing

Alternative 9: 19.1% accurate, 209.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 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 78.1%

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

4. Taylor expanded in u1 around 0 68.9%

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

   1. +-commutative 68.9%

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

   2. distribute-lft-in 68.9%

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

   3. *-rgt-identity 68.9%

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

   4. fma-define 68.9%

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

6. Simplified 68.9%

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

7. Taylor expanded in u1 around inf 19.3%

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

Reproduce

herbie shell --seed 2024141 
(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))))