HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.4% → 99.3%

Time: 8.4s

Alternatives: 4

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: 99.4% 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: 99.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{-2}{v}} \cdot \left(1 - u\right)\\ \left(\log \left({u}^{3} + {t\_0}^{3}\right) - \log \left(\left(u - t\_0\right) \cdot u + {t\_0}^{2}\right)\right) \cdot v + 1 \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (let* ((t_0 (* (exp (/ -2.0 v)) (- 1.0 u))))
   (+
    (*
     (-
      (log (+ (pow u 3.0) (pow t_0 3.0)))
      (log (+ (* (- u t_0) u) (pow t_0 2.0))))
     v)
    1.0)))

C

float code(float u, float v) {
	float t_0 = expf((-2.0f / v)) * (1.0f - u);
	return ((logf((powf(u, 3.0f) + powf(t_0, 3.0f))) - logf((((u - t_0) * u) + powf(t_0, 2.0f)))) * v) + 1.0f;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: t_0
    t_0 = exp(((-2.0e0) / v)) * (1.0e0 - u)
    code = ((log(((u ** 3.0e0) + (t_0 ** 3.0e0))) - log((((u - t_0) * u) + (t_0 ** 2.0e0)))) * v) + 1.0e0
end function

Julia

function code(u, v)
	t_0 = Float32(exp(Float32(Float32(-2.0) / v)) * Float32(Float32(1.0) - u))
	return Float32(Float32(Float32(log(Float32((u ^ Float32(3.0)) + (t_0 ^ Float32(3.0)))) - log(Float32(Float32(Float32(u - t_0) * u) + (t_0 ^ Float32(2.0))))) * v) + Float32(1.0))
end

MATLAB

function tmp = code(u, v)
	t_0 = exp((single(-2.0) / v)) * (single(1.0) - u);
	tmp = ((log(((u ^ single(3.0)) + (t_0 ^ single(3.0)))) - log((((u - t_0) * u) + (t_0 ^ single(2.0))))) * v) + 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. lift-+.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lift--.f32 N/A

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

   5. sub-neg N/A

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

   6. distribute-rgt-in N/A

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

   7. *-lft-identity N/A

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

   8. associate-+r+ N/A

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

   9. lower-+.f32 N/A

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

   10. lower-+.f32 N/A

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

   11. lower-*.f32 N/A

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

   12. lower-neg.f32 99.6

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

4. Applied rewrites 99.6%

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

   1. lift-log.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. lift-+.f32 N/A

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

   4. associate-+l+ N/A

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

   5. +-commutative N/A

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

   6. lift-*.f32 N/A

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

   7. distribute-rgt1-in N/A

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

   8. +-commutative N/A

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

   9. lift-neg.f32 N/A

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

   10. sub-neg N/A

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

   11. lift-exp.f32 N/A

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

   12. lift-/.f32 N/A

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

   13. flip3-+ N/A

       \[\leadsto 1 + v \cdot \log \color{blue}{\left(\frac{{\left(\left(1 - u\right) \cdot e^{\frac{-2}{v}}\right)}^{3} + {u}^{3}}{\left(\left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \cdot \left(\left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) + \left(u \cdot u - \left(\left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \cdot u\right)}\right)} \]

6. Applied rewrites 99.7%

   \[\leadsto 1 + v \cdot \color{blue}{\left(\log \left({\left(\left(1 - u\right) \cdot e^{\frac{-2}{v}}\right)}^{3} + {u}^{3}\right) - \log \left({\left(\left(1 - u\right) \cdot e^{\frac{-2}{v}}\right)}^{2} + \left(u \cdot u - \left(u \cdot \left(1 - u\right)\right) \cdot e^{\frac{-2}{v}}\right)\right)\right)} \]
7. Step-by-step derivation

   1. lift--.f32 N/A

      \[\leadsto 1 + v \cdot \left(\log \left({\left(\left(1 - u\right) \cdot e^{\frac{-2}{v}}\right)}^{3} + {u}^{3}\right) - \log \left({\left(\left(1 - u\right) \cdot e^{\frac{-2}{v}}\right)}^{2} + \color{blue}{\left(u \cdot u - \left(u \cdot \left(1 - u\right)\right) \cdot e^{\frac{-2}{v}}\right)}\right)\right) \]

   2. lift-*.f32 N/A

      \[\leadsto 1 + v \cdot \left(\log \left({\left(\left(1 - u\right) \cdot e^{\frac{-2}{v}}\right)}^{3} + {u}^{3}\right) - \log \left({\left(\left(1 - u\right) \cdot e^{\frac{-2}{v}}\right)}^{2} + \left(\color{blue}{u \cdot u} - \left(u \cdot \left(1 - u\right)\right) \cdot e^{\frac{-2}{v}}\right)\right)\right) \]

   3. lift-*.f32 N/A

      \[\leadsto 1 + v \cdot \left(\log \left({\left(\left(1 - u\right) \cdot e^{\frac{-2}{v}}\right)}^{3} + {u}^{3}\right) - \log \left({\left(\left(1 - u\right) \cdot e^{\frac{-2}{v}}\right)}^{2} + \left(u \cdot u - \color{blue}{\left(u \cdot \left(1 - u\right)\right) \cdot e^{\frac{-2}{v}}}\right)\right)\right) \]

