Beckmann Sample, normalization factor

Percentage Accurate: 97.8% → 98.3%

Time: 18.9s

Alternatives: 14

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: 98.3% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.4%

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

   1. +-commutative 97.4%

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

   2. fma-define 97.4%

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

3. Simplified 98.0%

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

5. Final simplification 98.0%

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

Alternative 2: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.4%

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

   1. associate-*l/ 98.1%

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

   2. *-un-lft-identity 98.1%

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

   3. associate--l- 98.0%

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

4. Applied egg-rr 98.0%

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

5. Final simplification 98.0%

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

Alternative 3: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.4%

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

   1. associate-+l+ 97.4%

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

   2. +-commutative 97.4%

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

   3. *-commutative 97.4%

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

   4. associate-*l* 97.4%

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

   5. fma-define 97.4%

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

3. Simplified 97.3%

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

5. Taylor expanded in c around 0 97.5%

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

6. Final simplification 97.5%

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

Alternative 4: 97.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.4%

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

   1. associate-+l+ 97.4%

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

   2. +-commutative 97.4%

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

   3. *-commutative 97.4%

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

   4. associate-*l* 97.4%

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

   5. fma-define 97.4%

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

3. Simplified 97.3%

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

5. Taylor expanded in c around 0 96.9%

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

6. Final simplification 96.9%

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

Alternative 5: 96.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\left(1 + c\right) + e^{-cosTheta \cdot cosTheta} \cdot \frac{\left(cosTheta \cdot -0.5 + \frac{1}{cosTheta}\right) + -1}{\sqrt{\pi}}} \end{array} \]

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (/
  1.0
  (+
   (+ 1.0 c)
   (*
    (exp (- (* cosTheta cosTheta)))
    (/ (+ (+ (* cosTheta -0.5) (/ 1.0 cosTheta)) -1.0) (sqrt PI))))))

C

float code(float cosTheta, float c) {
	return 1.0f / ((1.0f + c) + (expf(-(cosTheta * cosTheta)) * ((((cosTheta * -0.5f) + (1.0f / cosTheta)) + -1.0f) / sqrtf(((float) M_PI)))));
}

Julia

function code(cosTheta, c)
	return Float32(Float32(1.0) / Float32(Float32(Float32(1.0) + c) + Float32(exp(Float32(-Float32(cosTheta * cosTheta))) * Float32(Float32(Float32(Float32(cosTheta * Float32(-0.5)) + Float32(Float32(1.0) / cosTheta)) + Float32(-1.0)) / sqrt(Float32(pi))))))
end

MATLAB

function tmp = code(cosTheta, c)
	tmp = single(1.0) / ((single(1.0) + c) + (exp(-(cosTheta * cosTheta)) * ((((cosTheta * single(-0.5)) + (single(1.0) / cosTheta)) + single(-1.0)) / sqrt(single(pi)))));
end

Derivation

1. Initial program 97.4%

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

   1. associate-*l/ 98.1%

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

   2. *-un-lft-identity 98.1%

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

   3. associate--l- 98.0%

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

4. Applied egg-rr 98.0%

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

5. Taylor expanded in cosTheta around 0 96.3%

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

6. Final simplification 96.3%

   \[\leadsto \frac{1}{\left(1 + c\right) + e^{-cosTheta \cdot cosTheta} \cdot \frac{\left(cosTheta \cdot -0.5 + \frac{1}{cosTheta}\right) + -1}{\sqrt{\pi}}} \]
7. Add Preprocessing

