HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.5%

Time: 15.8s

Alternatives: 21

Speedup: 1.0×

Specification

\[\left(10^{-5} \leq u \land u \leq 1\right) \land \left(0 \leq v \land v \leq 109.746574\right)\]

Math

\[\begin{array}{l} \\ 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (+ 1.0 (* v (log (+ u (* (- 1.0 u) (exp (/ -2.0 v))))))))

C

float code(float u, float v) {
	return 1.0f + (v * logf((u + ((1.0f - u) * expf((-2.0f / v))))));
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    code = 1.0e0 + (v * log((u + ((1.0e0 - u) * exp(((-2.0e0) / v))))))
end function

Julia

function code(u, v)
	return Float32(Float32(1.0) + Float32(v * log(Float32(u + Float32(Float32(Float32(1.0) - u) * exp(Float32(Float32(-2.0) / v)))))))
end

MATLAB

function tmp = code(u, v)
	tmp = single(1.0) + (v * log((u + ((single(1.0) - u) * exp((single(-2.0) / v))))));
end

Initial Program: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (+ 1.0 (* v (log (+ u (* (- 1.0 u) (exp (/ -2.0 v))))))))

C

float code(float u, float v) {
	return 1.0f + (v * logf((u + ((1.0f - u) * expf((-2.0f / v))))));
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    code = 1.0e0 + (v * log((u + ((1.0e0 - u) * exp(((-2.0e0) / v))))))
end function

Julia

function code(u, v)
	return Float32(Float32(1.0) + Float32(v * log(Float32(u + Float32(Float32(Float32(1.0) - u) * exp(Float32(Float32(-2.0) / v)))))))
end

MATLAB

function tmp = code(u, v)
	tmp = single(1.0) + (v * log((u + ((single(1.0) - u) * exp((single(-2.0) / v))))));
end

Alternative 1: 99.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(v, \log \left(\mathsf{fma}\left(\mathsf{fma}\left(u, -u, 1\right), \frac{e^{\frac{-2}{v}}}{1 + u}, u\right)\right), 1\right) \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (fma v (log (fma (fma u (- u) 1.0) (/ (exp (/ -2.0 v)) (+ 1.0 u)) u)) 1.0))

C

float code(float u, float v) {
	return fmaf(v, logf(fmaf(fmaf(u, -u, 1.0f), (expf((-2.0f / v)) / (1.0f + u)), u)), 1.0f);
}

Julia

function code(u, v)
	return fma(v, log(fma(fma(u, Float32(-u), Float32(1.0)), Float32(exp(Float32(Float32(-2.0) / v)) / Float32(Float32(1.0) + u)), u)), Float32(1.0))
end

Derivation

1. Initial program 99.6%

   \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
2. Add Preprocessing

4. Applied rewrites 99.5%

   \[\leadsto 1 + v \cdot \log \color{blue}{\left(\mathsf{fma}\left(e^{\frac{-2}{v}} \cdot \left(1 - u \cdot u\right), \frac{1}{1 + u}, u\right)\right)} \]

5. Taylor expanded in v around 0

   \[\leadsto \color{blue}{1 + v \cdot \log \left(u + \frac{e^{\frac{-2}{v}} \cdot \left(1 - {u}^{2}\right)}{1 + u}\right)} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{v \cdot \log \left(u + \frac{e^{\frac{-2}{v}} \cdot \left(1 - {u}^{2}\right)}{1 + u}\right) + 1} \]

   2. lower-fma.f32 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(v, \log \left(u + \frac{e^{\frac{-2}{v}} \cdot \left(1 - {u}^{2}\right)}{1 + u}\right), 1\right)} \]

7. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(v, \log \left(\mathsf{fma}\left(\mathsf{fma}\left(u, -u, 1\right), \frac{e^{\frac{-2}{v}}}{u + 1}, u\right)\right), 1\right)} \]

