Beckmann Sample, normalization factor

Percentage Accurate: 97.8% → 98.5%

Time: 18.4s

Alternatives: 11

Speedup: 1.0×

Specification

\[\left(0 < cosTheta \land cosTheta < 0.9999\right) \land \left(-1 < c \land c < 1\right)\]

Math

\[\begin{array}{l} \\ \frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \end{array} \]

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (/
  1.0
  (+
   (+ 1.0 c)
   (*
    (* (/ 1.0 (sqrt PI)) (/ (sqrt (- (- 1.0 cosTheta) cosTheta)) cosTheta))
    (exp (* (- cosTheta) cosTheta))))))

C

float code(float cosTheta, float c) {
	return 1.0f / ((1.0f + c) + (((1.0f / sqrtf(((float) M_PI))) * (sqrtf(((1.0f - cosTheta) - cosTheta)) / cosTheta)) * expf((-cosTheta * cosTheta))));
}

Julia

function code(cosTheta, c)
	return Float32(Float32(1.0) / Float32(Float32(Float32(1.0) + c) + Float32(Float32(Float32(Float32(1.0) / sqrt(Float32(pi))) * Float32(sqrt(Float32(Float32(Float32(1.0) - cosTheta) - cosTheta)) / cosTheta)) * exp(Float32(Float32(-cosTheta) * cosTheta)))))
end

MATLAB

function tmp = code(cosTheta, c)
	tmp = single(1.0) / ((single(1.0) + c) + (((single(1.0) / sqrt(single(pi))) * (sqrt(((single(1.0) - cosTheta) - cosTheta)) / cosTheta)) * exp((-cosTheta * cosTheta))));
end

Initial Program: 97.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \end{array} \]

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (/
  1.0
  (+
   (+ 1.0 c)
   (*
    (* (/ 1.0 (sqrt PI)) (/ (sqrt (- (- 1.0 cosTheta) cosTheta)) cosTheta))
    (exp (* (- cosTheta) cosTheta))))))

C

float code(float cosTheta, float c) {
	return 1.0f / ((1.0f + c) + (((1.0f / sqrtf(((float) M_PI))) * (sqrtf(((1.0f - cosTheta) - cosTheta)) / cosTheta)) * expf((-cosTheta * cosTheta))));
}

Julia

function code(cosTheta, c)
	return Float32(Float32(1.0) / Float32(Float32(Float32(1.0) + c) + Float32(Float32(Float32(Float32(1.0) / sqrt(Float32(pi))) * Float32(sqrt(Float32(Float32(Float32(1.0) - cosTheta) - cosTheta)) / cosTheta)) * exp(Float32(Float32(-cosTheta) * cosTheta)))))
end

MATLAB

function tmp = code(cosTheta, c)
	tmp = single(1.0) / ((single(1.0) + c) + (((single(1.0) / sqrt(single(pi))) * (sqrt(((single(1.0) - cosTheta) - cosTheta)) / cosTheta)) * exp((-cosTheta * cosTheta))));
end

Alternative 1: 98.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 + \mathsf{fma}\left(\frac{\sqrt{1 + cosTheta \cdot -2}}{cosTheta \cdot \sqrt{\pi}}, {\left(e^{-cosTheta}\right)}^{cosTheta}, c\right)} \end{array} \]

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (/
  1.0
  (+
   1.0
   (fma
    (/ (sqrt (+ 1.0 (* cosTheta -2.0))) (* cosTheta (sqrt PI)))
    (pow (exp (- cosTheta)) cosTheta)
    c))))

C

float code(float cosTheta, float c) {
	return 1.0f / (1.0f + fmaf((sqrtf((1.0f + (cosTheta * -2.0f))) / (cosTheta * sqrtf(((float) M_PI)))), powf(expf(-cosTheta), cosTheta), c));
}

Julia

function code(cosTheta, c)
	return Float32(Float32(1.0) / Float32(Float32(1.0) + fma(Float32(sqrt(Float32(Float32(1.0) + Float32(cosTheta * Float32(-2.0)))) / Float32(cosTheta * sqrt(Float32(pi)))), (exp(Float32(-cosTheta)) ^ cosTheta), c)))
end

Derivation

1. Initial program 98.0%

   \[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
2. Step-by-step derivation

   1. associate-+l+ 98.0%

      \[\leadsto \frac{1}{\color{blue}{1 + \left(c + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}\right)}} \]

   2. +-commutative 98.0%

      \[\leadsto \frac{1}{1 + \color{blue}{\left(\left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta} + c\right)}} \]

   3. fma-define 98.0%

      \[\leadsto \frac{1}{1 + \color{blue}{\mathsf{fma}\left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}, e^{\left(-cosTheta\right) \cdot cosTheta}, c\right)}} \]

3. Simplified 98.6%

   \[\leadsto \color{blue}{\frac{1}{1 + \mathsf{fma}\left(\frac{\sqrt{1 + cosTheta \cdot -2}}{\sqrt{\pi} \cdot cosTheta}, {\left(e^{-cosTheta}\right)}^{cosTheta}, c\right)}} \]
4. Add Preprocessing

5. Final simplification 98.6%

   \[\leadsto \frac{1}{1 + \mathsf{fma}\left(\frac{\sqrt{1 + cosTheta \cdot -2}}{cosTheta \cdot \sqrt{\pi}}, {\left(e^{-cosTheta}\right)}^{cosTheta}, c\right)} \]
6. Add Preprocessing

Alternative 2: 97.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 + \left(c + \sqrt{\frac{1 + cosTheta \cdot -2}{\pi}} \cdot \frac{e^{-{cosTheta}^{2}}}{cosTheta}\right)} \end{array} \]

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (/
  1.0
  (+
   1.0
   (+
    c
    (*
     (sqrt (/ (+ 1.0 (* cosTheta -2.0)) PI))
     (/ (exp (- (pow cosTheta 2.0))) cosTheta))))))

C

float code(float cosTheta, float c) {
	return 1.0f / (1.0f + (c + (sqrtf(((1.0f + (cosTheta * -2.0f)) / ((float) M_PI))) * (expf(-powf(cosTheta, 2.0f)) / cosTheta))));
}

Julia

function code(cosTheta, c)
	return Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(c + Float32(sqrt(Float32(Float32(Float32(1.0) + Float32(cosTheta * Float32(-2.0))) / Float32(pi))) * Float32(exp(Float32(-(cosTheta ^ Float32(2.0)))) / cosTheta)))))
end

MATLAB

function tmp = code(cosTheta, c)
	tmp = single(1.0) / (single(1.0) + (c + (sqrt(((single(1.0) + (cosTheta * single(-2.0))) / single(pi))) * (exp(-(cosTheta ^ single(2.0))) / cosTheta))));
end

Derivation

1. Initial program 98.0%

   \[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
2. Step-by-step derivation

   1. associate-+l+ 98.0%

      \[\leadsto \frac{1}{\color{blue}{1 + \left(c + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}\right)}} \]

   2. +-commutative 98.0%

      \[\leadsto \frac{1}{1 + \color{blue}{\left(\left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta} + c\right)}} \]

   3. fma-define 98.0%

      \[\leadsto \frac{1}{1 + \color{blue}{\mathsf{fma}\left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}, e^{\left(-cosTheta\right) \cdot cosTheta}, c\right)}} \]

3. Simplified 98.6%

   \[\leadsto \color{blue}{\frac{1}{1 + \mathsf{fma}\left(\frac{\sqrt{1 + cosTheta \cdot -2}}{\sqrt{\pi} \cdot cosTheta}, {\left(e^{-cosTheta}\right)}^{cosTheta}, c\right)}} \]
4. Add Preprocessing

5. Taylor expanded in c around 0 98.2%

   \[\leadsto \frac{1}{1 + \color{blue}{\left(c + \frac{e^{-1 \cdot {cosTheta}^{2}}}{cosTheta} \cdot \sqrt{\frac{1 + -2 \cdot cosTheta}{\pi}}\right)}} \]

6. Final simplification 98.2%

   \[\leadsto \frac{1}{1 + \left(c + \sqrt{\frac{1 + cosTheta \cdot -2}{\pi}} \cdot \frac{e^{-{cosTheta}^{2}}}{cosTheta}\right)} \]
7. Add Preprocessing

Alternative 3: 97.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 + \sqrt{\frac{1 + cosTheta \cdot -2}{\pi}} \cdot \frac{e^{-{cosTheta}^{2}}}{cosTheta}} \end{array} \]

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (/
  1.0
  (+
   1.0
   (*
    (sqrt (/ (+ 1.0 (* cosTheta -2.0)) PI))
    (/ (exp (- (pow cosTheta 2.0))) cosTheta)))))

