Beckmann Sample, normalization factor

Percentage Accurate: 97.9% → 99.5%

Time: 18.0s

Alternatives: 13

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: 99.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \langle \left( \frac{1}{c + \left(1 + \frac{\sqrt{\frac{\mathsf{fma}\left(-2, cosTheta, 1\right)}{\pi}}}{cosTheta \cdot e^{{cosTheta}^{2}}}\right)} \right)_{\text{binary64}} \rangle_{\text{binary32}} \end{array} \]

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (cast
  (!
   :precision
   binary64
   (/
    1.0
    (+
     c
     (+
      1.0
      (/
       (sqrt (/ (fma -2.0 cosTheta 1.0) PI))
       (* cosTheta (exp (pow cosTheta 2.0))))))))))

C

float code(float cosTheta, float c) {
	double tmp = 1.0 / (((double) c) + (1.0 + (sqrt((fma(-2.0, cosTheta, 1.0) / ((double) M_PI))) / (((double) cosTheta) * exp(pow(cosTheta, 2.0))))));
	return (float) tmp;
}

Julia

function code(cosTheta, c)
	tmp = Float64(1.0 / Float64(Float64(c) + Float64(1.0 + Float64(sqrt(Float64(fma(-2.0, cosTheta, 1.0) / pi)) / Float64(Float64(cosTheta) * exp((cosTheta ^ 2.0)))))))
	return Float32(tmp)
end

Derivation

1. Initial program 99.5%

   \[\langle \frac{1}{\left(1 + c\right) + \frac{\frac{\sqrt{1 + cosTheta \cdot -2}}{\sqrt{\pi} \cdot cosTheta}}{{\left(e^{cosTheta}\right)}^{cosTheta}}} \rangle_{\text{binary64}} \]
2. Step-by-step derivation

   1. div-inv 99.5%

      \[\leadsto \langle \color{blue}{1} \cdot \frac{1}{\left(1 + c\right) + \frac{\frac{\sqrt{1 + cosTheta \cdot -2}}{\sqrt{\pi} \cdot cosTheta}}{{\left(e^{cosTheta}\right)}^{cosTheta}}} \rangle_{\text{binary64}} \]

   2. associate-+l+ 99.5%

      \[\leadsto \langle 1 \cdot \frac{1}{1 + \left(c + \frac{\frac{\sqrt{1 + cosTheta \cdot -2}}{\sqrt{\pi} \cdot cosTheta}}{{\left(e^{cosTheta}\right)}^{cosTheta}}\right)} \rangle_{\text{binary64}} \]

   3. associate-/r* 99.5%

      \[\leadsto \langle 1 \cdot \frac{1}{1 + \left(c + \frac{\frac{\frac{\sqrt{1 + cosTheta \cdot -2}}{\sqrt{\pi}}}{cosTheta}}{{\left(e^{cosTheta}\right)}^{cosTheta}}\right)} \rangle_{\text{binary64}} \]

   4. associate-/l/ 99.5%

      \[\leadsto \langle 1 \cdot \frac{1}{1 + \left(c + \frac{\frac{\sqrt{1 + cosTheta \cdot -2}}{\sqrt{\pi}}}{{\left(e^{cosTheta}\right)}^{cosTheta} \cdot cosTheta}\right)} \rangle_{\text{binary64}} \]

3. Applied egg-rr 99.5%

   \[\leadsto \langle \color{blue}{1} \cdot \frac{1}{1 + \left(c + \frac{\sqrt{\frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{\pi}}}{e^{{cosTheta}^{2}} \cdot cosTheta}\right)} \rangle_{\text{binary64}} \]
4. Step-by-step derivation

   1. *-lft-identity 99.5%

      \[\leadsto \langle \frac{\color{blue}{1}}{1 + \left(c + \frac{\sqrt{\frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{\pi}}}{e^{{cosTheta}^{2}} \cdot cosTheta}\right)} \rangle_{\text{binary64}} \]

   2. +-commutative 99.5%

      \[\leadsto \langle \frac{1}{\left(c + \frac{\sqrt{\frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{\pi}}}{e^{{cosTheta}^{2}} \cdot cosTheta}\right) + 1} \rangle_{\text{binary64}} \]

   3. associate-+l+ 99.5%

      \[\leadsto \langle \frac{1}{c + \left(\frac{\sqrt{\frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{\pi}}}{e^{{cosTheta}^{2}} \cdot cosTheta} + 1\right)} \rangle_{\text{binary64}} \]