8. Final simplification 99.6%

   \[\leadsto \mathsf{fma}\left(v, \log \left(\mathsf{fma}\left(\mathsf{fma}\left(u, -u, 1\right), \frac{e^{\frac{-2}{v}}}{1 + u}, u\right)\right), 1\right) \]
9. Add Preprocessing

Alternative 2: 97.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{fma}\left(v, \log \left(\left(-u\right) \cdot \mathsf{expm1}\left(\frac{-2}{v}\right)\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 + v \cdot \mathsf{fma}\left(u, \mathsf{expm1}\left(\frac{2}{v}\right), \frac{-2}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224)
   (fma v (log (* (- u) (expm1 (/ -2.0 v)))) 1.0)
   (+ 1.0 (* v (fma u (expm1 (/ 2.0 v)) (/ -2.0 v))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = fmaf(v, logf((-u * expm1f((-2.0f / v)))), 1.0f);
	} else {
		tmp = 1.0f + (v * fmaf(u, expm1f((2.0f / v)), (-2.0f / v)));
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.20000000298023224))
		tmp = fma(v, log(Float32(Float32(-u) * expm1(Float32(Float32(-2.0) / v)))), Float32(1.0));
	else
		tmp = Float32(Float32(1.0) + Float32(v * fma(u, expm1(Float32(Float32(2.0) / v)), Float32(Float32(-2.0) / v))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   1. Initial program 100.0%

      \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
   2. Add Preprocessing

   4. Applied rewrites 100.0%

      \[\leadsto 1 + v \cdot \log \color{blue}{\left(\mathsf{fma}\left(e^{\frac{-2}{v}} \cdot \left(1 - u \cdot u\right), \frac{1}{1 + u}, u\right)\right)} \]

   5. Taylor expanded in v around 0

      \[\leadsto \color{blue}{1 + v \cdot \log \left(u + \frac{e^{\frac{-2}{v}} \cdot \left(1 - {u}^{2}\right)}{1 + u}\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{v \cdot \log \left(u + \frac{e^{\frac{-2}{v}} \cdot \left(1 - {u}^{2}\right)}{1 + u}\right) + 1} \]

      2. lower-fma.f32 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(v, \log \left(u + \frac{e^{\frac{-2}{v}} \cdot \left(1 - {u}^{2}\right)}{1 + u}\right), 1\right)} \]

   7. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(v, \log \left(\mathsf{fma}\left(\mathsf{fma}\left(u, -u, 1\right), \frac{e^{\frac{-2}{v}}}{u + 1}, u\right)\right), 1\right)} \]

   8. Taylor expanded in u around -inf

      \[\leadsto \mathsf{fma}\left(v, \log \left(-1 \cdot \left(u \cdot \left(e^{\frac{-2}{v}} - 1\right)\right)\right), 1\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 99.7%

       \[\leadsto \mathsf{fma}\left(v, \log \left(\mathsf{expm1}\left(\frac{-2}{v}\right) \cdot \left(-u\right)\right), 1\right) \]

   if 0.200000003 < v

   1. Initial program 93.0%

      \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in u around 0

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

      1. sub-neg N/A

         \[\leadsto 1 + v \cdot \color{blue}{\left(u \cdot \left(\frac{1}{e^{\frac{-2}{v}}} - 1\right) + \left(\mathsf{neg}\left(2 \cdot \frac{1}{v}\right)\right)\right)} \]

      2. lower-fma.f32 N/A

         \[\leadsto 1 + v \cdot \color{blue}{\mathsf{fma}\left(u, \frac{1}{e^{\frac{-2}{v}}} - 1, \mathsf{neg}\left(2 \cdot \frac{1}{v}\right)\right)} \]

      3. rec-exp N/A

         \[\leadsto 1 + v \cdot \mathsf{fma}\left(u, \color{blue}{e^{\mathsf{neg}\left(\frac{-2}{v}\right)}} - 1, \mathsf{neg}\left(2 \cdot \frac{1}{v}\right)\right) \]