C

float code(float cosTheta, float c) {
	return 1.0f / (1.0f + (sqrtf(((1.0f + (cosTheta * -2.0f)) / ((float) M_PI))) * (expf(-powf(cosTheta, 2.0f)) / cosTheta)));
}

Julia

function code(cosTheta, c)
	return Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(sqrt(Float32(Float32(Float32(1.0) + Float32(cosTheta * Float32(-2.0))) / Float32(pi))) * Float32(exp(Float32(-(cosTheta ^ Float32(2.0)))) / cosTheta))))
end

MATLAB

function tmp = code(cosTheta, c)
	tmp = single(1.0) / (single(1.0) + (sqrt(((single(1.0) + (cosTheta * single(-2.0))) / single(pi))) * (exp(-(cosTheta ^ single(2.0))) / cosTheta)));
end

Derivation

1. Initial program 98.0%

   \[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. add-cube-cbrt 98.0%

      \[\leadsto \frac{1}{\left(1 + c\right) + \left(\color{blue}{\left(\left(\sqrt[3]{\frac{1}{\sqrt{\pi}}} \cdot \sqrt[3]{\frac{1}{\sqrt{\pi}}}\right) \cdot \sqrt[3]{\frac{1}{\sqrt{\pi}}}\right)} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   2. cbrt-unprod 98.0%

      \[\leadsto \frac{1}{\left(1 + c\right) + \left(\left(\color{blue}{\sqrt[3]{\frac{1}{\sqrt{\pi}} \cdot \frac{1}{\sqrt{\pi}}}} \cdot \sqrt[3]{\frac{1}{\sqrt{\pi}}}\right) \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   3. frac-times 98.0%

      \[\leadsto \frac{1}{\left(1 + c\right) + \left(\left(\sqrt[3]{\color{blue}{\frac{1 \cdot 1}{\sqrt{\pi} \cdot \sqrt{\pi}}}} \cdot \sqrt[3]{\frac{1}{\sqrt{\pi}}}\right) \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   4. metadata-eval 98.0%

      \[\leadsto \frac{1}{\left(1 + c\right) + \left(\left(\sqrt[3]{\frac{\color{blue}{1}}{\sqrt{\pi} \cdot \sqrt{\pi}}} \cdot \sqrt[3]{\frac{1}{\sqrt{\pi}}}\right) \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   5. add-sqr-sqrt 98.0%

      \[\leadsto \frac{1}{\left(1 + c\right) + \left(\left(\sqrt[3]{\frac{1}{\color{blue}{\pi}}} \cdot \sqrt[3]{\frac{1}{\sqrt{\pi}}}\right) \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   6. inv-pow 98.0%

      \[\leadsto \frac{1}{\left(1 + c\right) + \left(\left(\sqrt[3]{\frac{1}{\pi}} \cdot \sqrt[3]{\color{blue}{{\left(\sqrt{\pi}\right)}^{-1}}}\right) \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   7. sqrt-pow2 98.0%

      \[\leadsto \frac{1}{\left(1 + c\right) + \left(\left(\sqrt[3]{\frac{1}{\pi}} \cdot \sqrt[3]{\color{blue}{{\pi}^{\left(\frac{-1}{2}\right)}}}\right) \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   8. metadata-eval 98.0%

      \[\leadsto \frac{1}{\left(1 + c\right) + \left(\left(\sqrt[3]{\frac{1}{\pi}} \cdot \sqrt[3]{{\pi}^{\color{blue}{-0.5}}}\right) \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

4. Applied egg-rr 98.0%

   \[\leadsto \frac{1}{\left(1 + c\right) + \left(\color{blue}{\left(\sqrt[3]{\frac{1}{\pi}} \cdot \sqrt[3]{{\pi}^{-0.5}}\right)} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
5. Step-by-step derivation

   1. associate-*r/ 98.0%

      \[\leadsto \frac{1}{\left(1 + c\right) + \color{blue}{\frac{\left(\sqrt[3]{\frac{1}{\pi}} \cdot \sqrt[3]{{\pi}^{-0.5}}\right) \cdot \sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   2. cbrt-unprod 97.9%

      \[\leadsto \frac{1}{\left(1 + c\right) + \frac{\color{blue}{\sqrt[3]{\frac{1}{\pi} \cdot {\pi}^{-0.5}}} \cdot \sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   3. inv-pow 97.9%

      \[\leadsto \frac{1}{\left(1 + c\right) + \frac{\sqrt[3]{\color{blue}{{\pi}^{-1}} \cdot {\pi}^{-0.5}} \cdot \sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   4. metadata-eval 97.9%

      \[\leadsto \frac{1}{\left(1 + c\right) + \frac{\sqrt[3]{{\pi}^{\color{blue}{\left(-0.5 + -0.5\right)}} \cdot {\pi}^{-0.5}} \cdot \sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   5. pow-prod-up 97.9%

      \[\leadsto \frac{1}{\left(1 + c\right) + \frac{\sqrt[3]{\color{blue}{\left({\pi}^{-0.5} \cdot {\pi}^{-0.5}\right)} \cdot {\pi}^{-0.5}} \cdot \sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   6. add-cbrt-cube 97.9%

      \[\leadsto \frac{1}{\left(1 + c\right) + \frac{\color{blue}{{\pi}^{-0.5}} \cdot \sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

6. Applied egg-rr 97.9%

   \[\leadsto \frac{1}{\left(1 + c\right) + \color{blue}{\frac{{\pi}^{-0.5} \cdot \sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

7. Taylor expanded in c around 0 97.8%

   \[\leadsto \color{blue}{\frac{1}{1 + \frac{e^{-1 \cdot {cosTheta}^{2}}}{cosTheta} \cdot \sqrt{\frac{1 - 2 \cdot cosTheta}{\pi}}}} \]
8. Step-by-step derivation

   1. *-commutative 97.8%

      \[\leadsto \frac{1}{1 + \color{blue}{\sqrt{\frac{1 - 2 \cdot cosTheta}{\pi}} \cdot \frac{e^{-1 \cdot {cosTheta}^{2}}}{cosTheta}}} \]

   2. cancel-sign-sub-inv 97.8%

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

   3. metadata-eval 97.8%

      \[\leadsto \frac{1}{1 + \sqrt{\frac{1 + \color{blue}{-2} \cdot cosTheta}{\pi}} \cdot \frac{e^{-1 \cdot {cosTheta}^{2}}}{cosTheta}} \]

   4. neg-mul-1 97.8%

      \[\leadsto \frac{1}{1 + \sqrt{\frac{1 + -2 \cdot cosTheta}{\pi}} \cdot \frac{e^{\color{blue}{-{cosTheta}^{2}}}}{cosTheta}} \]

9. Simplified 97.8%

   \[\leadsto \color{blue}{\frac{1}{1 + \sqrt{\frac{1 + -2 \cdot cosTheta}{\pi}} \cdot \frac{e^{-{cosTheta}^{2}}}{cosTheta}}} \]

10. Final simplification 97.8%

    \[\leadsto \frac{1}{1 + \sqrt{\frac{1 + cosTheta \cdot -2}{\pi}} \cdot \frac{e^{-{cosTheta}^{2}}}{cosTheta}} \]
11. Add Preprocessing

Alternative 4: 96.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 + \left(c + \sqrt{cosTheta \cdot \left(\frac{1}{cosTheta \cdot \pi} - \frac{2}{\pi}\right)} \cdot \left(\left(1 - {cosTheta}^{2}\right) \cdot \frac{1}{cosTheta}\right)\right)} \end{array} \]

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (/
  1.0
  (+
   1.0
   (+
    c
    (*
     (sqrt (* cosTheta (- (/ 1.0 (* cosTheta PI)) (/ 2.0 PI))))
     (* (- 1.0 (pow cosTheta 2.0)) (/ 1.0 cosTheta)))))))

C

float code(float cosTheta, float c) {
	return 1.0f / (1.0f + (c + (sqrtf((cosTheta * ((1.0f / (cosTheta * ((float) M_PI))) - (2.0f / ((float) M_PI))))) * ((1.0f - powf(cosTheta, 2.0f)) * (1.0f / cosTheta)))));
}

Julia

function code(cosTheta, c)
	return Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(c + Float32(sqrt(Float32(cosTheta * Float32(Float32(Float32(1.0) / Float32(cosTheta * Float32(pi))) - Float32(Float32(2.0) / Float32(pi))))) * Float32(Float32(Float32(1.0) - (cosTheta ^ Float32(2.0))) * Float32(Float32(1.0) / cosTheta))))))
end