   4. fma-udef 99.5%

      \[\leadsto \langle \frac{1}{c + \left(\frac{\sqrt{\frac{cosTheta \cdot -2 + 1}{\pi}}}{e^{{cosTheta}^{2}} \cdot cosTheta} + 1\right)} \rangle_{\text{binary64}} \]

   5. *-commutative 99.5%

      \[\leadsto \langle \frac{1}{c + \left(\frac{\sqrt{\frac{-2 \cdot cosTheta + 1}{\pi}}}{e^{{cosTheta}^{2}} \cdot cosTheta} + 1\right)} \rangle_{\text{binary64}} \]

   6. fma-def 99.5%

      \[\leadsto \langle \frac{1}{c + \left(\frac{\sqrt{\frac{\mathsf{fma}\left(-2, cosTheta, 1\right)}{\pi}}}{e^{{cosTheta}^{2}} \cdot cosTheta} + 1\right)} \rangle_{\text{binary64}} \]

   7. *-commutative 99.5%

      \[\leadsto \langle \frac{1}{c + \left(\frac{\sqrt{\frac{\mathsf{fma}\left(-2, cosTheta, 1\right)}{\pi}}}{cosTheta \cdot e^{{cosTheta}^{2}}} + 1\right)} \rangle_{\text{binary64}} \]

5. Simplified 99.5%

   \[\leadsto \langle \frac{\color{blue}{1}}{c + \left(\frac{\sqrt{\frac{\mathsf{fma}\left(-2, cosTheta, 1\right)}{\pi}}}{cosTheta \cdot e^{{cosTheta}^{2}}} + 1\right)} \rangle_{\text{binary64}} \]

6. Final simplification 99.5%

   \[\leadsto \langle \frac{1}{c + \left(1 + \frac{\sqrt{\frac{\mathsf{fma}\left(-2, cosTheta, 1\right)}{\pi}}}{cosTheta \cdot e^{{cosTheta}^{2}}}\right)} \rangle_{\text{binary64}} \]

Alternative 2: 99.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\left(1 + c\right) + \langle \left( \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{1 - cosTheta \cdot 2}}{cosTheta}\right) \right)_{\text{binary64}} \rangle_{\text{binary32}} \cdot e^{-cosTheta \cdot cosTheta}} \end{array} \]

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (/
  1.0
  (+
   (+ 1.0 c)
   (*
    (cast
     (!
      :precision
      binary64
      (* (/ 1.0 (sqrt PI)) (/ (sqrt (- 1.0 (* cosTheta 2.0))) cosTheta))))
    (exp (- (* cosTheta cosTheta)))))))

C

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

Julia

function code(cosTheta, c)
	tmp = Float64(Float64(1.0 / sqrt(pi)) * Float64(sqrt(Float64(1.0 - Float64(Float64(cosTheta) * 2.0))) / Float64(cosTheta)))
	return Float32(Float32(1.0) / Float32(Float32(Float32(1.0) + c) + Float32(Float32(tmp) * exp(Float32(-Float32(cosTheta * cosTheta))))))
end

MATLAB

function tmp_2 = code(cosTheta, c)
	tmp = (1.0 / sqrt(pi)) * (sqrt((1.0 - (double(cosTheta) * 2.0))) / double(cosTheta));
	tmp_2 = single(1.0) / ((single(1.0) + c) + single((single(tmp) * double(exp(-(cosTheta * cosTheta))))));
end

Derivation

1. Initial program 97.9%

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

   1. rewrite-binary32/binary64 98.9%

      \[\leadsto \color{blue}{\frac{1}{\left(1 + c\right) + \langle \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \rangle_{\text{binary64}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}}} \]

3. Applied rewrite-once 98.9%

   \[\leadsto \frac{1}{\left(1 + c\right) + \color{blue}{\langle \color{blue}{\left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right)} \rangle_{\text{binary64}}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
4. Step-by-step derivation

   1. associate--r+ 98.9%

      \[\leadsto \frac{1}{\left(1 + c\right) + \langle \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{1 - \left(cosTheta + cosTheta\right)}}{cosTheta}\right) \rangle_{\text{binary64}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   2. count-2 98.9%

      \[\leadsto \frac{1}{\left(1 + c\right) + \langle \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{1 - 2 \cdot cosTheta}}{cosTheta}\right) \rangle_{\text{binary64}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

5. Simplified 98.9%

   \[\leadsto \frac{1}{\left(1 + c\right) + \color{blue}{\langle \color{blue}{\left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{1 - 2 \cdot cosTheta}}{cosTheta}\right)} \rangle_{\text{binary64}}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

