Beckmann Sample, normalization factor

Percentage Accurate: 97.9% → 98.4%

Time: 16.0s

Alternatives: 19

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: 97.9% 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: 98.4% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (/
  1.0
  (fma
   (/ (/ (sqrt (+ 1.0 (* cosTheta -2.0))) cosTheta) (sqrt PI))
   (/ 1.0 (exp (pow cosTheta 2.0)))
   (+ 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))), (1.0f / expf(powf(cosTheta, 2.0f))), (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))), Float32(Float32(1.0) / exp((cosTheta ^ Float32(2.0)))), Float32(Float32(1.0) + c)))
end

Derivation

1. Initial program 98.1%

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

   1. +-commutative 98.1%

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

   2. fma-define 98.0%

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

   3. associate-*l/ 98.7%

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

   4. *-lft-identity 98.7%

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

   5. sub-neg 98.7%

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

   6. associate--l+ 98.6%

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

   7. unsub-neg 98.6%

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

   8. neg-mul-1 98.6%

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

   9. neg-mul-1 98.6%

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

   10. distribute-rgt-out 98.6%

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

   11. metadata-eval 98.6%

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

   12. *-commutative 98.6%

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

   13. exp-prod 98.6%

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

3. Simplified 98.6%

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

   1. pow-neg 98.6%

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

   2. pow-exp 98.6%

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

   3. pow2 98.6%

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

6. Applied egg-rr 98.6%

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

Alternative 2: 98.4% 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 98.1%

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

   1. +-commutative 98.1%

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

   2. fma-define 98.0%

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

   3. associate-*l/ 98.7%

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

   4. *-lft-identity 98.7%

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

   5. sub-neg 98.7%

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

   6. associate--l+ 98.6%

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

   7. unsub-neg 98.6%

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

   8. neg-mul-1 98.6%

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

   9. neg-mul-1 98.6%

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

   10. distribute-rgt-out 98.6%

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

   11. metadata-eval 98.6%

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

   12. *-commutative 98.6%

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

   13. exp-prod 98.6%

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

3. Simplified 98.6%

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

Alternative 3: 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 98.1%

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

   1. associate-+l+ 98.1%

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

   2. +-commutative 98.1%

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

   3. fma-define 98.1%

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

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 4: 98.0% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

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))), Float32(Float32(1.0) / exp((cosTheta ^ Float32(2.0)))), Float32(1.0)))
end

Derivation

1. Initial program 98.1%

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

   1. +-commutative 98.1%

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

   2. fma-define 98.0%

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

   3. associate-*l/ 98.7%

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

   4. *-lft-identity 98.7%

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

   5. sub-neg 98.7%

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

   6. associate--l+ 98.6%

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

   7. unsub-neg 98.6%

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

   8. neg-mul-1 98.6%

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

   9. neg-mul-1 98.6%

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

   10. distribute-rgt-out 98.6%

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

   11. metadata-eval 98.6%

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

   12. *-commutative 98.6%

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

   13. exp-prod 98.6%

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

3. Simplified 98.6%

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

   1. pow-neg 98.6%

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

   2. pow-exp 98.6%

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

   3. pow2 98.6%

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

6. Applied egg-rr 98.6%

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

7. Taylor expanded in c around 0 98.5%

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

Alternative 5: 97.9% accurate, 1.0× speedup

Math

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

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(cosTheta * Float32(-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

Derivation

1. Initial program 98.1%

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

3. Final simplification 98.1%

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

Alternative 6: 97.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

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

3. Taylor expanded in cosTheta around 0 98.0%

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

4. Final simplification 98.0%

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

Alternative 7: 97.6% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (/
  1.0
  (+
   1.0
   (*
    (* (exp (- (pow cosTheta 2.0))) (/ 1.0 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)) * (1.0f / 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)))) * Float32(Float32(1.0) / 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))) * (single(1.0) / cosTheta)) * sqrt(((single(1.0) - (cosTheta * single(2.0))) / single(pi)))));
end

Derivation

1. Initial program 98.1%

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

3. Taylor expanded in c around 0 97.9%

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

   1. div-inv 97.9%

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

   2. mul-1-neg 97.9%

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

5. Applied egg-rr 97.9%

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

6. Final simplification 97.9%

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

Alternative 8: 97.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

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

3. Taylor expanded in c around 0 97.9%

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

   1. associate-*l/ 97.9%

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

   2. mul-1-neg 97.9%

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

   3. cancel-sign-sub-inv 97.9%

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

   4. metadata-eval 97.9%

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

5. Applied egg-rr 97.9%

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

6. Final simplification 97.9%

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

Alternative 9: 97.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Derivation

