Trowbridge-Reitz Sample, near normal, slope_y

Percentage Accurate: 98.4% → 98.3%

Time: 12.7s

Alternatives: 10

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: 98.4% 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, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

   1. clear-num 98.5%

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

   2. inv-pow 98.5%

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

4. Applied egg-rr 98.5%

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

5. Final simplification 98.5%

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

Alternative 2: 90.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.00800000037997961:\\ \;\;\;\;u2 \cdot \sqrt{\frac{39.47841760436263}{-1 + \frac{1}{u1}}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.00800000037997961)
   (* u2 (sqrt (/ 39.47841760436263 (+ -1.0 (/ 1.0 u1)))))
   (* (sin (* 6.28318530718 u2)) (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.00800000037997961f) {
		tmp = u2 * sqrtf((39.47841760436263f / (-1.0f + (1.0f / u1))));
	} 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) <= 0.00800000037997961e0) then
        tmp = u2 * sqrt((39.47841760436263e0 / ((-1.0e0) + (1.0e0 / u1))))
    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(0.00800000037997961))
		tmp = Float32(u2 * sqrt(Float32(Float32(39.47841760436263) / Float32(Float32(-1.0) + Float32(Float32(1.0) / u1)))));
	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(0.00800000037997961))
		tmp = u2 * sqrt((single(39.47841760436263) / (single(-1.0) + (single(1.0) / u1))));
	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) < 0.00800000038

   1. Initial program 98.7%

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

      1. clear-num 98.8%

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

      2. inv-pow 98.8%

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

   4. Applied egg-rr 98.8%

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

      1. unpow-1 98.8%

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

      2. div-sub 98.7%

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

      3. *-inverses 98.7%

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

      4. sub-neg 98.7%

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

      5. metadata-eval 98.7%

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

   6. Applied egg-rr 98.7%

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

   7. Taylor expanded in u2 around 0 97.0%

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

      1. associate-*r* 97.0%

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

      2. *-commutative 97.0%

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

      3. associate-*l* 97.1%

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

      4. sub-neg 97.1%

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

      5. metadata-eval 97.1%

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

      6. +-commutative 97.1%

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

   9. Simplified 97.1%

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

       1. add-sqr-sqrt 96.4%

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

       2. sqrt-unprod 97.1%

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

       3. *-commutative 97.1%

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

       4. *-commutative 97.1%

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

       5. swap-sqr 97.1%

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

       6. add-sqr-sqrt 97.2%

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

       7. +-commutative 97.2%

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

       8. metadata-eval 97.1%

          \[\leadsto u2 \cdot \sqrt{\frac{1}{\frac{1}{u1} + -1} \cdot \color{blue}{39.47841760436263}} \]

   11. Applied egg-rr 97.1%

       \[\leadsto u2 \cdot \color{blue}{\sqrt{\frac{1}{\frac{1}{u1} + -1} \cdot 39.47841760436263}} \]
   12. Step-by-step derivation

       1. associate-*l/ 97.3%

          \[\leadsto u2 \cdot \sqrt{\color{blue}{\frac{1 \cdot 39.47841760436263}{\frac{1}{u1} + -1}}} \]

       2. metadata-eval 97.3%

          \[\leadsto u2 \cdot \sqrt{\frac{\color{blue}{39.47841760436263}}{\frac{1}{u1} + -1}} \]

   13. Simplified 97.3%

       \[\leadsto u2 \cdot \color{blue}{\sqrt{\frac{39.47841760436263}{\frac{1}{u1} + -1}}} \]

   if 0.00800000038 < (*.f32 314159265359/50000000000 u2)

   1. Initial program 97.8%

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

   3. 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 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.00800000037997961:\\ \;\;\;\;u2 \cdot \sqrt{\frac{39.47841760436263}{-1 + \frac{1}{u1}}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.4% 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 98.4%

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

3. Final simplification 98.4%

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

Alternative 4: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

   1. clear-num 98.5%

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

   2. inv-pow 98.5%

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

4. Applied egg-rr 98.5%

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

   1. unpow-1 98.5%

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

   2. div-sub 98.4%

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

   3. *-inverses 98.4%

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

   4. sub-neg 98.4%

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

   5. metadata-eval 98.4%

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

6. Applied egg-rr 98.4%

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

   1. *-commutative 98.4%

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

   2. sqrt-div 98.2%

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

   3. metadata-eval 98.2%

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

   4. un-div-inv 98.5%

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

8. Applied egg-rr 98.5%

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

9. Final simplification 98.5%

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

Alternative 5: 81.7% 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 98.4%

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

3. Taylor expanded in u2 around 0 81.8%

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

4. Final simplification 81.8%

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

Alternative 6: 82.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ u2 \cdot \sqrt{\frac{39.47841760436263}{-1 + \frac{1}{u1}}} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* u2 (sqrt (/ 39.47841760436263 (+ -1.0 (/ 1.0 u1))))))

C

float code(float cosTheta_i, float u1, float u2) {
	return u2 * sqrtf((39.47841760436263f / (-1.0f + (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 = u2 * sqrt((39.47841760436263e0 / ((-1.0e0) + (1.0e0 / u1))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