6. Final simplification 98.9%

   \[\leadsto \frac{1}{\left(1 + c\right) + \langle \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{1 - cosTheta \cdot 2}}{cosTheta}\right) \rangle_{\text{binary64}} \cdot e^{-cosTheta \cdot cosTheta}} \]

Alternative 3: 98.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\left(1 + c\right) + \frac{\frac{\sqrt{1 + -2 \cdot cosTheta}}{\langle \left( cosTheta \cdot \sqrt{\pi} \right)_{\text{binary64}} \rangle_{\text{binary32}}}}{{\left(e^{cosTheta}\right)}^{cosTheta}}} \end{array} \]

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (/
  1.0
  (+
   (+ 1.0 c)
   (/
    (/
     (sqrt (+ 1.0 (* -2.0 cosTheta)))
     (cast (! :precision binary64 (* cosTheta (sqrt PI)))))
    (pow (exp cosTheta) cosTheta)))))

C

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

Julia

function code(cosTheta, c)
	tmp = Float64(Float64(cosTheta) * sqrt(pi))
	return Float32(Float32(1.0) / Float32(Float32(Float32(1.0) + c) + Float32(Float32(sqrt(Float32(Float32(1.0) + Float32(Float32(-2.0) * cosTheta))) / Float32(tmp)) / (exp(cosTheta) ^ cosTheta))))
end

MATLAB

function tmp_2 = code(cosTheta, c)
	tmp = double(cosTheta) * sqrt(pi);
	tmp_2 = single(1.0) / ((single(1.0) + c) + (single((double(sqrt((single(1.0) + (single(-2.0) * cosTheta)))) / single(tmp))) / (exp(cosTheta) ^ cosTheta)));
end

Derivation

1. Initial program 97.9%

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

   1. *-commutative 97.9%

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

   2. associate-*l* 97.9%

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

   3. /-rgt-identity 97.9%

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

   4. associate-/r/ 97.9%

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

   5. exp-neg 97.9%

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

   6. distribute-lft-neg-out 97.9%

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

   7. remove-double-neg 97.9%

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

   8. associate-*r/ 97.9%

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

   9. *-commutative 97.9%

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

3. Simplified 98.5%

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

   1. rewrite-binary32/binary64 98.9%

      \[\leadsto \color{blue}{\frac{1}{\left(1 + c\right) + \frac{\frac{\sqrt{1 + cosTheta \cdot -2}}{\langle \sqrt{\pi} \cdot cosTheta \rangle_{\text{binary64}}}}{{\left(e^{cosTheta}\right)}^{cosTheta}}}} \]

5. Applied rewrite-once 98.9%

   \[\leadsto \frac{1}{\left(1 + c\right) + \frac{\frac{\sqrt{1 + cosTheta \cdot -2}}{\color{blue}{\langle \color{blue}{\sqrt{\pi} \cdot cosTheta} \rangle_{\text{binary64}}}}}{{\left(e^{cosTheta}\right)}^{cosTheta}}} \]

6. Final simplification 98.9%

   \[\leadsto \frac{1}{\left(1 + c\right) + \frac{\frac{\sqrt{1 + -2 \cdot cosTheta}}{\langle cosTheta \cdot \sqrt{\pi} \rangle_{\text{binary64}}}}{{\left(e^{cosTheta}\right)}^{cosTheta}}} \]

Alternative 4: 98.5% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.9%

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

   1. *-commutative 97.9%

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

   2. associate-*l* 97.9%

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

   3. /-rgt-identity 97.9%

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

   4. associate-/r/ 97.9%

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

   5. exp-neg 97.9%

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

   6. distribute-lft-neg-out 97.9%

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

   7. remove-double-neg 97.9%

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

   8. associate-*r/ 97.9%

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

   9. *-commutative 97.9%

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

3. Simplified 98.5%

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

4. Final simplification 98.5%

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

Alternative 5: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.9%

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

   1. div-inv 96.5%

      \[\leadsto \frac{1}{\left(1 + c\right) + \color{blue}{\left(1 \cdot \frac{1}{\sqrt{\pi}}\right)} \cdot \left(\left(-1.5 \cdot cosTheta + \left(\left(cosTheta \cdot cosTheta\right) \cdot 0.5 + \frac{1}{cosTheta}\right)\right) - 1\right)} \]