1. Initial program 98.1%

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

3. Taylor expanded in c around 0 97.9%

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

   1. unpow2 97.9%

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

5. Applied egg-rr 97.9%

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

6. Final simplification 97.9%

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

Alternative 10: 96.7% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (/
  1.0
  (+
   1.0
   (*
    (sqrt (/ (- 1.0 (* cosTheta 2.0)) PI))
    (/ (- 1.0 (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))) * ((1.0f - 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(Float32(Float32(1.0) - (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))) * ((single(1.0) - (cosTheta ^ single(2.0))) / cosTheta)));
end

Derivation

1. Initial program 98.1%

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

3. Taylor expanded in c around 0 97.9%

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

4. Taylor expanded in cosTheta around 0 96.8%

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

   1. neg-mul-1 96.8%

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

   2. unsub-neg 96.8%

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

6. Simplified 96.8%

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

7. Final simplification 96.8%

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

Alternative 11: 95.7% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

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

   1. associate-+l+ 98.1%

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

   2. +-commutative 98.1%

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

   3. fma-define 98.1%

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

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 95.6%

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

   1. associate-*r* 95.6%

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

   2. mul-1-neg 95.6%

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

   3. mul-1-neg 95.6%

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

7. Simplified 95.6%

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

8. Final simplification 95.6%

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

Alternative 12: 95.7% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.1%

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

   1. associate-+l+ 98.1%

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

   2. +-commutative 98.1%

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

   3. fma-define 98.1%

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

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 95.6%

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

   1. mul-1-neg 95.6%

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

   2. unsub-neg 95.6%

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

   3. associate-*r* 95.6%

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

   4. associate-+r+ 95.6%

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

   5. mul-1-neg 95.6%

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

   6. unsub-neg 95.6%

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

7. Simplified 95.6%

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

8. Final simplification 95.6%

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

Alternative 13: 95.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

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

   1. +-commutative 98.1%

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

   2. fma-define 98.0%

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

   3. associate-*l/ 98.7%

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

   4. *-lft-identity 98.7%

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

   5. sub-neg 98.7%

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

   6. associate--l+ 98.6%

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

   7. unsub-neg 98.6%

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

   8. neg-mul-1 98.6%

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

   9. neg-mul-1 98.6%

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

   10. distribute-rgt-out 98.6%

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

   11. metadata-eval 98.6%

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

   12. *-commutative 98.6%

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

   13. exp-prod 98.6%

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

3. Simplified 98.6%

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

   1. pow-neg 98.6%

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

   2. pow-exp 98.6%

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

   3. pow2 98.6%

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

6. Applied egg-rr 98.6%

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

7. Taylor expanded in c around 0 98.5%

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

8. Taylor expanded in cosTheta around 0 95.5%

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

   1. +-commutative 95.5%

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

   2. *-commutative 95.5%

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

   3. mul-1-neg 95.5%

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

   4. sub-neg 95.5%

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

   5. *-commutative 95.5%

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

   6. +-commutative 95.5%

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

10. Simplified 95.5%

    \[\leadsto \color{blue}{cosTheta \cdot \left(\sqrt{\pi} - \left(cosTheta \cdot \pi\right) \cdot \left(1 - \sqrt{\frac{1}{\pi}}\right)\right)} \]
11. Add Preprocessing