MATLAB

function tmp = code(cosTheta, c)
	tmp = single(1.0) / (single(1.0) + (c + (sqrt((cosTheta * ((single(1.0) / (cosTheta * single(pi))) - (single(2.0) / single(pi))))) * ((single(1.0) - (cosTheta ^ single(2.0))) * (single(1.0) / cosTheta)))));
end

Derivation

1. Initial program 98.0%

   \[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
2. Step-by-step derivation

   1. associate-+l+ 98.0%

      \[\leadsto \frac{1}{\color{blue}{1 + \left(c + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}\right)}} \]

   2. +-commutative 98.0%

      \[\leadsto \frac{1}{1 + \color{blue}{\left(\left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta} + c\right)}} \]

   3. fma-define 98.0%

      \[\leadsto \frac{1}{1 + \color{blue}{\mathsf{fma}\left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}, e^{\left(-cosTheta\right) \cdot cosTheta}, c\right)}} \]

3. Simplified 98.6%

   \[\leadsto \color{blue}{\frac{1}{1 + \mathsf{fma}\left(\frac{\sqrt{1 + cosTheta \cdot -2}}{\sqrt{\pi} \cdot cosTheta}, {\left(e^{-cosTheta}\right)}^{cosTheta}, c\right)}} \]
4. Add Preprocessing

5. Taylor expanded in c around 0 98.2%

   \[\leadsto \frac{1}{1 + \color{blue}{\left(c + \frac{e^{-1 \cdot {cosTheta}^{2}}}{cosTheta} \cdot \sqrt{\frac{1 + -2 \cdot cosTheta}{\pi}}\right)}} \]

6. Taylor expanded in cosTheta around inf 98.1%

   \[\leadsto \frac{1}{1 + \left(c + \frac{e^{-1 \cdot {cosTheta}^{2}}}{cosTheta} \cdot \sqrt{\color{blue}{cosTheta \cdot \left(\frac{1}{cosTheta \cdot \pi} - 2 \cdot \frac{1}{\pi}\right)}}\right)} \]
7. Step-by-step derivation

   1. *-commutative 98.1%

      \[\leadsto \frac{1}{1 + \left(c + \frac{e^{-1 \cdot {cosTheta}^{2}}}{cosTheta} \cdot \sqrt{cosTheta \cdot \left(\frac{1}{\color{blue}{\pi \cdot cosTheta}} - 2 \cdot \frac{1}{\pi}\right)}\right)} \]

   2. associate-*r/ 98.1%

      \[\leadsto \frac{1}{1 + \left(c + \frac{e^{-1 \cdot {cosTheta}^{2}}}{cosTheta} \cdot \sqrt{cosTheta \cdot \left(\frac{1}{\pi \cdot cosTheta} - \color{blue}{\frac{2 \cdot 1}{\pi}}\right)}\right)} \]

   3. metadata-eval 98.1%

      \[\leadsto \frac{1}{1 + \left(c + \frac{e^{-1 \cdot {cosTheta}^{2}}}{cosTheta} \cdot \sqrt{cosTheta \cdot \left(\frac{1}{\pi \cdot cosTheta} - \frac{\color{blue}{2}}{\pi}\right)}\right)} \]

8. Simplified 98.1%

   \[\leadsto \frac{1}{1 + \left(c + \frac{e^{-1 \cdot {cosTheta}^{2}}}{cosTheta} \cdot \sqrt{\color{blue}{cosTheta \cdot \left(\frac{1}{\pi \cdot cosTheta} - \frac{2}{\pi}\right)}}\right)} \]
9. Step-by-step derivation

   1. add-exp-log 90.1%

      \[\leadsto \frac{1}{1 + \left(c + \color{blue}{e^{\log \left(\frac{e^{-1 \cdot {cosTheta}^{2}}}{cosTheta} \cdot \sqrt{cosTheta \cdot \left(\frac{1}{\pi \cdot cosTheta} - \frac{2}{\pi}\right)}\right)}}\right)} \]

   2. *-commutative 90.1%

      \[\leadsto \frac{1}{1 + \left(c + e^{\log \color{blue}{\left(\sqrt{cosTheta \cdot \left(\frac{1}{\pi \cdot cosTheta} - \frac{2}{\pi}\right)} \cdot \frac{e^{-1 \cdot {cosTheta}^{2}}}{cosTheta}\right)}}\right)} \]

   3. log-prod 89.9%

      \[\leadsto \frac{1}{1 + \left(c + e^{\color{blue}{\log \left(\sqrt{cosTheta \cdot \left(\frac{1}{\pi \cdot cosTheta} - \frac{2}{\pi}\right)}\right) + \log \left(\frac{e^{-1 \cdot {cosTheta}^{2}}}{cosTheta}\right)}}\right)} \]

   4. *-commutative 89.9%

      \[\leadsto \frac{1}{1 + \left(c + e^{\log \left(\sqrt{cosTheta \cdot \left(\frac{1}{\color{blue}{cosTheta \cdot \pi}} - \frac{2}{\pi}\right)}\right) + \log \left(\frac{e^{-1 \cdot {cosTheta}^{2}}}{cosTheta}\right)}\right)} \]

   5. mul-1-neg 89.9%

      \[\leadsto \frac{1}{1 + \left(c + e^{\log \left(\sqrt{cosTheta \cdot \left(\frac{1}{cosTheta \cdot \pi} - \frac{2}{\pi}\right)}\right) + \log \left(\frac{e^{\color{blue}{-{cosTheta}^{2}}}}{cosTheta}\right)}\right)} \]

   6. unpow2 89.9%

      \[\leadsto \frac{1}{1 + \left(c + e^{\log \left(\sqrt{cosTheta \cdot \left(\frac{1}{cosTheta \cdot \pi} - \frac{2}{\pi}\right)}\right) + \log \left(\frac{e^{-\color{blue}{cosTheta \cdot cosTheta}}}{cosTheta}\right)}\right)} \]

   7. distribute-lft-neg-out 89.9%

      \[\leadsto \frac{1}{1 + \left(c + e^{\log \left(\sqrt{cosTheta \cdot \left(\frac{1}{cosTheta \cdot \pi} - \frac{2}{\pi}\right)}\right) + \log \left(\frac{e^{\color{blue}{\left(-cosTheta\right) \cdot cosTheta}}}{cosTheta}\right)}\right)} \]

10. Applied egg-rr 89.9%

    \[\leadsto \frac{1}{1 + \left(c + \color{blue}{e^{\log \left(\sqrt{cosTheta \cdot \left(\frac{1}{cosTheta \cdot \pi} - \frac{2}{\pi}\right)}\right) + \left(\left(-{cosTheta}^{2}\right) - \log cosTheta\right)}}\right)} \]
11. Step-by-step derivation

    1. exp-sum 90.1%

       \[\leadsto \frac{1}{1 + \left(c + \color{blue}{e^{\log \left(\sqrt{cosTheta \cdot \left(\frac{1}{cosTheta \cdot \pi} - \frac{2}{\pi}\right)}\right)} \cdot e^{\left(-{cosTheta}^{2}\right) - \log cosTheta}}\right)} \]

    2. rem-exp-log 90.1%

       \[\leadsto \frac{1}{1 + \left(c + \color{blue}{\sqrt{cosTheta \cdot \left(\frac{1}{cosTheta \cdot \pi} - \frac{2}{\pi}\right)}} \cdot e^{\left(-{cosTheta}^{2}\right) - \log cosTheta}\right)} \]

    3. *-commutative 90.1%

       \[\leadsto \frac{1}{1 + \left(c + \sqrt{cosTheta \cdot \left(\frac{1}{\color{blue}{\pi \cdot cosTheta}} - \frac{2}{\pi}\right)} \cdot e^{\left(-{cosTheta}^{2}\right) - \log cosTheta}\right)} \]

12. Simplified 90.1%

    \[\leadsto \frac{1}{1 + \left(c + \color{blue}{\sqrt{cosTheta \cdot \left(\frac{1}{\pi \cdot cosTheta} - \frac{2}{\pi}\right)} \cdot e^{\left(-{cosTheta}^{2}\right) - \log cosTheta}}\right)} \]

13. Taylor expanded in cosTheta around 0 89.1%

    \[\leadsto \frac{1}{1 + \left(c + \sqrt{cosTheta \cdot \left(\frac{1}{\pi \cdot cosTheta} - \frac{2}{\pi}\right)} \cdot \color{blue}{\left(e^{-\log cosTheta} + -1 \cdot \left({cosTheta}^{2} \cdot e^{-\log cosTheta}\right)\right)}\right)} \]
14. Step-by-step derivation