   2. pow1/2 96.5%

      \[\leadsto \frac{1}{\left(1 + c\right) + \left(1 \cdot \frac{1}{\color{blue}{{\pi}^{0.5}}}\right) \cdot \left(\left(-1.5 \cdot cosTheta + \left(\left(cosTheta \cdot cosTheta\right) \cdot 0.5 + \frac{1}{cosTheta}\right)\right) - 1\right)} \]

   3. pow-flip 96.5%

      \[\leadsto \frac{1}{\left(1 + c\right) + \left(1 \cdot \color{blue}{{\pi}^{\left(-0.5\right)}}\right) \cdot \left(\left(-1.5 \cdot cosTheta + \left(\left(cosTheta \cdot cosTheta\right) \cdot 0.5 + \frac{1}{cosTheta}\right)\right) - 1\right)} \]

   4. metadata-eval 96.5%

      \[\leadsto \frac{1}{\left(1 + c\right) + \left(1 \cdot {\pi}^{\color{blue}{-0.5}}\right) \cdot \left(\left(-1.5 \cdot cosTheta + \left(\left(cosTheta \cdot cosTheta\right) \cdot 0.5 + \frac{1}{cosTheta}\right)\right) - 1\right)} \]

3. Applied egg-rr 97.9%

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

   1. *-lft-identity 96.5%

      \[\leadsto \frac{1}{\left(1 + c\right) + \color{blue}{{\pi}^{-0.5}} \cdot \left(\left(-1.5 \cdot cosTheta + \left(\left(cosTheta \cdot cosTheta\right) \cdot 0.5 + \frac{1}{cosTheta}\right)\right) - 1\right)} \]

5. Simplified 97.9%

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

6. Final simplification 97.9%

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

Alternative 6: 97.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.9%

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

2. Taylor expanded in c around 0 97.8%

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

   1. mul-1-neg 97.8%

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

   2. exp-neg 97.8%

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

   3. unpow2 97.8%

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

   4. exp-prod 97.8%

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

   5. associate-/r* 97.8%

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

   6. exp-prod 97.8%

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

   7. unpow2 97.8%

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

   8. *-commutative 97.8%

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

   9. cancel-sign-sub-inv 97.8%

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

   10. metadata-eval 97.8%

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

4. Simplified 97.9%

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

5. Final simplification 97.9%

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

Alternative 7: 96.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\left(1 + c\right) + {\pi}^{-0.5} \cdot \left(\left(-1 + \mathsf{fma}\left(0.5, cosTheta \cdot cosTheta, \frac{1}{cosTheta}\right)\right) + cosTheta \cdot -1.5\right)} \end{array} \]

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (/
  1.0
  (+
   (+ 1.0 c)
   (*
    (pow PI -0.5)
    (+
     (+ -1.0 (fma 0.5 (* cosTheta cosTheta) (/ 1.0 cosTheta)))
     (* cosTheta -1.5))))))

C

float code(float cosTheta, float c) {
	return 1.0f / ((1.0f + c) + (powf(((float) M_PI), -0.5f) * ((-1.0f + fmaf(0.5f, (cosTheta * cosTheta), (1.0f / cosTheta))) + (cosTheta * -1.5f))));
}

Julia

function code(cosTheta, c)
	return Float32(Float32(1.0) / Float32(Float32(Float32(1.0) + c) + Float32((Float32(pi) ^ Float32(-0.5)) * Float32(Float32(Float32(-1.0) + fma(Float32(0.5), Float32(cosTheta * cosTheta), Float32(Float32(1.0) / cosTheta))) + Float32(cosTheta * Float32(-1.5))))))
end

Derivation

1. Initial program 97.9%

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

   1. associate-*l* 97.9%

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

   2. associate--l- 97.9%

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

   3. *-commutative 97.9%

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

   4. exp-prod 97.9%

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

3. Simplified 97.9%

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

4. Taylor expanded in cosTheta around 0 96.5%

   \[\leadsto \frac{1}{\left(1 + c\right) + \frac{1}{\sqrt{\pi}} \cdot \color{blue}{\left(\left(-1.5 \cdot cosTheta + \left(0.5 \cdot {cosTheta}^{2} + \frac{1}{cosTheta}\right)\right) - 1\right)}} \]
5. Step-by-step derivation

   1. associate--l+ 96.5%

      \[\leadsto \frac{1}{\left(1 + c\right) + \frac{1}{\sqrt{\pi}} \cdot \color{blue}{\left(-1.5 \cdot cosTheta + \left(\left(0.5 \cdot {cosTheta}^{2} + \frac{1}{cosTheta}\right) - 1\right)\right)}} \]

