Trowbridge-Reitz Sample, near normal, slope_x

Percentage Accurate: 99.0% → 98.9%

Time: 11.9s

Alternatives: 12

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: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return powf(((1.0f / u1) + -1.0f), -0.5f) * cosf((u2 * 6.28318530718f));
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = (((1.0e0 / u1) + (-1.0e0)) ** (-0.5e0)) * cos((u2 * 6.28318530718e0))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.2%

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

   1. sqrt-div 98.8%

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

   2. associate-*l/ 98.9%

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

4. Applied egg-rr 98.9%

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

   1. associate-*l/ 98.8%

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

   2. sqrt-div 99.2%

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

   3. clear-num 99.0%

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

   4. sqrt-div 98.7%

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

   5. metadata-eval 98.7%

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

   6. inv-pow 98.7%

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

   7. sqrt-pow2 99.1%

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

   8. div-sub 99.2%

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

   9. *-inverses 99.2%

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

   10. sub-neg 99.2%

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

   11. metadata-eval 99.2%

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

   12. metadata-eval 99.2%

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

   13. *-commutative 99.2%

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

6. Applied egg-rr 99.2%

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

7. Final simplification 99.2%

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

Alternative 2: 93.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(u2 \cdot 6.28318530718\right)\\ \mathbf{if}\;t\_0 \leq 0.9999886155128479:\\ \;\;\;\;t\_0 \cdot \sqrt{u1 \cdot \left(1 + u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* u2 6.28318530718))))
   (if (<= t_0 0.9999886155128479)
     (* t_0 (sqrt (* u1 (+ 1.0 u1))))
     (sqrt (/ u1 (- 1.0 u1))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = cosf((u2 * 6.28318530718f));
	float tmp;
	if (t_0 <= 0.9999886155128479f) {
		tmp = t_0 * sqrtf((u1 * (1.0f + u1)));
	} else {
		tmp = sqrtf((u1 / (1.0f - u1)));
	}
	return tmp;
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    real(4) :: t_0
    real(4) :: tmp
    t_0 = cos((u2 * 6.28318530718e0))
    if (t_0 <= 0.9999886155128479e0) then
        tmp = t_0 * sqrt((u1 * (1.0e0 + u1)))
    else
        tmp = sqrt((u1 / (1.0e0 - u1)))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	t_0 = cos(Float32(u2 * Float32(6.28318530718)))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.9999886155128479))
		tmp = Float32(t_0 * sqrt(Float32(u1 * Float32(Float32(1.0) + u1))));
	else
		tmp = sqrt(Float32(u1 / Float32(Float32(1.0) - u1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	t_0 = cos((u2 * single(6.28318530718)));
	tmp = single(0.0);
	if (t_0 <= single(0.9999886155128479))
		tmp = t_0 * sqrt((u1 * (single(1.0) + u1)));
	else
		tmp = sqrt((u1 / (single(1.0) - u1)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2)) < 0.999988616

   1. Initial program 98.5%

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

   3. Taylor expanded in u1 around 0 86.6%

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

      1. +-commutative 45.1%

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

   5. Simplified 86.6%

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

   if 0.999988616 < (cos.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2))

   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 97.7%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(u2 \cdot 6.28318530718\right) \leq 0.9999886155128479:\\ \;\;\;\;\cos \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1 \cdot \left(1 + u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * 6.28318530718f) <= 0.00800000037997961f) {
		tmp = sqrtf((u1 / (1.0f - u1)));
	} else {
		tmp = cosf((u2 * 6.28318530718f)) * (1.0f / powf(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) :: tmp
    if ((u2 * 6.28318530718e0) <= 0.00800000037997961e0) then
        tmp = sqrt((u1 / (1.0e0 - u1)))
    else
        tmp = cos((u2 * 6.28318530718e0)) * (1.0e0 / (u1 ** (-0.5e0)))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(u2 * Float32(6.28318530718)) <= Float32(0.00800000037997961))
		tmp = sqrt(Float32(u1 / Float32(Float32(1.0) - u1)));
	else
		tmp = Float32(cos(Float32(u2 * Float32(6.28318530718))) * Float32(Float32(1.0) / (u1 ^ Float32(-0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((u2 * single(6.28318530718)) <= single(0.00800000037997961))
		tmp = sqrt((u1 / (single(1.0) - u1)));
	else
		tmp = cos((u2 * single(6.28318530718))) * (single(1.0) / (u1 ^ single(-0.5)));
	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 99.5%