Alternative 6: 95.7% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.4%

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

   1. associate-*l/ 98.1%

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

   2. *-un-lft-identity 98.1%

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

   3. associate--l- 98.0%

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

4. Applied egg-rr 98.0%

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

5. Taylor expanded in cosTheta around 0 95.7%

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

6. Final simplification 95.7%

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

Alternative 7: 95.3% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.4%

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

   1. +-commutative 97.4%

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

   2. fma-define 97.4%

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

3. Simplified 98.0%

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

5. Taylor expanded in c around 0 96.9%

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

   1. +-commutative 96.9%

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

   2. fma-define 97.1%

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

   3. neg-mul-1 97.1%

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

   4. +-commutative 97.1%

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

   5. *-commutative 97.1%

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

   6. fma-undefine 97.1%

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

7. Simplified 97.1%

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

8. Taylor expanded in cosTheta around 0 94.6%

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

   1. inv-pow 94.6%

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

   2. sqrt-div 94.6%

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

   3. metadata-eval 94.6%

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

   4. inv-pow 94.6%

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

   5. unpow-prod-down 95.2%

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

   6. inv-pow 95.2%

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

   7. associate-/r* 95.3%

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

10. Applied egg-rr 95.3%

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

11. Final simplification 95.3%

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

Alternative 8: 95.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 + \left(\frac{1}{cosTheta \cdot \sqrt{\pi}} - {\pi}^{-0.5}\right)} \end{array} \]

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (/ 1.0 (+ 1.0 (- (/ 1.0 (* cosTheta (sqrt PI))) (pow PI -0.5)))))

C

float code(float cosTheta, float c) {
	return 1.0f / (1.0f + ((1.0f / (cosTheta * sqrtf(((float) M_PI)))) - powf(((float) M_PI), -0.5f)));
}

Julia

function code(cosTheta, c)
	return Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(Float32(1.0) / Float32(cosTheta * sqrt(Float32(pi)))) - (Float32(pi) ^ Float32(-0.5)))))
end

MATLAB

function tmp = code(cosTheta, c)
	tmp = single(1.0) / (single(1.0) + ((single(1.0) / (cosTheta * sqrt(single(pi)))) - (single(pi) ^ single(-0.5))));
end

Derivation

1. Initial program 97.4%

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

   1. +-commutative 97.4%

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

   2. fma-define 97.4%

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

3. Simplified 98.0%

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

5. Taylor expanded in c around 0 96.9%

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

   1. +-commutative 96.9%

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

   2. fma-define 97.1%

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

   3. neg-mul-1 97.1%

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

   4. +-commutative 97.1%

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

   5. *-commutative 97.1%

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

   6. fma-undefine 97.1%

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

7. Simplified 97.1%

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

8. Taylor expanded in cosTheta around 0 94.6%

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

   1. +-commutative 94.6%

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

   2. add-sqr-sqrt 94.4%

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

   3. fma-define 94.4%

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

10. Applied egg-rr 94.8%

    \[\leadsto \frac{1}{1 + \color{blue}{\mathsf{fma}\left({\left(cosTheta \cdot \sqrt{\pi}\right)}^{-0.5}, {\left(cosTheta \cdot \sqrt{\pi}\right)}^{-0.5}, -{\pi}^{-0.5}\right)}} \]
11. Step-by-step derivation

    1. fma-neg 94.7%

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

    2. pow-sqr 95.2%

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

    3. metadata-eval 95.2%

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

    4. unpow-1 95.2%

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

12. Simplified 95.2%

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

13. Final simplification 95.2%

    \[\leadsto \frac{1}{1 + \left(\frac{1}{cosTheta \cdot \sqrt{\pi}} - {\pi}^{-0.5}\right)} \]
14. Add Preprocessing

Alternative 9: 94.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\mathsf{fma}\left({\pi}^{-0.5}, \frac{1}{cosTheta} + -1, 1\right)} \end{array} \]

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (/ 1.0 (fma (pow PI -0.5) (+ (/ 1.0 cosTheta) -1.0) 1.0)))

C

float code(float cosTheta, float c) {
	return 1.0f / fmaf(powf(((float) M_PI), -0.5f), ((1.0f / cosTheta) + -1.0f), 1.0f);
}