   2. distribute-lft-in 96.5%

      \[\leadsto \frac{1}{\left(1 + c\right) + \color{blue}{\left(\frac{1}{\sqrt{\pi}} \cdot \left(-1.5 \cdot cosTheta\right) + \frac{1}{\sqrt{\pi}} \cdot \left(\left(0.5 \cdot {cosTheta}^{2} + \frac{1}{cosTheta}\right) - 1\right)\right)}} \]

   3. inv-pow 96.5%

      \[\leadsto \frac{1}{\left(1 + c\right) + \left(\color{blue}{{\left(\sqrt{\pi}\right)}^{-1}} \cdot \left(-1.5 \cdot cosTheta\right) + \frac{1}{\sqrt{\pi}} \cdot \left(\left(0.5 \cdot {cosTheta}^{2} + \frac{1}{cosTheta}\right) - 1\right)\right)} \]

   4. sqrt-pow2 96.5%

      \[\leadsto \frac{1}{\left(1 + c\right) + \left(\color{blue}{{\pi}^{\left(\frac{-1}{2}\right)}} \cdot \left(-1.5 \cdot cosTheta\right) + \frac{1}{\sqrt{\pi}} \cdot \left(\left(0.5 \cdot {cosTheta}^{2} + \frac{1}{cosTheta}\right) - 1\right)\right)} \]

   5. metadata-eval 96.5%

      \[\leadsto \frac{1}{\left(1 + c\right) + \left({\pi}^{\color{blue}{-0.5}} \cdot \left(-1.5 \cdot cosTheta\right) + \frac{1}{\sqrt{\pi}} \cdot \left(\left(0.5 \cdot {cosTheta}^{2} + \frac{1}{cosTheta}\right) - 1\right)\right)} \]

   6. *-commutative 96.5%

      \[\leadsto \frac{1}{\left(1 + c\right) + \left({\pi}^{-0.5} \cdot \color{blue}{\left(cosTheta \cdot -1.5\right)} + \frac{1}{\sqrt{\pi}} \cdot \left(\left(0.5 \cdot {cosTheta}^{2} + \frac{1}{cosTheta}\right) - 1\right)\right)} \]

   7. inv-pow 96.5%

      \[\leadsto \frac{1}{\left(1 + c\right) + \left({\pi}^{-0.5} \cdot \left(cosTheta \cdot -1.5\right) + \color{blue}{{\left(\sqrt{\pi}\right)}^{-1}} \cdot \left(\left(0.5 \cdot {cosTheta}^{2} + \frac{1}{cosTheta}\right) - 1\right)\right)} \]

   8. sqrt-pow2 96.5%

      \[\leadsto \frac{1}{\left(1 + c\right) + \left({\pi}^{-0.5} \cdot \left(cosTheta \cdot -1.5\right) + \color{blue}{{\pi}^{\left(\frac{-1}{2}\right)}} \cdot \left(\left(0.5 \cdot {cosTheta}^{2} + \frac{1}{cosTheta}\right) - 1\right)\right)} \]

   9. metadata-eval 96.5%

      \[\leadsto \frac{1}{\left(1 + c\right) + \left({\pi}^{-0.5} \cdot \left(cosTheta \cdot -1.5\right) + {\pi}^{\color{blue}{-0.5}} \cdot \left(\left(0.5 \cdot {cosTheta}^{2} + \frac{1}{cosTheta}\right) - 1\right)\right)} \]

   10. sub-neg 96.5%

       \[\leadsto \frac{1}{\left(1 + c\right) + \left({\pi}^{-0.5} \cdot \left(cosTheta \cdot -1.5\right) + {\pi}^{-0.5} \cdot \color{blue}{\left(\left(0.5 \cdot {cosTheta}^{2} + \frac{1}{cosTheta}\right) + \left(-1\right)\right)}\right)} \]

   11. fma-def 96.5%

       \[\leadsto \frac{1}{\left(1 + c\right) + \left({\pi}^{-0.5} \cdot \left(cosTheta \cdot -1.5\right) + {\pi}^{-0.5} \cdot \left(\color{blue}{\mathsf{fma}\left(0.5, {cosTheta}^{2}, \frac{1}{cosTheta}\right)} + \left(-1\right)\right)\right)} \]

   12. unpow2 96.5%

       \[\leadsto \frac{1}{\left(1 + c\right) + \left({\pi}^{-0.5} \cdot \left(cosTheta \cdot -1.5\right) + {\pi}^{-0.5} \cdot \left(\mathsf{fma}\left(0.5, \color{blue}{cosTheta \cdot cosTheta}, \frac{1}{cosTheta}\right) + \left(-1\right)\right)\right)} \]