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

   3. Taylor expanded in u2 around 0 96.8%

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

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

   1. Initial program 98.4%

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

      1. sqrt-div 98.4%

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

      2. associate-*l/ 98.5%

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

   4. Applied egg-rr 98.5%

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

      1. associate-*l/ 98.4%

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

      2. sqrt-div 98.4%

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

      3. clear-num 98.3%

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

      4. sqrt-div 98.4%

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

      5. metadata-eval 98.4%

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

      6. inv-pow 98.4%

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

      7. sqrt-pow2 98.3%

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

      8. div-sub 98.5%

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

      9. *-inverses 98.5%

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

      10. sub-neg 98.5%

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

      11. metadata-eval 98.5%

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

      12. metadata-eval 98.5%

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

      13. *-commutative 98.5%

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

   6. Applied egg-rr 98.5%

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

   7. Taylor expanded in u1 around 0 74.9%

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

      1. add-sqr-sqrt 75.0%

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

      2. unpow-prod-down 74.7%

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

      3. inv-pow 74.7%

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

      4. sqrt-pow1 74.7%

         \[\leadsto \left({\color{blue}{\left({u1}^{\left(\frac{-1}{2}\right)}\right)}}^{-0.5} \cdot {\left(\sqrt{\frac{1}{u1}}\right)}^{-0.5}\right) \cdot \cos \left(u2 \cdot 6.28318530718\right) \]

      5. metadata-eval 74.7%

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

      6. inv-pow 74.7%

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

      7. sqrt-pow1 74.7%

         \[\leadsto \left({\left({u1}^{-0.5}\right)}^{-0.5} \cdot {\color{blue}{\left({u1}^{\left(\frac{-1}{2}\right)}\right)}}^{-0.5}\right) \cdot \cos \left(u2 \cdot 6.28318530718\right) \]

      8. metadata-eval 74.7%

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

   9. Applied egg-rr 74.7%

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

       1. pow-sqr 75.0%

          \[\leadsto \color{blue}{{\left({u1}^{-0.5}\right)}^{\left(2 \cdot -0.5\right)}} \cdot \cos \left(u2 \cdot 6.28318530718\right) \]

       2. metadata-eval 75.0%

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

       3. unpow-1 75.0%

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

   11. Simplified 75.0%

       \[\leadsto \color{blue}{\frac{1}{{u1}^{-0.5}}} \cdot \cos \left(u2 \cdot 6.28318530718\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.0%

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

Alternative 4: 89.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * 6.28318530718f) <= 0.00800000037997961f) {
		tmp = sqrtf((u1 / (1.0f - u1)));
	} else {
		tmp = cosf((u2 * 6.28318530718f)) * 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 ((u2 * 6.28318530718e0) <= 0.00800000037997961e0) then
        tmp = sqrt((u1 / (1.0e0 - u1)))
    else
        tmp = cos((u2 * 6.28318530718e0)) * sqrt(u1)
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(u2 * Float32(6.28318530718)) <= Float32(0.00800000037997961))
		tmp = sqrt(Float32(u1 / Float32(Float32(1.0) - u1)));
	else
		tmp = Float32(cos(Float32(u2 * Float32(6.28318530718))) * sqrt(u1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((u2 * single(6.28318530718)) <= single(0.00800000037997961))
		tmp = sqrt((u1 / (single(1.0) - u1)));
	else
		tmp = cos((u2 * single(6.28318530718))) * 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 99.5%

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

   3. Taylor expanded in u2 around 0 96.8%

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

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

   1. Initial program 98.4%

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

   3. Taylor expanded in u1 around 0 74.9%

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

4. Final simplification 90.0%

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

Alternative 5: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.2%

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

3. Final simplification 99.2%

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

Alternative 6: 71.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.2%

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

3. Taylor expanded in u2 around 0 80.4%

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

4. Taylor expanded in u1 around 0 72.8%

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

   1. +-commutative 72.8%

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

6. Simplified 72.8%

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

7. Final simplification 72.8%

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

Alternative 7: 71.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{u1 + u1 \cdot u1} \end{array} \]