Julia

function code(cosTheta, c)
	return Float32(Float32(1.0) / fma((Float32(pi) ^ Float32(-0.5)), Float32(Float32(Float32(1.0) / cosTheta) + Float32(-1.0)), Float32(1.0)))
end

Derivation

1. Initial program 97.4%

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

   1. +-commutative 97.4%

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

   2. fma-define 97.4%

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

3. Simplified 98.0%

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

5. Taylor expanded in c around 0 96.9%

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

   1. +-commutative 96.9%

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

   2. fma-define 97.1%

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

   3. neg-mul-1 97.1%

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

   4. +-commutative 97.1%

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

   5. *-commutative 97.1%

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

   6. fma-undefine 97.1%

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

7. Simplified 97.1%

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

8. Taylor expanded in cosTheta around 0 94.6%

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

   1. *-un-lft-identity 94.6%

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

   2. +-commutative 94.6%

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

   3. distribute-rgt-out 94.7%

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

   4. fma-define 94.8%

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

   5. inv-pow 94.8%

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

   6. sqrt-pow1 94.8%

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

   7. metadata-eval 94.8%

      \[\leadsto 1 \cdot \frac{1}{\mathsf{fma}\left({\pi}^{\color{blue}{-0.5}}, -1 + \frac{1}{cosTheta}, 1\right)} \]

10. Applied egg-rr 94.8%

    \[\leadsto \color{blue}{1 \cdot \frac{1}{\mathsf{fma}\left({\pi}^{-0.5}, -1 + \frac{1}{cosTheta}, 1\right)}} \]
11. Step-by-step derivation

    1. *-lft-identity 94.8%

       \[\leadsto \color{blue}{\frac{1}{\mathsf{fma}\left({\pi}^{-0.5}, -1 + \frac{1}{cosTheta}, 1\right)}} \]

12. Simplified 94.8%

    \[\leadsto \color{blue}{\frac{1}{\mathsf{fma}\left({\pi}^{-0.5}, -1 + \frac{1}{cosTheta}, 1\right)}} \]

13. Final simplification 94.8%

    \[\leadsto \frac{1}{\mathsf{fma}\left({\pi}^{-0.5}, \frac{1}{cosTheta} + -1, 1\right)} \]
14. Add Preprocessing

Alternative 10: 94.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 + \left(c + {\pi}^{-0.5} \cdot \left(\frac{1}{cosTheta} + -1\right)\right)} \end{array} \]

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (/ 1.0 (+ 1.0 (+ c (* (pow PI -0.5) (+ (/ 1.0 cosTheta) -1.0))))))

C

float code(float cosTheta, float c) {
	return 1.0f / (1.0f + (c + (powf(((float) M_PI), -0.5f) * ((1.0f / cosTheta) + -1.0f))));
}

Julia

function code(cosTheta, c)
	return Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(c + Float32((Float32(pi) ^ Float32(-0.5)) * Float32(Float32(Float32(1.0) / cosTheta) + Float32(-1.0))))))
end

MATLAB

function tmp = code(cosTheta, c)
	tmp = single(1.0) / (single(1.0) + (c + ((single(pi) ^ single(-0.5)) * ((single(1.0) / cosTheta) + single(-1.0)))));
end

Derivation

1. Initial program 97.4%

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

   1. associate-+l+ 97.4%

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

   2. +-commutative 97.4%

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

   3. *-commutative 97.4%

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

   4. associate-*l* 97.4%

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

   5. fma-define 97.4%

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

3. Simplified 97.3%

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

5. Taylor expanded in cosTheta around 0 94.8%

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

   1. +-commutative 94.8%

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

   2. distribute-rgt-out 94.7%

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

   3. fma-define 94.8%

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

7. Simplified 94.8%

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

   1. fma-undefine 94.7%

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

   2. inv-pow 94.7%

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

   3. sqrt-pow1 94.7%

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

   4. metadata-eval 94.7%

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

   5. *-commutative 94.7%

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

9. Applied egg-rr 94.7%

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

10. Final simplification 94.7%

    \[\leadsto \frac{1}{1 + \left(c + {\pi}^{-0.5} \cdot \left(\frac{1}{cosTheta} + -1\right)\right)} \]
11. Add Preprocessing