   4. lift-*.f32 N/A

      \[\leadsto 1 + v \cdot \left(\log \left({\left(\left(1 - u\right) \cdot e^{\frac{-2}{v}}\right)}^{3} + {u}^{3}\right) - \log \left({\left(\left(1 - u\right) \cdot e^{\frac{-2}{v}}\right)}^{2} + \left(u \cdot u - \color{blue}{\left(u \cdot \left(1 - u\right)\right)} \cdot e^{\frac{-2}{v}}\right)\right)\right) \]

   5. associate-*l* N/A

      \[\leadsto 1 + v \cdot \left(\log \left({\left(\left(1 - u\right) \cdot e^{\frac{-2}{v}}\right)}^{3} + {u}^{3}\right) - \log \left({\left(\left(1 - u\right) \cdot e^{\frac{-2}{v}}\right)}^{2} + \left(u \cdot u - \color{blue}{u \cdot \left(\left(1 - u\right) \cdot e^{\frac{-2}{v}}\right)}\right)\right)\right) \]

   6. lift-*.f32 N/A

      \[\leadsto 1 + v \cdot \left(\log \left({\left(\left(1 - u\right) \cdot e^{\frac{-2}{v}}\right)}^{3} + {u}^{3}\right) - \log \left({\left(\left(1 - u\right) \cdot e^{\frac{-2}{v}}\right)}^{2} + \left(u \cdot u - u \cdot \color{blue}{\left(\left(1 - u\right) \cdot e^{\frac{-2}{v}}\right)}\right)\right)\right) \]

   7. distribute-lft-out-- N/A

      \[\leadsto 1 + v \cdot \left(\log \left({\left(\left(1 - u\right) \cdot e^{\frac{-2}{v}}\right)}^{3} + {u}^{3}\right) - \log \left({\left(\left(1 - u\right) \cdot e^{\frac{-2}{v}}\right)}^{2} + \color{blue}{u \cdot \left(u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right)}\right)\right) \]

   8. lower-*.f32 N/A

      \[\leadsto 1 + v \cdot \left(\log \left({\left(\left(1 - u\right) \cdot e^{\frac{-2}{v}}\right)}^{3} + {u}^{3}\right) - \log \left({\left(\left(1 - u\right) \cdot e^{\frac{-2}{v}}\right)}^{2} + \color{blue}{u \cdot \left(u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right)}\right)\right) \]

   9. lower--.f32 99.7

      \[\leadsto 1 + v \cdot \left(\log \left({\left(\left(1 - u\right) \cdot e^{\frac{-2}{v}}\right)}^{3} + {u}^{3}\right) - \log \left({\left(\left(1 - u\right) \cdot e^{\frac{-2}{v}}\right)}^{2} + u \cdot \color{blue}{\left(u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right)}\right)\right) \]

   10. lift-*.f32 N/A

       \[\leadsto 1 + v \cdot \left(\log \left({\left(\left(1 - u\right) \cdot e^{\frac{-2}{v}}\right)}^{3} + {u}^{3}\right) - \log \left({\left(\left(1 - u\right) \cdot e^{\frac{-2}{v}}\right)}^{2} + u \cdot \left(u - \color{blue}{\left(1 - u\right) \cdot e^{\frac{-2}{v}}}\right)\right)\right) \]

   11. *-commutative N/A

       \[\leadsto 1 + v \cdot \left(\log \left({\left(\left(1 - u\right) \cdot e^{\frac{-2}{v}}\right)}^{3} + {u}^{3}\right) - \log \left({\left(\left(1 - u\right) \cdot e^{\frac{-2}{v}}\right)}^{2} + u \cdot \left(u - \color{blue}{e^{\frac{-2}{v}} \cdot \left(1 - u\right)}\right)\right)\right) \]

   12. lower-*.f32 99.7

       \[\leadsto 1 + v \cdot \left(\log \left({\left(\left(1 - u\right) \cdot e^{\frac{-2}{v}}\right)}^{3} + {u}^{3}\right) - \log \left({\left(\left(1 - u\right) \cdot e^{\frac{-2}{v}}\right)}^{2} + u \cdot \left(u - \color{blue}{e^{\frac{-2}{v}} \cdot \left(1 - u\right)}\right)\right)\right) \]

8. Applied rewrites 99.7%

   \[\leadsto 1 + v \cdot \left(\log \left({\left(\left(1 - u\right) \cdot e^{\frac{-2}{v}}\right)}^{3} + {u}^{3}\right) - \log \left({\left(\left(1 - u\right) \cdot e^{\frac{-2}{v}}\right)}^{2} + \color{blue}{u \cdot \left(u - e^{\frac{-2}{v}} \cdot \left(1 - u\right)\right)}\right)\right) \]

9. Final simplification 99.7%

   \[\leadsto \left(\log \left({u}^{3} + {\left(e^{\frac{-2}{v}} \cdot \left(1 - u\right)\right)}^{3}\right) - \log \left(\left(u - e^{\frac{-2}{v}} \cdot \left(1 - u\right)\right) \cdot u + {\left(e^{\frac{-2}{v}} \cdot \left(1 - u\right)\right)}^{2}\right)\right) \cdot v + 1 \]
10. Add Preprocessing

Alternative 2: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(u, v)
	tmp = (log((u - ((u - single(1.0)) * exp((single(-2.0) / v))))) * v) + 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. Final simplification 99.6%

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

Alternative 3: 86.3% accurate, 231.0× 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. Taylor expanded in v around 0

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

5. Applied rewrites 90.1%

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

Alternative 4: 6.1% accurate, 231.0× 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. Taylor expanded in u around 0

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

5. Applied rewrites 5.3%

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

Reproduce

herbie shell --seed 2024268 
(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))))))))