Beckmann Sample, normalization factor

Percentage Accurate: 97.8% → 97.9%

Time: 2.3s

Alternatives: 7

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: 97.9% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.8%

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

2. Taylor expanded in c around 0

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

   1. lower-+.f32 N/A

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

   2. lower-+.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. lower-exp.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-pow.f32 N/A

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

   8. lower-sqrt.f32 N/A

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

   9. lower-/.f32 N/A

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

   10. lower--.f32 N/A

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

   11. lower-*.f32 N/A

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

   12. lift-PI.f32 97.9

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

4. Applied rewrites 97.9%

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

Alternative 2: 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

Derivation

1. Initial program 97.8%

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

Alternative 3: 95.8% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.8%

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

2. Taylor expanded in cosTheta around 0

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

   1. lower-*.f32 N/A

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

   2. lower-+.f32 N/A

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

   3. lift-sqrt.f32 N/A

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

   4. lift-PI.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. lift-PI.f32 N/A

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

   9. lower-+.f32 N/A

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

   10. lower-+.f32 N/A

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

4. Applied rewrites 95.8%

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

Alternative 4: 95.7% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.8%

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

2. Taylor expanded in cosTheta around 0

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

   1. lower-*.f32 N/A

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

   2. lower-+.f32 N/A

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

   3. lift-sqrt.f32 N/A

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

   4. lift-PI.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. lift-PI.f32 N/A

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

   9. lower-+.f32 N/A

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

   10. lower-+.f32 N/A

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

4. Applied rewrites 95.8%

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

5. Taylor expanded in c around 0

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

   1. lift-/.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-sqrt.f32 N/A

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

   4. lift-*.f32 95.7

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

7. Applied rewrites 95.7%

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

Alternative 5: 95.2% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.8%

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

2. Taylor expanded in c around 0

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

   1. lower-+.f32 N/A

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

   2. lower-+.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. lower-exp.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-pow.f32 N/A

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

   8. lower-sqrt.f32 N/A

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

   9. lower-/.f32 N/A

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

   10. lower--.f32 N/A

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

   11. lower-*.f32 N/A

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

   12. lift-PI.f32 97.9

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

4. Applied rewrites 97.9%

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

5. Taylor expanded in cosTheta around 0

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

7. Applied rewrites 95.2%

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

Alternative 6: 93.0% accurate, 7.8× 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 97.8%

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

2. Taylor expanded in cosTheta around 0

   \[\leadsto \color{blue}{cosTheta \cdot \sqrt{\mathsf{PI}\left(\right)}} \]
3. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto cosTheta \cdot \color{blue}{\sqrt{\mathsf{PI}\left(\right)}} \]

   2. lift-sqrt.f32 N/A

      \[\leadsto cosTheta \cdot \sqrt{\mathsf{PI}\left(\right)} \]

   3. lift-PI.f32 93.0

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

4. Applied rewrites 93.0%

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

Alternative 7: 5.0% accurate, 10.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(4) function code(costheta, c)
use fmin_fmax_functions
    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 97.8%

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

2. Taylor expanded in c around inf

   \[\leadsto \color{blue}{\frac{1}{c}} \]
3. Step-by-step derivation

   1. lower-/.f32 5.0

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

4. Applied rewrites 5.0%

   \[\leadsto \color{blue}{\frac{1}{c}} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025132 
(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))))))