   13. metadata-eval 96.5%

       \[\leadsto \frac{1}{\left(1 + c\right) + \left({\pi}^{-0.5} \cdot \left(cosTheta \cdot -1.5\right) + {\pi}^{-0.5} \cdot \left(\mathsf{fma}\left(0.5, cosTheta \cdot cosTheta, \frac{1}{cosTheta}\right) + \color{blue}{-1}\right)\right)} \]

6. Applied egg-rr 96.5%

   \[\leadsto \frac{1}{\left(1 + c\right) + \color{blue}{\left({\pi}^{-0.5} \cdot \left(cosTheta \cdot -1.5\right) + {\pi}^{-0.5} \cdot \left(\mathsf{fma}\left(0.5, cosTheta \cdot cosTheta, \frac{1}{cosTheta}\right) + -1\right)\right)}} \]
7. Step-by-step derivation

   1. distribute-lft-out 96.5%

      \[\leadsto \frac{1}{\left(1 + c\right) + \color{blue}{{\pi}^{-0.5} \cdot \left(cosTheta \cdot -1.5 + \left(\mathsf{fma}\left(0.5, cosTheta \cdot cosTheta, \frac{1}{cosTheta}\right) + -1\right)\right)}} \]

   2. +-commutative 96.5%

      \[\leadsto \frac{1}{\left(1 + c\right) + {\pi}^{-0.5} \cdot \left(cosTheta \cdot -1.5 + \color{blue}{\left(-1 + \mathsf{fma}\left(0.5, cosTheta \cdot cosTheta, \frac{1}{cosTheta}\right)\right)}\right)} \]

8. Simplified 96.5%

   \[\leadsto \frac{1}{\left(1 + c\right) + \color{blue}{{\pi}^{-0.5} \cdot \left(cosTheta \cdot -1.5 + \left(-1 + \mathsf{fma}\left(0.5, cosTheta \cdot cosTheta, \frac{1}{cosTheta}\right)\right)\right)}} \]

9. Final simplification 96.5%

   \[\leadsto \frac{1}{\left(1 + c\right) + {\pi}^{-0.5} \cdot \left(\left(-1 + \mathsf{fma}\left(0.5, cosTheta \cdot cosTheta, \frac{1}{cosTheta}\right)\right) + cosTheta \cdot -1.5\right)} \]

Alternative 8: 96.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\left(1 + c\right) + {\pi}^{-0.5} \cdot \left(-1 + \left(cosTheta \cdot -1.5 + \left(\frac{1}{cosTheta} + \left(cosTheta \cdot cosTheta\right) \cdot 0.5\right)\right)\right)} \end{array} \]

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (/
  1.0
  (+
   (+ 1.0 c)
   (*
    (pow PI -0.5)
    (+
     -1.0
     (+
      (* cosTheta -1.5)
      (+ (/ 1.0 cosTheta) (* (* cosTheta cosTheta) 0.5))))))))

C

float code(float cosTheta, float c) {
	return 1.0f / ((1.0f + c) + (powf(((float) M_PI), -0.5f) * (-1.0f + ((cosTheta * -1.5f) + ((1.0f / cosTheta) + ((cosTheta * cosTheta) * 0.5f))))));
}

Julia

function code(cosTheta, c)
	return Float32(Float32(1.0) / Float32(Float32(Float32(1.0) + c) + Float32((Float32(pi) ^ Float32(-0.5)) * Float32(Float32(-1.0) + Float32(Float32(cosTheta * Float32(-1.5)) + Float32(Float32(Float32(1.0) / cosTheta) + Float32(Float32(cosTheta * cosTheta) * Float32(0.5))))))))
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(-1.5)) + ((single(1.0) / cosTheta) + ((cosTheta * cosTheta) * single(0.5)))))));
end

Derivation

1. Initial program 97.9%

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

   1. associate-*l* 97.9%

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

   2. associate--l- 97.9%

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

   3. *-commutative 97.9%

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

   4. exp-prod 97.9%

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

3. Simplified 97.9%

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

4. Taylor expanded in cosTheta around 0 96.5%

   \[\leadsto \frac{1}{\left(1 + c\right) + \frac{1}{\sqrt{\pi}} \cdot \color{blue}{\left(\left(-1.5 \cdot cosTheta + \left(0.5 \cdot {cosTheta}^{2} + \frac{1}{cosTheta}\right)\right) - 1\right)}} \]
5. Step-by-step derivation