    1. associate-*r* 89.1%

       \[\leadsto \frac{1}{1 + \left(c + \sqrt{cosTheta \cdot \left(\frac{1}{\pi \cdot cosTheta} - \frac{2}{\pi}\right)} \cdot \left(e^{-\log cosTheta} + \color{blue}{\left(-1 \cdot {cosTheta}^{2}\right) \cdot e^{-\log cosTheta}}\right)\right)} \]

    2. neg-mul-1 89.1%

       \[\leadsto \frac{1}{1 + \left(c + \sqrt{cosTheta \cdot \left(\frac{1}{\pi \cdot cosTheta} - \frac{2}{\pi}\right)} \cdot \left(e^{-\log cosTheta} + \color{blue}{\left(-{cosTheta}^{2}\right)} \cdot e^{-\log cosTheta}\right)\right)} \]

    3. distribute-rgt1-in 89.1%

       \[\leadsto \frac{1}{1 + \left(c + \sqrt{cosTheta \cdot \left(\frac{1}{\pi \cdot cosTheta} - \frac{2}{\pi}\right)} \cdot \color{blue}{\left(\left(\left(-{cosTheta}^{2}\right) + 1\right) \cdot e^{-\log cosTheta}\right)}\right)} \]

    4. exp-neg 89.1%

       \[\leadsto \frac{1}{1 + \left(c + \sqrt{cosTheta \cdot \left(\frac{1}{\pi \cdot cosTheta} - \frac{2}{\pi}\right)} \cdot \left(\left(\left(-{cosTheta}^{2}\right) + 1\right) \cdot \color{blue}{\frac{1}{e^{\log cosTheta}}}\right)\right)} \]

    5. rem-exp-log 97.0%

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

15. Simplified 97.0%

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

16. Final simplification 97.0%

    \[\leadsto \frac{1}{1 + \left(c + \sqrt{cosTheta \cdot \left(\frac{1}{cosTheta \cdot \pi} - \frac{2}{\pi}\right)} \cdot \left(\left(1 - {cosTheta}^{2}\right) \cdot \frac{1}{cosTheta}\right)\right)} \]
17. Add Preprocessing

Alternative 5: 95.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ cosTheta \cdot \left(\sqrt{\pi} + \pi \cdot \left(cosTheta \cdot \left(\sqrt{\frac{1}{\pi}} + \left(-1 - c\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (*
  cosTheta
  (+ (sqrt PI) (* PI (* cosTheta (+ (sqrt (/ 1.0 PI)) (- -1.0 c)))))))

C

float code(float cosTheta, float c) {
	return cosTheta * (sqrtf(((float) M_PI)) + (((float) M_PI) * (cosTheta * (sqrtf((1.0f / ((float) M_PI))) + (-1.0f - c)))));
}

Julia

function code(cosTheta, c)
	return Float32(cosTheta * Float32(sqrt(Float32(pi)) + Float32(Float32(pi) * Float32(cosTheta * Float32(sqrt(Float32(Float32(1.0) / Float32(pi))) + Float32(Float32(-1.0) - c))))))
end

MATLAB

function tmp = code(cosTheta, c)
	tmp = cosTheta * (sqrt(single(pi)) + (single(pi) * (cosTheta * (sqrt((single(1.0) / single(pi))) + (single(-1.0) - c)))));
end

Derivation

1. Initial program 98.0%

   \[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
2. Step-by-step derivation

   1. associate-+l+ 98.0%

      \[\leadsto \frac{1}{\color{blue}{1 + \left(c + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}\right)}} \]

   2. +-commutative 98.0%

      \[\leadsto \frac{1}{1 + \color{blue}{\left(\left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta} + c\right)}} \]

   3. fma-define 98.0%

      \[\leadsto \frac{1}{1 + \color{blue}{\mathsf{fma}\left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}, e^{\left(-cosTheta\right) \cdot cosTheta}, c\right)}} \]

3. Simplified 98.6%

   \[\leadsto \color{blue}{\frac{1}{1 + \mathsf{fma}\left(\frac{\sqrt{1 + cosTheta \cdot -2}}{\sqrt{\pi} \cdot cosTheta}, {\left(e^{-cosTheta}\right)}^{cosTheta}, c\right)}} \]
4. Add Preprocessing

5. Taylor expanded in cosTheta around 0 96.0%

   \[\leadsto \color{blue}{cosTheta \cdot \left(\sqrt{\pi} + -1 \cdot \left(cosTheta \cdot \left(\pi \cdot \left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)\right)\right)} \]
6. Step-by-step derivation

   1. mul-1-neg 96.0%

      \[\leadsto cosTheta \cdot \left(\sqrt{\pi} + \color{blue}{\left(-cosTheta \cdot \left(\pi \cdot \left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)\right)}\right) \]

   2. unsub-neg 96.0%

      \[\leadsto cosTheta \cdot \color{blue}{\left(\sqrt{\pi} - cosTheta \cdot \left(\pi \cdot \left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)\right)} \]

   3. associate-*r* 96.0%

      \[\leadsto cosTheta \cdot \left(\sqrt{\pi} - \color{blue}{\left(cosTheta \cdot \pi\right) \cdot \left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\pi}}\right)\right)}\right) \]

   4. *-commutative 96.0%

      \[\leadsto cosTheta \cdot \left(\sqrt{\pi} - \color{blue}{\left(\pi \cdot cosTheta\right)} \cdot \left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right) \]

   5. associate-*l* 96.0%

      \[\leadsto cosTheta \cdot \left(\sqrt{\pi} - \color{blue}{\pi \cdot \left(cosTheta \cdot \left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)}\right) \]

   6. associate-+r+ 96.0%

      \[\leadsto cosTheta \cdot \left(\sqrt{\pi} - \pi \cdot \left(cosTheta \cdot \color{blue}{\left(\left(1 + c\right) + -1 \cdot \sqrt{\frac{1}{\pi}}\right)}\right)\right) \]

   7. mul-1-neg 96.0%

      \[\leadsto cosTheta \cdot \left(\sqrt{\pi} - \pi \cdot \left(cosTheta \cdot \left(\left(1 + c\right) + \color{blue}{\left(-\sqrt{\frac{1}{\pi}}\right)}\right)\right)\right) \]

   8. unsub-neg 96.0%

      \[\leadsto cosTheta \cdot \left(\sqrt{\pi} - \pi \cdot \left(cosTheta \cdot \color{blue}{\left(\left(1 + c\right) - \sqrt{\frac{1}{\pi}}\right)}\right)\right) \]

7. Simplified 96.0%

   \[\leadsto \color{blue}{cosTheta \cdot \left(\sqrt{\pi} - \pi \cdot \left(cosTheta \cdot \left(\left(1 + c\right) - \sqrt{\frac{1}{\pi}}\right)\right)\right)} \]

8. Final simplification 96.0%

   \[\leadsto cosTheta \cdot \left(\sqrt{\pi} + \pi \cdot \left(cosTheta \cdot \left(\sqrt{\frac{1}{\pi}} + \left(-1 - c\right)\right)\right)\right) \]
9. Add Preprocessing

Alternative 6: 95.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 + \left(c + \sqrt{cosTheta \cdot \left(\frac{1}{cosTheta \cdot \pi} - \frac{2}{\pi}\right)} \cdot \frac{1}{cosTheta}\right)} \end{array} \]

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (/
  1.0
  (+
   1.0
   (+
    c
    (*
     (sqrt (* cosTheta (- (/ 1.0 (* cosTheta PI)) (/ 2.0 PI))))
     (/ 1.0 cosTheta))))))

C

float code(float cosTheta, float c) {
	return 1.0f / (1.0f + (c + (sqrtf((cosTheta * ((1.0f / (cosTheta * ((float) M_PI))) - (2.0f / ((float) M_PI))))) * (1.0f / cosTheta))));
}

Julia

function code(cosTheta, c)
	return Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(c + Float32(sqrt(Float32(cosTheta * Float32(Float32(Float32(1.0) / Float32(cosTheta * Float32(pi))) - Float32(Float32(2.0) / Float32(pi))))) * Float32(Float32(1.0) / cosTheta)))))
end

MATLAB

function tmp = code(cosTheta, c)
	tmp = single(1.0) / (single(1.0) + (c + (sqrt((cosTheta * ((single(1.0) / (cosTheta * single(pi))) - (single(2.0) / single(pi))))) * (single(1.0) / cosTheta))));
end

Derivation

1. Initial program 98.0%

   \[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
2. Step-by-step derivation