   1. clear-num 98.5%

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

   2. inv-pow 98.5%

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

4. Applied egg-rr 98.5%

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

   1. unpow-1 98.5%

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

   2. div-sub 98.4%

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

   3. *-inverses 98.4%

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

   4. sub-neg 98.4%

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

   5. metadata-eval 98.4%

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

6. Applied egg-rr 98.4%

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

7. Taylor expanded in u2 around 0 81.8%

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

   1. associate-*r* 81.8%

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

   2. *-commutative 81.8%

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

   3. associate-*l* 81.8%

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

   4. sub-neg 81.8%

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

   5. metadata-eval 81.8%

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

   6. +-commutative 81.8%

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

9. Simplified 81.8%

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

    1. add-sqr-sqrt 81.4%

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

    2. sqrt-unprod 81.8%

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

    3. *-commutative 81.8%

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

    4. *-commutative 81.8%

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

    5. swap-sqr 81.9%

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

    6. add-sqr-sqrt 81.9%

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

    7. +-commutative 81.9%

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

    8. metadata-eval 81.9%

       \[\leadsto u2 \cdot \sqrt{\frac{1}{\frac{1}{u1} + -1} \cdot \color{blue}{39.47841760436263}} \]

11. Applied egg-rr 81.9%

    \[\leadsto u2 \cdot \color{blue}{\sqrt{\frac{1}{\frac{1}{u1} + -1} \cdot 39.47841760436263}} \]
12. Step-by-step derivation

    1. associate-*l/ 82.0%

       \[\leadsto u2 \cdot \sqrt{\color{blue}{\frac{1 \cdot 39.47841760436263}{\frac{1}{u1} + -1}}} \]

    2. metadata-eval 82.0%

       \[\leadsto u2 \cdot \sqrt{\frac{\color{blue}{39.47841760436263}}{\frac{1}{u1} + -1}} \]

13. Simplified 82.0%

    \[\leadsto u2 \cdot \color{blue}{\sqrt{\frac{39.47841760436263}{\frac{1}{u1} + -1}}} \]

14. Final simplification 82.0%

    \[\leadsto u2 \cdot \sqrt{\frac{39.47841760436263}{-1 + \frac{1}{u1}}} \]
15. Add Preprocessing

Alternative 7: 65.1% 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 98.4%

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

3. Taylor expanded in u2 around 0 81.8%

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

4. Taylor expanded in u1 around 0 68.0%

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

5. Final simplification 68.0%

   \[\leadsto 6.28318530718 \cdot \left(u2 \cdot \sqrt{u1}\right) \]
6. Add Preprocessing

Alternative 8: 65.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

   1. clear-num 98.5%

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

   2. inv-pow 98.5%

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

4. Applied egg-rr 98.5%

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

   1. unpow-1 98.5%

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

   2. div-sub 98.4%

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

   3. *-inverses 98.4%

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

   4. sub-neg 98.4%

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

   5. metadata-eval 98.4%

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

6. Applied egg-rr 98.4%

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

7. Taylor expanded in u2 around 0 81.8%

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

   1. associate-*r* 81.8%

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

   2. *-commutative 81.8%

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

   3. associate-*l* 81.8%

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

   4. sub-neg 81.8%

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

   5. metadata-eval 81.8%

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

   6. +-commutative 81.8%

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

9. Simplified 81.8%

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

10. Taylor expanded in u1 around 0 68.0%

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

11. Final simplification 68.0%

    \[\leadsto u2 \cdot \left(6.28318530718 \cdot \sqrt{u1}\right) \]
12. Add Preprocessing

Alternative 9: 4.8% accurate, 41.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

3. Taylor expanded in u1 around 0 88.5%

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

   1. +-commutative 88.5%

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

   2. unpow2 88.5%

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

   3. fma-def 88.5%

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

5. Simplified 88.5%

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

6. Taylor expanded in u2 around 0 75.4%

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

   1. associate-*r* 75.4%

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

   2. +-commutative 75.4%

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

   3. unpow2 75.4%

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

   4. rem-square-sqrt 75.3%

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

   5. hypot-def 75.3%

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

8. Simplified 75.3%

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

9. Taylor expanded in u1 around -inf 4.7%

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

10. Final simplification 4.7%

    \[\leadsto -6.28318530718 \cdot \left(u1 \cdot u2\right) \]
11. Add Preprocessing

Alternative 10: 19.4% accurate, 41.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

3. Taylor expanded in u1 around 0 88.5%

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

   1. +-commutative 88.5%

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

   2. unpow2 88.5%

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

   3. fma-def 88.5%

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

5. Simplified 88.5%

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

6. Taylor expanded in u2 around 0 75.4%

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

   1. associate-*r* 75.4%

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

   2. +-commutative 75.4%

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

   3. unpow2 75.4%

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

   4. rem-square-sqrt 75.3%

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

   5. hypot-def 75.3%

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

8. Simplified 75.3%

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

9. Taylor expanded in u1 around inf 19.0%

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

    1. *-commutative 19.0%

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

11. Simplified 19.0%

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

12. Final simplification 19.0%

    \[\leadsto 6.28318530718 \cdot \left(u1 \cdot u2\right) \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024020 
(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))))