Alternative 11: 94.8% accurate, 2.9× speedup

Math

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

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (/ 1.0 (+ 1.0 (* (pow PI -0.5) (+ (/ 1.0 cosTheta) -1.0)))))

C

float code(float cosTheta, float c) {
	return 1.0f / (1.0f + (powf(((float) M_PI), -0.5f) * ((1.0f / cosTheta) + -1.0f)));
}

Julia

function code(cosTheta, c)
	return Float32(Float32(1.0) / Float32(Float32(1.0) + Float32((Float32(pi) ^ Float32(-0.5)) * Float32(Float32(Float32(1.0) / cosTheta) + Float32(-1.0)))))
end

MATLAB

function tmp = code(cosTheta, c)
	tmp = single(1.0) / (single(1.0) + ((single(pi) ^ single(-0.5)) * ((single(1.0) / cosTheta) + single(-1.0))));
end

Derivation

1. Initial program 97.4%

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

   1. +-commutative 97.4%

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

   2. fma-define 97.4%

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

3. Simplified 98.0%

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

5. Taylor expanded in c around 0 96.9%

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

   1. +-commutative 96.9%

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

   2. fma-define 97.1%

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

   3. neg-mul-1 97.1%

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

   4. +-commutative 97.1%

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

   5. *-commutative 97.1%

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

   6. fma-undefine 97.1%

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

7. Simplified 97.1%

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

8. Taylor expanded in cosTheta around 0 94.6%

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

   1. *-un-lft-identity 94.6%

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

   2. distribute-rgt-out 94.7%

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

   3. inv-pow 94.7%

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

   4. sqrt-pow1 94.7%

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

   5. metadata-eval 94.7%

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

10. Applied egg-rr 94.7%

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

    1. *-lft-identity 94.7%

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

12. Simplified 94.7%

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

13. Final simplification 94.7%

    \[\leadsto \frac{1}{1 + {\pi}^{-0.5} \cdot \left(\frac{1}{cosTheta} + -1\right)} \]
14. Add Preprocessing

Alternative 12: 92.9% 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 97.4%

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

   1. +-commutative 97.4%

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

   2. fma-define 97.4%

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

3. Simplified 98.0%

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

5. Taylor expanded in cosTheta around 0 93.5%

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

6. Final simplification 93.5%

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

Alternative 13: 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 97.4%

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

   1. associate-+l+ 97.4%

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

   2. +-commutative 97.4%

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

   3. *-commutative 97.4%

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

   4. associate-*l* 97.4%

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

   5. fma-define 97.4%

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

3. Simplified 97.3%

   \[\leadsto \color{blue}{\frac{1}{1 + \mathsf{fma}\left(\frac{\sqrt{\mathsf{fma}\left(cosTheta, -2, 1\right)}}{cosTheta}, \frac{e^{-cosTheta \cdot cosTheta}}{\sqrt{\pi}}, c\right)}} \]
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 14: 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 97.4%

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

   1. associate-+l+ 97.4%

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

   2. +-commutative 97.4%

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

   3. *-commutative 97.4%

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

   4. associate-*l* 97.4%

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

   5. fma-define 97.4%

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

3. Simplified 97.3%

   \[\leadsto \color{blue}{\frac{1}{1 + \mathsf{fma}\left(\frac{\sqrt{\mathsf{fma}\left(cosTheta, -2, 1\right)}}{cosTheta}, \frac{e^{-cosTheta \cdot cosTheta}}{\sqrt{\pi}}, c\right)}} \]
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 2024053 
(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))))))