Beckmann Sample, normalization factor

Percentage Accurate: 97.8% → 98.5%

Time: 28.9s

Alternatives: 9

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 97.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 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. frac-times 98.6%

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

   2. *-un-lft-identity 98.6%

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

   3. associate--r+ 98.6%

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

3. Applied egg-rr 98.6%

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

4. Final simplification 98.6%

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

Alternative 2: 97.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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(1.0) / Float32(cosTheta * exp((cosTheta ^ Float32(2.0))))))))
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 * exp((cosTheta ^ single(2.0)))))));
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%

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

   3. associate-*r* 98.1%

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

   4. *-commutative 98.1%

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

   5. associate-/r/ 98.0%

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

3. Simplified 98.6%

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

4. Taylor expanded in c around 0 98.0%

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

5. Final simplification 98.0%

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

Alternative 3: 97.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(cosTheta, c)
	return Float32(Float32(1.0) / fma(Float32(exp(Float32(cosTheta * Float32(-cosTheta))) / cosTheta), sqrt(Float32(Float32(Float32(1.0) + Float32(cosTheta * Float32(-2.0))) / Float32(pi))), 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. frac-times 98.6%

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

   2. *-un-lft-identity 98.6%

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

   3. associate--r+ 98.6%

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

3. Applied egg-rr 98.6%

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

4. Taylor expanded in c around 0 98.0%

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

   1. fma-def 98.0%

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

   2. mul-1-neg 98.0%

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

   3. unpow2 98.0%

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

   4. distribute-rgt-neg-out 98.0%

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

   5. cancel-sign-sub-inv 98.0%

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

   6. metadata-eval 98.0%

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

6. Simplified 98.0%

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

7. Final simplification 98.0%

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

Alternative 4: 97.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

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

   3. associate-*r* 98.1%

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

   4. *-commutative 98.1%

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

   5. associate-/r/ 98.0%

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

3. Simplified 98.6%

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

4. Taylor expanded in c around 0 98.0%

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

   1. un-div-inv 98.0%

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

   2. *-commutative 98.0%

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

   3. unpow2 98.0%

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

6. Applied egg-rr 98.0%

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

7. Final simplification 98.0%

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

Alternative 5: 94.9% accurate, 2.0× 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%

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

   3. associate-*r* 98.1%

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

   4. *-commutative 98.1%

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

   5. associate-/r/ 98.0%

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

3. Simplified 98.6%

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

4. Taylor expanded in c around 0 98.0%

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

5. Taylor expanded in cosTheta around 0 95.6%

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

   1. distribute-rgt-out 95.6%

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

7. Simplified 95.6%

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

8. Final simplification 95.6%

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

Alternative 6: 93.0% 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 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. frac-times 98.6%

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

   2. *-un-lft-identity 98.6%

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

   3. associate--r+ 98.6%

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

3. Applied egg-rr 98.6%

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

4. Taylor expanded in cosTheta around 0 93.2%

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

5. Final simplification 93.2%

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

Alternative 7: 10.9% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta, c)
	tmp = -c * (single(pi) * (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. 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%

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

   3. associate-*r* 98.1%

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

   4. *-commutative 98.1%

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

   5. associate-/r/ 98.0%

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

3. Simplified 98.6%

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

4. Taylor expanded in cosTheta around 0 92.9%

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

5. Taylor expanded in cosTheta around 0 92.9%

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

   1. fma-def 93.0%

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

   2. associate-*r* 93.0%

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

   3. *-commutative 93.0%

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

   4. unpow2 93.0%

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

   5. +-commutative 93.0%

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

   6. distribute-lft-in 93.0%

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

   7. metadata-eval 93.0%

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

   8. mul-1-neg 93.0%

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

   9. unsub-neg 93.0%

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

7. Simplified 93.0%

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

8. Taylor expanded in c around inf 11.0%

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

   1. associate-*r* 11.0%

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

   2. mul-1-neg 11.0%

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

   3. *-commutative 11.0%

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

   4. unpow2 11.0%

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

10. Simplified 11.0%

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

11. Final simplification 11.0%

    \[\leadsto \left(-c\right) \cdot \left(\pi \cdot \left(cosTheta \cdot cosTheta\right)\right) \]

Alternative 8: 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 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%

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

   3. associate-*r* 98.1%

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

   4. *-commutative 98.1%

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

   5. associate-/r/ 98.0%

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

3. Simplified 98.6%

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

4. Taylor expanded in cosTheta around inf 10.4%

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

5. Taylor expanded in c around 0 10.4%

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

   1. +-commutative 10.4%

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

   2. mul-1-neg 10.4%

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

   3. unsub-neg 10.4%

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

7. Simplified 10.4%

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

8. Final simplification 10.4%

   \[\leadsto 1 - c \]

Alternative 9: 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 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%

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

   3. associate-*r* 98.1%

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

   4. *-commutative 98.1%

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

   5. associate-/r/ 98.0%

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

3. Simplified 98.6%

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

4. Taylor expanded in cosTheta around inf 10.4%

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

5. Taylor expanded in c around 0 10.4%

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

6. Final simplification 10.4%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023278 
(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))))))