   1. add-exp-log_binary32 96.5%

      \[\leadsto \color{blue}{\frac{1}{\left(1 + c\right) + \frac{1}{\sqrt{\pi}} \cdot \left(\left(-1.5 \cdot cosTheta + \left(e^{\log \left(0.5 \cdot {cosTheta}^{2}\right)} + \frac{1}{cosTheta}\right)\right) - 1\right)}} \]

6. Applied rewrite-once 96.5%

   \[\leadsto \frac{1}{\left(1 + c\right) + \frac{1}{\sqrt{\pi}} \cdot \left(\left(-1.5 \cdot cosTheta + \left(\color{blue}{e^{\log \left(0.5 \cdot {cosTheta}^{2}\right)}} + \frac{1}{cosTheta}\right)\right) - 1\right)} \]
7. Step-by-step derivation

   1. rem-exp-log 96.5%

      \[\leadsto \frac{1}{\left(1 + c\right) + \frac{1}{\sqrt{\pi}} \cdot \left(\left(-1.5 \cdot cosTheta + \left(\color{blue}{0.5 \cdot {cosTheta}^{2}} + \frac{1}{cosTheta}\right)\right) - 1\right)} \]

   2. *-commutative 96.5%

      \[\leadsto \frac{1}{\left(1 + c\right) + \frac{1}{\sqrt{\pi}} \cdot \left(\left(-1.5 \cdot cosTheta + \left(\color{blue}{{cosTheta}^{2} \cdot 0.5} + \frac{1}{cosTheta}\right)\right) - 1\right)} \]

   3. unpow2 96.5%

      \[\leadsto \frac{1}{\left(1 + c\right) + \frac{1}{\sqrt{\pi}} \cdot \left(\left(-1.5 \cdot cosTheta + \left(\color{blue}{\left(cosTheta \cdot cosTheta\right)} \cdot 0.5 + \frac{1}{cosTheta}\right)\right) - 1\right)} \]

8. Simplified 96.5%

   \[\leadsto \frac{1}{\left(1 + c\right) + \frac{1}{\sqrt{\pi}} \cdot \left(\left(-1.5 \cdot cosTheta + \left(\color{blue}{\left(cosTheta \cdot cosTheta\right) \cdot 0.5} + \frac{1}{cosTheta}\right)\right) - 1\right)} \]
9. Step-by-step derivation

   1. div-inv 96.5%

      \[\leadsto \frac{1}{\left(1 + c\right) + \color{blue}{\left(1 \cdot \frac{1}{\sqrt{\pi}}\right)} \cdot \left(\left(-1.5 \cdot cosTheta + \left(\left(cosTheta \cdot cosTheta\right) \cdot 0.5 + \frac{1}{cosTheta}\right)\right) - 1\right)} \]

   2. pow1/2 96.5%

      \[\leadsto \frac{1}{\left(1 + c\right) + \left(1 \cdot \frac{1}{\color{blue}{{\pi}^{0.5}}}\right) \cdot \left(\left(-1.5 \cdot cosTheta + \left(\left(cosTheta \cdot cosTheta\right) \cdot 0.5 + \frac{1}{cosTheta}\right)\right) - 1\right)} \]

   3. pow-flip 96.5%

      \[\leadsto \frac{1}{\left(1 + c\right) + \left(1 \cdot \color{blue}{{\pi}^{\left(-0.5\right)}}\right) \cdot \left(\left(-1.5 \cdot cosTheta + \left(\left(cosTheta \cdot cosTheta\right) \cdot 0.5 + \frac{1}{cosTheta}\right)\right) - 1\right)} \]

   4. metadata-eval 96.5%

      \[\leadsto \frac{1}{\left(1 + c\right) + \left(1 \cdot {\pi}^{\color{blue}{-0.5}}\right) \cdot \left(\left(-1.5 \cdot cosTheta + \left(\left(cosTheta \cdot cosTheta\right) \cdot 0.5 + \frac{1}{cosTheta}\right)\right) - 1\right)} \]

10. Applied egg-rr 96.5%

    \[\leadsto \frac{1}{\left(1 + c\right) + \color{blue}{\left(1 \cdot {\pi}^{-0.5}\right)} \cdot \left(\left(-1.5 \cdot cosTheta + \left(\left(cosTheta \cdot cosTheta\right) \cdot 0.5 + \frac{1}{cosTheta}\right)\right) - 1\right)} \]
11. Step-by-step derivation