      4. distribute-neg-frac N/A

         \[\leadsto 1 + v \cdot \mathsf{fma}\left(u, e^{\color{blue}{\frac{\mathsf{neg}\left(-2\right)}{v}}} - 1, \mathsf{neg}\left(2 \cdot \frac{1}{v}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto 1 + v \cdot \mathsf{fma}\left(u, e^{\frac{\color{blue}{2}}{v}} - 1, \mathsf{neg}\left(2 \cdot \frac{1}{v}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto 1 + v \cdot \mathsf{fma}\left(u, e^{\frac{\color{blue}{2 \cdot 1}}{v}} - 1, \mathsf{neg}\left(2 \cdot \frac{1}{v}\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto 1 + v \cdot \mathsf{fma}\left(u, e^{\color{blue}{2 \cdot \frac{1}{v}}} - 1, \mathsf{neg}\left(2 \cdot \frac{1}{v}\right)\right) \]

      8. lower-expm1.f32 N/A

         \[\leadsto 1 + v \cdot \mathsf{fma}\left(u, \color{blue}{\mathsf{expm1}\left(2 \cdot \frac{1}{v}\right)}, \mathsf{neg}\left(2 \cdot \frac{1}{v}\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto 1 + v \cdot \mathsf{fma}\left(u, \mathsf{expm1}\left(\color{blue}{\frac{2 \cdot 1}{v}}\right), \mathsf{neg}\left(2 \cdot \frac{1}{v}\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto 1 + v \cdot \mathsf{fma}\left(u, \mathsf{expm1}\left(\frac{\color{blue}{2}}{v}\right), \mathsf{neg}\left(2 \cdot \frac{1}{v}\right)\right) \]

      11. lower-/.f32 N/A

          \[\leadsto 1 + v \cdot \mathsf{fma}\left(u, \mathsf{expm1}\left(\color{blue}{\frac{2}{v}}\right), \mathsf{neg}\left(2 \cdot \frac{1}{v}\right)\right) \]

      12. associate-*r/ N/A

          \[\leadsto 1 + v \cdot \mathsf{fma}\left(u, \mathsf{expm1}\left(\frac{2}{v}\right), \mathsf{neg}\left(\color{blue}{\frac{2 \cdot 1}{v}}\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto 1 + v \cdot \mathsf{fma}\left(u, \mathsf{expm1}\left(\frac{2}{v}\right), \mathsf{neg}\left(\frac{\color{blue}{2}}{v}\right)\right) \]

      14. distribute-neg-frac N/A

          \[\leadsto 1 + v \cdot \mathsf{fma}\left(u, \mathsf{expm1}\left(\frac{2}{v}\right), \color{blue}{\frac{\mathsf{neg}\left(2\right)}{v}}\right) \]

      15. metadata-eval N/A

          \[\leadsto 1 + v \cdot \mathsf{fma}\left(u, \mathsf{expm1}\left(\frac{2}{v}\right), \frac{\color{blue}{-2}}{v}\right) \]

      16. lower-/.f32 65.9

          \[\leadsto 1 + v \cdot \mathsf{fma}\left(u, \mathsf{expm1}\left(\frac{2}{v}\right), \color{blue}{\frac{-2}{v}}\right) \]

   5. Applied rewrites 65.9%

      \[\leadsto 1 + v \cdot \color{blue}{\mathsf{fma}\left(u, \mathsf{expm1}\left(\frac{2}{v}\right), \frac{-2}{v}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{fma}\left(v, \log \left(\left(-u\right) \cdot \mathsf{expm1}\left(\frac{-2}{v}\right)\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 + v \cdot \mathsf{fma}\left(u, \mathsf{expm1}\left(\frac{2}{v}\right), \frac{-2}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024227 
(FPCore (u v)
  :name "HairBSDF, sample_f, cosTheta"
  :precision binary32
  :pre (and (and (<= 1e-5 u) (<= u 1.0)) (and (<= 0.0 v) (<= v 109.746574)))
  (+ 1.0 (* v (log (+ u (* (- 1.0 u) (exp (/ -2.0 v))))))))