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 + (u1 * 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 + (u1 * u1)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.2%

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

3. Taylor expanded in u2 around 0 80.4%

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

4. Taylor expanded in u1 around 0 72.8%

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

   1. distribute-rgt-in 73.0%

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

   2. *-lft-identity 73.0%

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

   3. unpow2 73.0%

      \[\leadsto \sqrt{u1 + \color{blue}{{u1}^{2}}} \]

6. Simplified 73.0%

   \[\leadsto \sqrt{\color{blue}{u1 + {u1}^{2}}} \]
7. Step-by-step derivation

   1. unpow2 73.0%

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

8. Applied egg-rr 73.0%

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

9. Final simplification 73.0%

   \[\leadsto \sqrt{u1 + u1 \cdot u1} \]
10. Add Preprocessing

Alternative 8: 79.8% 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 99.2%

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

3. Taylor expanded in u2 around 0 80.4%

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

4. Final simplification 80.4%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \]
5. Add Preprocessing

Alternative 9: 63.0% 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 99.2%

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

3. Taylor expanded in u2 around 0 80.4%

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

4. Taylor expanded in u1 around 0 64.9%

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

5. Final simplification 64.9%

   \[\leadsto \sqrt{u1} \]
6. Add Preprocessing

Alternative 10: 20.2% accurate, 29.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.2%

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

3. Taylor expanded in u2 around 0 80.4%

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

4. Taylor expanded in u1 around 0 72.8%

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

   1. +-commutative 72.8%

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

6. Simplified 72.8%

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

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. +-commutative 20.2%

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

   2. associate-*r/ 20.2%

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

   3. metadata-eval 20.2%

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

   4. remove-double-neg 20.2%

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

   5. distribute-frac-neg2 20.2%

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

   6. distribute-neg-frac2 20.2%

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

   7. metadata-eval 20.2%

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

   8. associate-*r/ 20.2%

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

   9. metadata-eval 20.2%

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

   10. distribute-neg-in 20.2%

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

   11. +-commutative 20.2%

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

   12. sub-neg 20.2%

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

   13. rem-square-sqrt -0.0%

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

   14. unpow2 -0.0%

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

   15. sub-neg -0.0%

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

   16. unpow2 -0.0%

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

   17. rem-square-sqrt 20.2%

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

   18. distribute-neg-in 20.2%

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

9. Simplified 20.2%

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

10. Final simplification 20.2%

    \[\leadsto u1 \cdot \left(1 + \frac{0.5}{u1}\right) \]
11. Add Preprocessing

Alternative 11: 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 99.2%

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

3. Taylor expanded in u2 around 0 80.4%

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

4. Taylor expanded in u1 around 0 72.8%

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

   1. +-commutative 72.8%

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

6. Simplified 72.8%

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

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-rgt-in 20.2%

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

   2. *-lft-identity 20.2%

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

   3. associate-*l* 20.2%

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

   4. lft-mult-inverse 20.2%

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

   5. metadata-eval 20.2%

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

9. Simplified 20.2%

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

10. Final simplification 20.2%

    \[\leadsto u1 + 0.5 \]
11. Add Preprocessing

Alternative 12: 4.5% accurate, 104.5× 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 Float32(-u1)
end

MATLAB

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

Derivation

1. Initial program 99.2%

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

3. Taylor expanded in u2 around 0 80.4%

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

4. Taylor expanded in u1 around 0 72.8%

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

   1. distribute-rgt-in 73.0%

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

   2. *-lft-identity 73.0%

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

   3. unpow2 73.0%

      \[\leadsto \sqrt{u1 + \color{blue}{{u1}^{2}}} \]

6. Simplified 73.0%

   \[\leadsto \sqrt{\color{blue}{u1 + {u1}^{2}}} \]

7. Taylor expanded in u1 around -inf 4.2%

   \[\leadsto \color{blue}{-1 \cdot u1} \]
8. Step-by-step derivation

   1. mul-1-neg 4.2%

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

9. Simplified 4.2%

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

10. Final simplification 4.2%

    \[\leadsto -u1 \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024112 
(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))))