   1. associate-+l+ 98.0%

      \[\leadsto \frac{1}{\color{blue}{1 + \left(c + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}\right)}} \]

   2. +-commutative 98.0%

      \[\leadsto \frac{1}{1 + \color{blue}{\left(\left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta} + c\right)}} \]

   3. fma-define 98.0%

      \[\leadsto \frac{1}{1 + \color{blue}{\mathsf{fma}\left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}, e^{\left(-cosTheta\right) \cdot cosTheta}, c\right)}} \]

3. Simplified 98.6%

   \[\leadsto \color{blue}{\frac{1}{1 + \mathsf{fma}\left(\frac{\sqrt{1 + cosTheta \cdot -2}}{\sqrt{\pi} \cdot cosTheta}, {\left(e^{-cosTheta}\right)}^{cosTheta}, c\right)}} \]
4. Add Preprocessing

5. Taylor expanded in c around 0 98.2%

   \[\leadsto \frac{1}{1 + \color{blue}{\left(c + \frac{e^{-1 \cdot {cosTheta}^{2}}}{cosTheta} \cdot \sqrt{\frac{1 + -2 \cdot cosTheta}{\pi}}\right)}} \]

6. Taylor expanded in cosTheta around inf 98.1%

   \[\leadsto \frac{1}{1 + \left(c + \frac{e^{-1 \cdot {cosTheta}^{2}}}{cosTheta} \cdot \sqrt{\color{blue}{cosTheta \cdot \left(\frac{1}{cosTheta \cdot \pi} - 2 \cdot \frac{1}{\pi}\right)}}\right)} \]
7. Step-by-step derivation

   1. *-commutative 98.1%

      \[\leadsto \frac{1}{1 + \left(c + \frac{e^{-1 \cdot {cosTheta}^{2}}}{cosTheta} \cdot \sqrt{cosTheta \cdot \left(\frac{1}{\color{blue}{\pi \cdot cosTheta}} - 2 \cdot \frac{1}{\pi}\right)}\right)} \]

   2. associate-*r/ 98.1%

      \[\leadsto \frac{1}{1 + \left(c + \frac{e^{-1 \cdot {cosTheta}^{2}}}{cosTheta} \cdot \sqrt{cosTheta \cdot \left(\frac{1}{\pi \cdot cosTheta} - \color{blue}{\frac{2 \cdot 1}{\pi}}\right)}\right)} \]

   3. metadata-eval 98.1%

      \[\leadsto \frac{1}{1 + \left(c + \frac{e^{-1 \cdot {cosTheta}^{2}}}{cosTheta} \cdot \sqrt{cosTheta \cdot \left(\frac{1}{\pi \cdot cosTheta} - \frac{\color{blue}{2}}{\pi}\right)}\right)} \]

8. Simplified 98.1%

   \[\leadsto \frac{1}{1 + \left(c + \frac{e^{-1 \cdot {cosTheta}^{2}}}{cosTheta} \cdot \sqrt{\color{blue}{cosTheta \cdot \left(\frac{1}{\pi \cdot cosTheta} - \frac{2}{\pi}\right)}}\right)} \]
9. Step-by-step derivation

   1. add-exp-log 90.1%

      \[\leadsto \frac{1}{1 + \left(c + \color{blue}{e^{\log \left(\frac{e^{-1 \cdot {cosTheta}^{2}}}{cosTheta} \cdot \sqrt{cosTheta \cdot \left(\frac{1}{\pi \cdot cosTheta} - \frac{2}{\pi}\right)}\right)}}\right)} \]

   2. *-commutative 90.1%

      \[\leadsto \frac{1}{1 + \left(c + e^{\log \color{blue}{\left(\sqrt{cosTheta \cdot \left(\frac{1}{\pi \cdot cosTheta} - \frac{2}{\pi}\right)} \cdot \frac{e^{-1 \cdot {cosTheta}^{2}}}{cosTheta}\right)}}\right)} \]

   3. log-prod 89.9%

      \[\leadsto \frac{1}{1 + \left(c + e^{\color{blue}{\log \left(\sqrt{cosTheta \cdot \left(\frac{1}{\pi \cdot cosTheta} - \frac{2}{\pi}\right)}\right) + \log \left(\frac{e^{-1 \cdot {cosTheta}^{2}}}{cosTheta}\right)}}\right)} \]

   4. *-commutative 89.9%

      \[\leadsto \frac{1}{1 + \left(c + e^{\log \left(\sqrt{cosTheta \cdot \left(\frac{1}{\color{blue}{cosTheta \cdot \pi}} - \frac{2}{\pi}\right)}\right) + \log \left(\frac{e^{-1 \cdot {cosTheta}^{2}}}{cosTheta}\right)}\right)} \]

   5. mul-1-neg 89.9%

      \[\leadsto \frac{1}{1 + \left(c + e^{\log \left(\sqrt{cosTheta \cdot \left(\frac{1}{cosTheta \cdot \pi} - \frac{2}{\pi}\right)}\right) + \log \left(\frac{e^{\color{blue}{-{cosTheta}^{2}}}}{cosTheta}\right)}\right)} \]

   6. unpow2 89.9%

      \[\leadsto \frac{1}{1 + \left(c + e^{\log \left(\sqrt{cosTheta \cdot \left(\frac{1}{cosTheta \cdot \pi} - \frac{2}{\pi}\right)}\right) + \log \left(\frac{e^{-\color{blue}{cosTheta \cdot cosTheta}}}{cosTheta}\right)}\right)} \]

   7. distribute-lft-neg-out 89.9%

      \[\leadsto \frac{1}{1 + \left(c + e^{\log \left(\sqrt{cosTheta \cdot \left(\frac{1}{cosTheta \cdot \pi} - \frac{2}{\pi}\right)}\right) + \log \left(\frac{e^{\color{blue}{\left(-cosTheta\right) \cdot cosTheta}}}{cosTheta}\right)}\right)} \]

10. Applied egg-rr 89.9%

    \[\leadsto \frac{1}{1 + \left(c + \color{blue}{e^{\log \left(\sqrt{cosTheta \cdot \left(\frac{1}{cosTheta \cdot \pi} - \frac{2}{\pi}\right)}\right) + \left(\left(-{cosTheta}^{2}\right) - \log cosTheta\right)}}\right)} \]
11. Step-by-step derivation

    1. exp-sum 90.1%

       \[\leadsto \frac{1}{1 + \left(c + \color{blue}{e^{\log \left(\sqrt{cosTheta \cdot \left(\frac{1}{cosTheta \cdot \pi} - \frac{2}{\pi}\right)}\right)} \cdot e^{\left(-{cosTheta}^{2}\right) - \log cosTheta}}\right)} \]

    2. rem-exp-log 90.1%

       \[\leadsto \frac{1}{1 + \left(c + \color{blue}{\sqrt{cosTheta \cdot \left(\frac{1}{cosTheta \cdot \pi} - \frac{2}{\pi}\right)}} \cdot e^{\left(-{cosTheta}^{2}\right) - \log cosTheta}\right)} \]

    3. *-commutative 90.1%

       \[\leadsto \frac{1}{1 + \left(c + \sqrt{cosTheta \cdot \left(\frac{1}{\color{blue}{\pi \cdot cosTheta}} - \frac{2}{\pi}\right)} \cdot e^{\left(-{cosTheta}^{2}\right) - \log cosTheta}\right)} \]

12. Simplified 90.1%

    \[\leadsto \frac{1}{1 + \left(c + \color{blue}{\sqrt{cosTheta \cdot \left(\frac{1}{\pi \cdot cosTheta} - \frac{2}{\pi}\right)} \cdot e^{\left(-{cosTheta}^{2}\right) - \log cosTheta}}\right)} \]

13. Taylor expanded in cosTheta around 0 87.7%

    \[\leadsto \frac{1}{1 + \left(c + \sqrt{cosTheta \cdot \left(\frac{1}{\pi \cdot cosTheta} - \frac{2}{\pi}\right)} \cdot \color{blue}{e^{-\log cosTheta}}\right)} \]
14. Step-by-step derivation

    1. exp-neg 87.8%

       \[\leadsto \frac{1}{1 + \left(c + \sqrt{cosTheta \cdot \left(\frac{1}{\pi \cdot cosTheta} - \frac{2}{\pi}\right)} \cdot \color{blue}{\frac{1}{e^{\log cosTheta}}}\right)} \]

    2. rem-exp-log 95.6%

       \[\leadsto \frac{1}{1 + \left(c + \sqrt{cosTheta \cdot \left(\frac{1}{\pi \cdot cosTheta} - \frac{2}{\pi}\right)} \cdot \frac{1}{\color{blue}{cosTheta}}\right)} \]

