HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.5%

Time: 12.3s

Alternatives: 5

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, 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

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
3. Add Preprocessing

Alternative 2: 95.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float u, float v) {
	return 1.0f + (v * logf((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 + exp(((-2.0e0) / v)))))
end function

Julia

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

MATLAB

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

3. Taylor expanded in u around 0 96.7%

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

Alternative 3: 94.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ 1 + v \cdot \log u \end{array} \]

FPCore

(FPCore (u v) :precision binary32 (+ 1.0 (* v (log u))))

C

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

Fortran

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

Julia

function code(u, v)
	return Float32(Float32(1.0) + Float32(v * log(u)))
end

MATLAB

function tmp = code(u, v)
	tmp = single(1.0) + (v * log(u));
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

3. Taylor expanded in u around 0 96.7%

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

4. Taylor expanded in u around inf 95.2%

   \[\leadsto 1 + v \cdot \log \color{blue}{u} \]
5. Add Preprocessing

Alternative 4: 86.7% accurate, 213.0× speedup

Math

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

FPCore

(FPCore (u v) :precision binary32 1.0)

C

float code(float u, float v) {
	return 1.0f;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    code = 1.0e0
end function

Julia

function code(u, v)
	return Float32(1.0)
end

MATLAB

function tmp = code(u, v)
	tmp = single(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
3. Step-by-step derivation

   1. *-commutative 99.6%

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

   2. sub-neg 99.6%

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

   3. distribute-rgt-in 99.6%

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

   4. *-un-lft-identity 99.6%

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

4. Applied egg-rr 99.6%

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

   1. associate-+r+ 99.6%

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

   2. distribute-lft-neg-out 99.6%

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

   3. unsub-neg 99.6%

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

   4. add-sqr-sqrt 99.6%

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

   5. sqrt-unprod 99.6%

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

   6. sqr-neg 99.6%

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

   7. sqrt-unprod -0.0%

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

   8. add-sqr-sqrt 96.2%

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

   9. *-commutative 96.2%

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

   10. add-sqr-sqrt -0.0%

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

   11. sqrt-unprod 99.6%

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

   12. sqr-neg 99.6%

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

   13. sqrt-unprod 99.6%

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

   14. add-sqr-sqrt 99.6%

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

6. Applied egg-rr 99.6%

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

   1. add-log-exp 97.3%

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

   2. exp-sum 97.1%

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

   3. exp-1-e 97.1%

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

   4. *-commutative 97.1%

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

   5. exp-to-pow 97.1%

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

   6. *-commutative 97.1%

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

8. Applied egg-rr 97.1%

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

   1. *-commutative 97.1%

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

   2. associate-+r- 97.1%

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

10. Simplified 97.1%

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

11. Taylor expanded in v around 0 84.5%

    \[\leadsto \color{blue}{\log e} \]
12. Step-by-step derivation

    1. log-E 86.7%

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

13. Simplified 86.7%

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

Alternative 5: 5.8% accurate, 213.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

(FPCore (u v) :precision binary32 -1.0)

C

float code(float u, float v) {
	return -1.0f;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    code = -1.0e0
end function

Julia

function code(u, v)
	return Float32(-1.0)
end

MATLAB

function tmp = code(u, v)
	tmp = single(-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. Step-by-step derivation

   1. +-commutative 99.6%

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

   2. fma-define 99.6%

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

   3. +-commutative 99.6%

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

   4. fma-define 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in u around 0 5.4%

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

Reproduce

herbie shell --seed 2024139 
(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))))))))