Alternative 14: 95.3% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

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

   1. associate-+l+ 98.1%

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

   2. +-commutative 98.1%

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

   3. fma-define 98.1%

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

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 94.8%

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

   1. mul-1-neg 94.8%

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

7. Simplified 94.8%

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

8. Taylor expanded in c around 0 94.7%

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

   1. mul-1-neg 94.7%

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

   2. unsub-neg 94.7%

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

10. Simplified 94.7%

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

    1. clear-num 94.7%

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

    2. inv-pow 94.7%

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

    3. sqrt-div 94.7%

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

    4. metadata-eval 94.7%

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

    5. pow1/2 94.7%

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

    6. pow-flip 94.7%

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

    7. metadata-eval 94.7%

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

    8. sqrt-div 94.7%

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

    9. metadata-eval 94.7%

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

    10. pow1/2 94.7%

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

    11. pow-flip 94.7%

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

    12. metadata-eval 94.7%

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

12. Applied egg-rr 94.7%

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

    1. unpow-1 94.7%

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

    2. *-lft-identity 94.7%

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

    3. distribute-rgt-out-- 94.7%

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

14. Simplified 94.7%

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

15. Taylor expanded in cosTheta around 0 95.3%

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

    1. distribute-rgt1-in 95.2%

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

17. Simplified 95.2%

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

18. Final simplification 95.2%

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

Alternative 15: 94.9% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

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

   1. associate-+l+ 98.1%

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

   2. +-commutative 98.1%

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

   3. fma-define 98.1%

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

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 94.8%

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

   1. mul-1-neg 94.8%

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

7. Simplified 94.8%

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

8. Taylor expanded in c around 0 94.8%

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

   1. associate--l+ 94.8%

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

   2. sub-neg 94.8%

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

   3. mul-1-neg 94.8%

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

   4. +-commutative 94.8%

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

   5. distribute-rgt-out 94.8%

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

10. Simplified 94.8%

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

11. Final simplification 94.8%

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

Alternative 16: 95.0% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

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

   1. associate-+l+ 98.1%

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

   2. +-commutative 98.1%

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

   3. fma-define 98.1%

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

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 94.8%

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

   1. mul-1-neg 94.8%

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

7. Simplified 94.8%

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

8. Taylor expanded in c around 0 94.7%

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

   1. mul-1-neg 94.7%

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

   2. unsub-neg 94.7%

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

10. Simplified 94.7%

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

    1. clear-num 94.7%

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

    2. inv-pow 94.7%

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

    3. sqrt-div 94.7%

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

    4. metadata-eval 94.7%

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

    5. pow1/2 94.7%

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

    6. pow-flip 94.7%

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

    7. metadata-eval 94.7%

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

    8. sqrt-div 94.7%

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

    9. metadata-eval 94.7%

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

    10. pow1/2 94.7%

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

    11. pow-flip 94.7%

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

    12. metadata-eval 94.7%

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

12. Applied egg-rr 94.7%

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

    1. unpow-1 94.7%

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

    2. *-lft-identity 94.7%

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

    3. distribute-rgt-out-- 94.7%

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

14. Simplified 94.7%

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

    1. *-un-lft-identity 94.7%

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

    2. +-commutative 94.7%

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

    3. associate-/r/ 94.7%

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

    4. fma-define 94.7%

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

    5. *-commutative 94.7%

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

16. Applied egg-rr 94.7%

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

    1. *-lft-identity 94.7%

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

    2. fma-undefine 94.7%

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

    3. associate-*l/ 94.7%

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

    4. *-lft-identity 94.7%

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

    5. associate-/l* 94.8%

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

18. Simplified 94.8%

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

19. Final simplification 94.8%

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

Alternative 17: 94.9% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

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

   1. associate-+l+ 98.1%

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

   2. +-commutative 98.1%

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

   3. fma-define 98.1%

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

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 94.8%

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

   1. mul-1-neg 94.8%

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

7. Simplified 94.8%

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

8. Taylor expanded in c around 0 94.7%

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

   1. mul-1-neg 94.7%

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

   2. unsub-neg 94.7%

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

10. Simplified 94.7%

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

    1. div-sub 94.8%

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

    2. sqrt-div 94.8%

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

    3. metadata-eval 94.8%

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

    4. pow1/2 94.8%

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

    5. pow-flip 94.8%

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

    6. metadata-eval 94.8%

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

    7. sqrt-div 94.8%

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

    8. metadata-eval 94.8%

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

    9. pow1/2 94.8%

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

    10. pow-flip 94.8%

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

    11. metadata-eval 94.8%

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

12. Applied egg-rr 94.8%

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

    1. div-sub 94.7%

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

    2. *-lft-identity 94.7%

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

    3. distribute-rgt-out-- 94.7%

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

14. Simplified 94.7%

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

15. Final simplification 94.7%

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

Alternative 18: 93.0% 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 98.1%

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

   1. associate-+l+ 98.1%

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

   2. +-commutative 98.1%

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

   3. fma-define 98.1%

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

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 94.8%

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

   1. mul-1-neg 94.8%

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

7. Simplified 94.8%

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

8. Taylor expanded in cosTheta around 0 92.8%

   \[\leadsto \color{blue}{cosTheta \cdot \sqrt{\pi}} \]
9. Add Preprocessing

Alternative 19: 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 98.1%

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

   1. associate-+l+ 98.1%

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

   2. +-commutative 98.1%

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

   3. fma-define 98.1%

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

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 10.8%

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

6. Taylor expanded in c around 0 10.8%

   \[\leadsto \color{blue}{1} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024151 
(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))))))