15. Simplified 95.6%

    \[\leadsto \frac{1}{1 + \left(c + \sqrt{cosTheta \cdot \left(\frac{1}{\pi \cdot cosTheta} - \frac{2}{\pi}\right)} \cdot \color{blue}{\frac{1}{cosTheta}}\right)} \]

16. Final simplification 95.6%

    \[\leadsto \frac{1}{1 + \left(c + \sqrt{cosTheta \cdot \left(\frac{1}{cosTheta \cdot \pi} - \frac{2}{\pi}\right)} \cdot \frac{1}{cosTheta}\right)} \]
17. Add Preprocessing

Alternative 7: 94.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 + \left(c + \sqrt{\frac{1}{\pi}} \cdot \left(\frac{1}{cosTheta} + cosTheta \cdot \frac{-1}{cosTheta}\right)\right)} \end{array} \]

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (/
  1.0
  (+
   1.0
   (+
    c
    (*
     (sqrt (/ 1.0 PI))
     (+ (/ 1.0 cosTheta) (* cosTheta (/ -1.0 cosTheta))))))))

C

float code(float cosTheta, float c) {
	return 1.0f / (1.0f + (c + (sqrtf((1.0f / ((float) M_PI))) * ((1.0f / cosTheta) + (cosTheta * (-1.0f / cosTheta))))));
}

Julia

function code(cosTheta, c)
	return Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(c + Float32(sqrt(Float32(Float32(1.0) / Float32(pi))) * Float32(Float32(Float32(1.0) / cosTheta) + Float32(cosTheta * Float32(Float32(-1.0) / cosTheta)))))))
end

MATLAB

function tmp = code(cosTheta, c)
	tmp = single(1.0) / (single(1.0) + (c + (sqrt((single(1.0) / single(pi))) * ((single(1.0) / cosTheta) + (cosTheta * (single(-1.0) / cosTheta))))));
end

Derivation

1. Initial program 98.0%

   \[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
2. Step-by-step derivation

   1. associate-+l+ 98.0%

      \[\leadsto \frac{1}{\color{blue}{1 + \left(c + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}\right)}} \]

   2. +-commutative 98.0%

      \[\leadsto \frac{1}{1 + \color{blue}{\left(\left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta} + c\right)}} \]

   3. fma-define 98.0%

      \[\leadsto \frac{1}{1 + \color{blue}{\mathsf{fma}\left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}, e^{\left(-cosTheta\right) \cdot cosTheta}, c\right)}} \]

3. Simplified 98.6%

   \[\leadsto \color{blue}{\frac{1}{1 + \mathsf{fma}\left(\frac{\sqrt{1 + cosTheta \cdot -2}}{\sqrt{\pi} \cdot cosTheta}, {\left(e^{-cosTheta}\right)}^{cosTheta}, c\right)}} \]
4. Add Preprocessing

5. Taylor expanded in c around 0 98.2%

   \[\leadsto \frac{1}{1 + \color{blue}{\left(c + \frac{e^{-1 \cdot {cosTheta}^{2}}}{cosTheta} \cdot \sqrt{\frac{1 + -2 \cdot cosTheta}{\pi}}\right)}} \]

6. Taylor expanded in cosTheta around inf 98.1%

   \[\leadsto \frac{1}{1 + \left(c + \frac{e^{-1 \cdot {cosTheta}^{2}}}{cosTheta} \cdot \sqrt{\color{blue}{cosTheta \cdot \left(\frac{1}{cosTheta \cdot \pi} - 2 \cdot \frac{1}{\pi}\right)}}\right)} \]
7. Step-by-step derivation

   1. *-commutative 98.1%

      \[\leadsto \frac{1}{1 + \left(c + \frac{e^{-1 \cdot {cosTheta}^{2}}}{cosTheta} \cdot \sqrt{cosTheta \cdot \left(\frac{1}{\color{blue}{\pi \cdot cosTheta}} - 2 \cdot \frac{1}{\pi}\right)}\right)} \]

   2. associate-*r/ 98.1%

      \[\leadsto \frac{1}{1 + \left(c + \frac{e^{-1 \cdot {cosTheta}^{2}}}{cosTheta} \cdot \sqrt{cosTheta \cdot \left(\frac{1}{\pi \cdot cosTheta} - \color{blue}{\frac{2 \cdot 1}{\pi}}\right)}\right)} \]

   3. metadata-eval 98.1%

      \[\leadsto \frac{1}{1 + \left(c + \frac{e^{-1 \cdot {cosTheta}^{2}}}{cosTheta} \cdot \sqrt{cosTheta \cdot \left(\frac{1}{\pi \cdot cosTheta} - \frac{\color{blue}{2}}{\pi}\right)}\right)} \]

8. Simplified 98.1%

   \[\leadsto \frac{1}{1 + \left(c + \frac{e^{-1 \cdot {cosTheta}^{2}}}{cosTheta} \cdot \sqrt{\color{blue}{cosTheta \cdot \left(\frac{1}{\pi \cdot cosTheta} - \frac{2}{\pi}\right)}}\right)} \]
9. Step-by-step derivation

   1. add-exp-log 90.1%

      \[\leadsto \frac{1}{1 + \left(c + \color{blue}{e^{\log \left(\frac{e^{-1 \cdot {cosTheta}^{2}}}{cosTheta} \cdot \sqrt{cosTheta \cdot \left(\frac{1}{\pi \cdot cosTheta} - \frac{2}{\pi}\right)}\right)}}\right)} \]

   2. *-commutative 90.1%

      \[\leadsto \frac{1}{1 + \left(c + e^{\log \color{blue}{\left(\sqrt{cosTheta \cdot \left(\frac{1}{\pi \cdot cosTheta} - \frac{2}{\pi}\right)} \cdot \frac{e^{-1 \cdot {cosTheta}^{2}}}{cosTheta}\right)}}\right)} \]

   3. log-prod 89.9%

      \[\leadsto \frac{1}{1 + \left(c + e^{\color{blue}{\log \left(\sqrt{cosTheta \cdot \left(\frac{1}{\pi \cdot cosTheta} - \frac{2}{\pi}\right)}\right) + \log \left(\frac{e^{-1 \cdot {cosTheta}^{2}}}{cosTheta}\right)}}\right)} \]

   4. *-commutative 89.9%

      \[\leadsto \frac{1}{1 + \left(c + e^{\log \left(\sqrt{cosTheta \cdot \left(\frac{1}{\color{blue}{cosTheta \cdot \pi}} - \frac{2}{\pi}\right)}\right) + \log \left(\frac{e^{-1 \cdot {cosTheta}^{2}}}{cosTheta}\right)}\right)} \]

   5. mul-1-neg 89.9%

      \[\leadsto \frac{1}{1 + \left(c + e^{\log \left(\sqrt{cosTheta \cdot \left(\frac{1}{cosTheta \cdot \pi} - \frac{2}{\pi}\right)}\right) + \log \left(\frac{e^{\color{blue}{-{cosTheta}^{2}}}}{cosTheta}\right)}\right)} \]

   6. unpow2 89.9%

      \[\leadsto \frac{1}{1 + \left(c + e^{\log \left(\sqrt{cosTheta \cdot \left(\frac{1}{cosTheta \cdot \pi} - \frac{2}{\pi}\right)}\right) + \log \left(\frac{e^{-\color{blue}{cosTheta \cdot cosTheta}}}{cosTheta}\right)}\right)} \]

   7. distribute-lft-neg-out 89.9%

      \[\leadsto \frac{1}{1 + \left(c + e^{\log \left(\sqrt{cosTheta \cdot \left(\frac{1}{cosTheta \cdot \pi} - \frac{2}{\pi}\right)}\right) + \log \left(\frac{e^{\color{blue}{\left(-cosTheta\right) \cdot cosTheta}}}{cosTheta}\right)}\right)} \]

10. Applied egg-rr 89.9%

    \[\leadsto \frac{1}{1 + \left(c + \color{blue}{e^{\log \left(\sqrt{cosTheta \cdot \left(\frac{1}{cosTheta \cdot \pi} - \frac{2}{\pi}\right)}\right) + \left(\left(-{cosTheta}^{2}\right) - \log cosTheta\right)}}\right)} \]
11. Step-by-step derivation

    1. exp-sum 90.1%

       \[\leadsto \frac{1}{1 + \left(c + \color{blue}{e^{\log \left(\sqrt{cosTheta \cdot \left(\frac{1}{cosTheta \cdot \pi} - \frac{2}{\pi}\right)}\right)} \cdot e^{\left(-{cosTheta}^{2}\right) - \log cosTheta}}\right)} \]