    1. *-lft-identity 96.5%

       \[\leadsto \frac{1}{\left(1 + c\right) + \color{blue}{{\pi}^{-0.5}} \cdot \left(\left(-1.5 \cdot cosTheta + \left(\left(cosTheta \cdot cosTheta\right) \cdot 0.5 + \frac{1}{cosTheta}\right)\right) - 1\right)} \]

12. Simplified 96.5%

    \[\leadsto \frac{1}{\left(1 + c\right) + \color{blue}{{\pi}^{-0.5}} \cdot \left(\left(-1.5 \cdot cosTheta + \left(\left(cosTheta \cdot cosTheta\right) \cdot 0.5 + \frac{1}{cosTheta}\right)\right) - 1\right)} \]

13. Final simplification 96.5%

    \[\leadsto \frac{1}{\left(1 + c\right) + {\pi}^{-0.5} \cdot \left(-1 + \left(cosTheta \cdot -1.5 + \left(\frac{1}{cosTheta} + \left(cosTheta \cdot cosTheta\right) \cdot 0.5\right)\right)\right)} \]

Alternative 9: 96.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\left(1 + c\right) + \frac{1}{\sqrt{\pi}} \cdot \left(-1 + \left(\frac{1}{cosTheta} + cosTheta \cdot -1.5\right)\right)} \end{array} \]

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (/
  1.0
  (+
   (+ 1.0 c)
   (* (/ 1.0 (sqrt PI)) (+ -1.0 (+ (/ 1.0 cosTheta) (* cosTheta -1.5)))))))

C

float code(float cosTheta, float c) {
	return 1.0f / ((1.0f + c) + ((1.0f / sqrtf(((float) M_PI))) * (-1.0f + ((1.0f / cosTheta) + (cosTheta * -1.5f)))));
}

Julia

function code(cosTheta, c)
	return Float32(Float32(1.0) / Float32(Float32(Float32(1.0) + c) + Float32(Float32(Float32(1.0) / sqrt(Float32(pi))) * Float32(Float32(-1.0) + Float32(Float32(Float32(1.0) / cosTheta) + Float32(cosTheta * Float32(-1.5)))))))
end

MATLAB

function tmp = code(cosTheta, c)
	tmp = single(1.0) / ((single(1.0) + c) + ((single(1.0) / sqrt(single(pi))) * (single(-1.0) + ((single(1.0) / cosTheta) + (cosTheta * single(-1.5))))));
end

Derivation

1. Initial program 97.9%

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

   1. associate-*l* 97.9%

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

   2. associate--l- 97.9%

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

   3. *-commutative 97.9%

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

   4. exp-prod 97.9%

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

3. Simplified 97.9%

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

4. Taylor expanded in cosTheta around 0 95.8%

   \[\leadsto \frac{1}{\left(1 + c\right) + \frac{1}{\sqrt{\pi}} \cdot \color{blue}{\left(\left(-1.5 \cdot cosTheta + \frac{1}{cosTheta}\right) - 1\right)}} \]

5. Final simplification 95.8%

   \[\leadsto \frac{1}{\left(1 + c\right) + \frac{1}{\sqrt{\pi}} \cdot \left(-1 + \left(\frac{1}{cosTheta} + cosTheta \cdot -1.5\right)\right)} \]

Alternative 10: 95.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.9%

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

   1. associate-*l* 97.9%

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

   2. associate--l- 97.9%

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

   3. *-commutative 97.9%

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

   4. exp-prod 97.9%

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

3. Simplified 97.9%

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

4. Taylor expanded in cosTheta around 0 94.4%

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

5. Final simplification 94.4%

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

Alternative 11: 93.1% accurate, 2.1× 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 97.9%

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

2. Taylor expanded in cosTheta around 0 92.8%

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

3. Final simplification 92.8%

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

Alternative 12: 10.8% accurate, 140.3× 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 97.9%

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

2. Taylor expanded in cosTheta around inf 10.8%

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

3. Taylor expanded in c around 0 10.8%

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

   1. neg-mul-1 10.8%

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

   2. sub-neg 10.8%

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

5. Simplified 10.8%

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

6. Final simplification 10.8%

   \[\leadsto 1 - c \]

Alternative 13: 10.8% accurate, 421.0× 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 97.9%

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

2. Taylor expanded in cosTheta around inf 10.8%

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

3. Taylor expanded in c around 0 10.8%

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

4. Final simplification 10.8%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023297 
(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))))))