    2. rem-exp-log 90.1%

       \[\leadsto \frac{1}{1 + \left(c + \color{blue}{\sqrt{cosTheta \cdot \left(\frac{1}{cosTheta \cdot \pi} - \frac{2}{\pi}\right)}} \cdot e^{\left(-{cosTheta}^{2}\right) - \log cosTheta}\right)} \]

    3. *-commutative 90.1%

       \[\leadsto \frac{1}{1 + \left(c + \sqrt{cosTheta \cdot \left(\frac{1}{\color{blue}{\pi \cdot cosTheta}} - \frac{2}{\pi}\right)} \cdot e^{\left(-{cosTheta}^{2}\right) - \log cosTheta}\right)} \]

12. Simplified 90.1%

    \[\leadsto \frac{1}{1 + \left(c + \color{blue}{\sqrt{cosTheta \cdot \left(\frac{1}{\pi \cdot cosTheta} - \frac{2}{\pi}\right)} \cdot e^{\left(-{cosTheta}^{2}\right) - \log cosTheta}}\right)} \]

13. Taylor expanded in cosTheta around 0 87.5%

    \[\leadsto \frac{1}{1 + \left(c + \color{blue}{\left(-1 \cdot \left(\left(cosTheta \cdot e^{-\log cosTheta}\right) \cdot \sqrt{\frac{1}{\pi}}\right) + \sqrt{\frac{1}{\pi}} \cdot e^{-\log cosTheta}\right)}\right)} \]
14. Step-by-step derivation

    1. +-commutative 87.5%

       \[\leadsto \frac{1}{1 + \left(c + \color{blue}{\left(\sqrt{\frac{1}{\pi}} \cdot e^{-\log cosTheta} + -1 \cdot \left(\left(cosTheta \cdot e^{-\log cosTheta}\right) \cdot \sqrt{\frac{1}{\pi}}\right)\right)}\right)} \]

    2. *-commutative 87.5%

       \[\leadsto \frac{1}{1 + \left(c + \left(\color{blue}{e^{-\log cosTheta} \cdot \sqrt{\frac{1}{\pi}}} + -1 \cdot \left(\left(cosTheta \cdot e^{-\log cosTheta}\right) \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)} \]

    3. associate-*r* 87.5%

       \[\leadsto \frac{1}{1 + \left(c + \left(e^{-\log cosTheta} \cdot \sqrt{\frac{1}{\pi}} + \color{blue}{\left(-1 \cdot \left(cosTheta \cdot e^{-\log cosTheta}\right)\right) \cdot \sqrt{\frac{1}{\pi}}}\right)\right)} \]

    4. distribute-rgt-out 87.5%

       \[\leadsto \frac{1}{1 + \left(c + \color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(e^{-\log cosTheta} + -1 \cdot \left(cosTheta \cdot e^{-\log cosTheta}\right)\right)}\right)} \]

    5. exp-neg 87.6%

       \[\leadsto \frac{1}{1 + \left(c + \sqrt{\frac{1}{\pi}} \cdot \left(\color{blue}{\frac{1}{e^{\log cosTheta}}} + -1 \cdot \left(cosTheta \cdot e^{-\log cosTheta}\right)\right)\right)} \]

    6. rem-exp-log 95.4%

       \[\leadsto \frac{1}{1 + \left(c + \sqrt{\frac{1}{\pi}} \cdot \left(\frac{1}{\color{blue}{cosTheta}} + -1 \cdot \left(cosTheta \cdot e^{-\log cosTheta}\right)\right)\right)} \]

    7. mul-1-neg 95.4%

       \[\leadsto \frac{1}{1 + \left(c + \sqrt{\frac{1}{\pi}} \cdot \left(\frac{1}{cosTheta} + \color{blue}{\left(-cosTheta \cdot e^{-\log cosTheta}\right)}\right)\right)} \]

    8. exp-neg 95.4%

       \[\leadsto \frac{1}{1 + \left(c + \sqrt{\frac{1}{\pi}} \cdot \left(\frac{1}{cosTheta} + \left(-cosTheta \cdot \color{blue}{\frac{1}{e^{\log cosTheta}}}\right)\right)\right)} \]

    9. rem-exp-log 95.4%

       \[\leadsto \frac{1}{1 + \left(c + \sqrt{\frac{1}{\pi}} \cdot \left(\frac{1}{cosTheta} + \left(-cosTheta \cdot \frac{1}{\color{blue}{cosTheta}}\right)\right)\right)} \]

15. Simplified 95.4%

    \[\leadsto \frac{1}{1 + \left(c + \color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(\frac{1}{cosTheta} + \left(-cosTheta \cdot \frac{1}{cosTheta}\right)\right)}\right)} \]

16. Final simplification 95.4%

    \[\leadsto \frac{1}{1 + \left(c + \sqrt{\frac{1}{\pi}} \cdot \left(\frac{1}{cosTheta} + cosTheta \cdot \frac{-1}{cosTheta}\right)\right)} \]
17. Add Preprocessing

Alternative 8: 95.0% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 + \left(c + \frac{\sqrt{\frac{1}{\pi}} \cdot \left(1 - cosTheta\right)}{cosTheta}\right)} \end{array} \]

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (/ 1.0 (+ 1.0 (+ c (/ (* (sqrt (/ 1.0 PI)) (- 1.0 cosTheta)) cosTheta)))))

C

float code(float cosTheta, float c) {
	return 1.0f / (1.0f + (c + ((sqrtf((1.0f / ((float) M_PI))) * (1.0f - cosTheta)) / cosTheta)));
}

Julia

function code(cosTheta, c)
	return Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(c + Float32(Float32(sqrt(Float32(Float32(1.0) / Float32(pi))) * Float32(Float32(1.0) - cosTheta)) / cosTheta))))
end

MATLAB

function tmp = code(cosTheta, c)
	tmp = single(1.0) / (single(1.0) + (c + ((sqrt((single(1.0) / single(pi))) * (single(1.0) - cosTheta)) / cosTheta)));
end

Derivation

1. Initial program 98.0%

   \[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
2. Step-by-step derivation

   1. associate-+l+ 98.0%

      \[\leadsto \frac{1}{\color{blue}{1 + \left(c + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}\right)}} \]

   2. +-commutative 98.0%

      \[\leadsto \frac{1}{1 + \color{blue}{\left(\left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta} + c\right)}} \]

   3. fma-define 98.0%

      \[\leadsto \frac{1}{1 + \color{blue}{\mathsf{fma}\left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}, e^{\left(-cosTheta\right) \cdot cosTheta}, c\right)}} \]

3. Simplified 98.6%

   \[\leadsto \color{blue}{\frac{1}{1 + \mathsf{fma}\left(\frac{\sqrt{1 + cosTheta \cdot -2}}{\sqrt{\pi} \cdot cosTheta}, {\left(e^{-cosTheta}\right)}^{cosTheta}, c\right)}} \]
4. Add Preprocessing

5. Taylor expanded in c around 0 98.2%

   \[\leadsto \frac{1}{1 + \color{blue}{\left(c + \frac{e^{-1 \cdot {cosTheta}^{2}}}{cosTheta} \cdot \sqrt{\frac{1 + -2 \cdot cosTheta}{\pi}}\right)}} \]

6. Taylor expanded in cosTheta around inf 98.1%

   \[\leadsto \frac{1}{1 + \left(c + \frac{e^{-1 \cdot {cosTheta}^{2}}}{cosTheta} \cdot \sqrt{\color{blue}{cosTheta \cdot \left(\frac{1}{cosTheta \cdot \pi} - 2 \cdot \frac{1}{\pi}\right)}}\right)} \]
7. Step-by-step derivation

   1. *-commutative 98.1%

      \[\leadsto \frac{1}{1 + \left(c + \frac{e^{-1 \cdot {cosTheta}^{2}}}{cosTheta} \cdot \sqrt{cosTheta \cdot \left(\frac{1}{\color{blue}{\pi \cdot cosTheta}} - 2 \cdot \frac{1}{\pi}\right)}\right)} \]

   2. associate-*r/ 98.1%

      \[\leadsto \frac{1}{1 + \left(c + \frac{e^{-1 \cdot {cosTheta}^{2}}}{cosTheta} \cdot \sqrt{cosTheta \cdot \left(\frac{1}{\pi \cdot cosTheta} - \color{blue}{\frac{2 \cdot 1}{\pi}}\right)}\right)} \]

   3. metadata-eval 98.1%

      \[\leadsto \frac{1}{1 + \left(c + \frac{e^{-1 \cdot {cosTheta}^{2}}}{cosTheta} \cdot \sqrt{cosTheta \cdot \left(\frac{1}{\pi \cdot cosTheta} - \frac{\color{blue}{2}}{\pi}\right)}\right)} \]

8. Simplified 98.1%

   \[\leadsto \frac{1}{1 + \left(c + \frac{e^{-1 \cdot {cosTheta}^{2}}}{cosTheta} \cdot \sqrt{\color{blue}{cosTheta \cdot \left(\frac{1}{\pi \cdot cosTheta} - \frac{2}{\pi}\right)}}\right)} \]

9. Taylor expanded in cosTheta around 0 95.4%

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

    1. associate-*r* 95.4%

       \[\leadsto \frac{1}{1 + \left(c + \frac{\sqrt{\frac{1}{\pi}} + \color{blue}{\left(-1 \cdot cosTheta\right) \cdot \sqrt{\frac{1}{\pi}}}}{cosTheta}\right)} \]

    2. distribute-rgt1-in 95.4%

       \[\leadsto \frac{1}{1 + \left(c + \frac{\color{blue}{\left(-1 \cdot cosTheta + 1\right) \cdot \sqrt{\frac{1}{\pi}}}}{cosTheta}\right)} \]

    3. mul-1-neg 95.4%

       \[\leadsto \frac{1}{1 + \left(c + \frac{\left(\color{blue}{\left(-cosTheta\right)} + 1\right) \cdot \sqrt{\frac{1}{\pi}}}{cosTheta}\right)} \]

11. Simplified 95.4%

    \[\leadsto \frac{1}{1 + \left(c + \color{blue}{\frac{\left(\left(-cosTheta\right) + 1\right) \cdot \sqrt{\frac{1}{\pi}}}{cosTheta}}\right)} \]

12. Final simplification 95.4%

    \[\leadsto \frac{1}{1 + \left(c + \frac{\sqrt{\frac{1}{\pi}} \cdot \left(1 - cosTheta\right)}{cosTheta}\right)} \]
13. Add Preprocessing

Alternative 9: 92.8% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ cosTheta \cdot \sqrt{\pi} \end{array} \]

FPCore

(FPCore (cosTheta c) :precision binary32 (* cosTheta (sqrt PI)))

C

float code(float cosTheta, float c) {
	return cosTheta * sqrtf(((float) M_PI));
}

Julia

function code(cosTheta, c)
	return Float32(cosTheta * sqrt(Float32(pi)))
end

MATLAB

function tmp = code(cosTheta, c)
	tmp = cosTheta * sqrt(single(pi));
end

Derivation

1. Initial program 98.0%

   \[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
2. Step-by-step derivation

   1. associate-+l+ 98.0%

      \[\leadsto \frac{1}{\color{blue}{1 + \left(c + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}\right)}} \]

   2. +-commutative 98.0%

      \[\leadsto \frac{1}{1 + \color{blue}{\left(\left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta} + c\right)}} \]

   3. fma-define 98.0%

      \[\leadsto \frac{1}{1 + \color{blue}{\mathsf{fma}\left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}, e^{\left(-cosTheta\right) \cdot cosTheta}, c\right)}} \]

3. Simplified 98.6%

   \[\leadsto \color{blue}{\frac{1}{1 + \mathsf{fma}\left(\frac{\sqrt{1 + cosTheta \cdot -2}}{\sqrt{\pi} \cdot cosTheta}, {\left(e^{-cosTheta}\right)}^{cosTheta}, c\right)}} \]
4. Add Preprocessing

5. Taylor expanded in c around 0 98.2%

   \[\leadsto \frac{1}{1 + \color{blue}{\left(c + \frac{e^{-1 \cdot {cosTheta}^{2}}}{cosTheta} \cdot \sqrt{\frac{1 + -2 \cdot cosTheta}{\pi}}\right)}} \]

6. Taylor expanded in cosTheta around 0 93.0%

   \[\leadsto \color{blue}{cosTheta \cdot \sqrt{\pi}} \]

7. Final simplification 93.0%

   \[\leadsto cosTheta \cdot \sqrt{\pi} \]
8. Add Preprocessing

Alternative 10: 10.8% accurate, 107.3× speedup

Math

\[\begin{array}{l} \\ 1 - c \end{array} \]

FPCore

(FPCore (cosTheta c) :precision binary32 (- 1.0 c))

C

float code(float cosTheta, float c) {
	return 1.0f - c;
}

Fortran

real(4) function code(costheta, c)
    real(4), intent (in) :: costheta
    real(4), intent (in) :: c
    code = 1.0e0 - c
end function

Julia

function code(cosTheta, c)
	return Float32(Float32(1.0) - c)
end

MATLAB

function tmp = code(cosTheta, c)
	tmp = single(1.0) - c;
end

Derivation

1. Initial program 98.0%

   \[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
2. Step-by-step derivation

   1. associate-+l+ 98.0%

      \[\leadsto \frac{1}{\color{blue}{1 + \left(c + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}\right)}} \]

   2. +-commutative 98.0%

      \[\leadsto \frac{1}{1 + \color{blue}{\left(\left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta} + c\right)}} \]

   3. fma-define 98.0%

      \[\leadsto \frac{1}{1 + \color{blue}{\mathsf{fma}\left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}, e^{\left(-cosTheta\right) \cdot cosTheta}, c\right)}} \]

3. Simplified 98.6%

   \[\leadsto \color{blue}{\frac{1}{1 + \mathsf{fma}\left(\frac{\sqrt{1 + cosTheta \cdot -2}}{\sqrt{\pi} \cdot cosTheta}, {\left(e^{-cosTheta}\right)}^{cosTheta}, c\right)}} \]
4. Add Preprocessing

5. Taylor expanded in c around inf 10.7%

   \[\leadsto \frac{1}{1 + \color{blue}{c}} \]

6. Taylor expanded in c around 0 10.7%

   \[\leadsto \color{blue}{1 + -1 \cdot c} \]
7. Step-by-step derivation

   1. mul-1-neg 10.7%

      \[\leadsto 1 + \color{blue}{\left(-c\right)} \]

   2. unsub-neg 10.7%

      \[\leadsto \color{blue}{1 - c} \]

8. Simplified 10.7%

   \[\leadsto \color{blue}{1 - c} \]

9. Final simplification 10.7%

   \[\leadsto 1 - c \]
10. Add Preprocessing

Alternative 11: 10.8% accurate, 322.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (cosTheta c) :precision binary32 1.0)

C

float code(float cosTheta, float c) {
	return 1.0f;
}

Fortran

real(4) function code(costheta, c)
    real(4), intent (in) :: costheta
    real(4), intent (in) :: c
    code = 1.0e0
end function

Julia

function code(cosTheta, c)
	return Float32(1.0)
end

MATLAB

function tmp = code(cosTheta, c)
	tmp = single(1.0);
end

Derivation

1. Initial program 98.0%

   \[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
2. Step-by-step derivation

   1. associate-+l+ 98.0%

      \[\leadsto \frac{1}{\color{blue}{1 + \left(c + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}\right)}} \]

   2. +-commutative 98.0%

      \[\leadsto \frac{1}{1 + \color{blue}{\left(\left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta} + c\right)}} \]

   3. fma-define 98.0%

      \[\leadsto \frac{1}{1 + \color{blue}{\mathsf{fma}\left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}, e^{\left(-cosTheta\right) \cdot cosTheta}, c\right)}} \]

3. Simplified 98.6%

   \[\leadsto \color{blue}{\frac{1}{1 + \mathsf{fma}\left(\frac{\sqrt{1 + cosTheta \cdot -2}}{\sqrt{\pi} \cdot cosTheta}, {\left(e^{-cosTheta}\right)}^{cosTheta}, c\right)}} \]
4. Add Preprocessing

5. Taylor expanded in c around inf 10.7%

   \[\leadsto \frac{1}{1 + \color{blue}{c}} \]

6. Taylor expanded in c around 0 10.7%

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

7. Final simplification 10.7%

   \[\leadsto 1 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024131 
(FPCore (cosTheta c)
  :name "Beckmann Sample, normalization factor"
  :precision binary32
  :pre (and (and (< 0.0 cosTheta) (< cosTheta 0.9999)) (and (< -1.0 c) (< c 1.0)))
  (/ 1.0 (+ (+ 1.0 c) (* (* (/ 1.0 (sqrt PI)) (/ (sqrt (- (- 1.0 cosTheta) cosTheta)) cosTheta)) (exp (* (- cosTheta) cosTheta))))))