HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.5%

Time: 10.4s

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 99.7%

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

Alternative 2: 94.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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)) - 1.0e0))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in u around -inf

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

   1. mul-1-neg N/A

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

   2. distribute-lft-neg-in N/A

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

   3. lower-*.f32 N/A

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

   4. lower-neg.f32 N/A

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

   5. metadata-eval N/A

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

   6. associate-*r/ N/A

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

   7. metadata-eval N/A

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

   8. lower-expm1.f32 N/A

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

   9. metadata-eval N/A

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

   10. associate-*r/ N/A

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

   11. metadata-eval N/A

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

   12. lower-/.f32 48.3

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

5. Applied rewrites 47.6%

   \[\leadsto 1 + v \cdot \log \color{blue}{\left(\left(-u\right) \cdot \mathsf{expm1}\left(\frac{-2}{v}\right)\right)} \]
6. Step-by-step derivation

7. Applied rewrites 95.6%

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

Alternative 3: 43.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.7%

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

4. Applied rewrites 99.7%

   \[\leadsto 1 + v \cdot \log \color{blue}{\left(\frac{{u}^{6} - {\left(e^{\frac{-2}{v}} \cdot \left(1 - u\right)\right)}^{6}}{\left(\left({\left(\left(\left(1 - u\right) \cdot u\right) \cdot e^{\frac{-2}{v}}\right)}^{2} + {\left(e^{\frac{-2}{v}} \cdot \left(1 - u\right)\right)}^{4}\right) + {u}^{4}\right) \cdot \left(u - e^{\frac{-2}{v}} \cdot \left(1 - u\right)\right)}\right)} \]
5. Step-by-step derivation

   1. lift-/.f32 N/A

      \[\leadsto 1 + v \cdot \log \color{blue}{\left(\frac{{u}^{6} - {\left(e^{\frac{-2}{v}} \cdot \left(1 - u\right)\right)}^{6}}{\left(\left({\left(\left(\left(1 - u\right) \cdot u\right) \cdot e^{\frac{-2}{v}}\right)}^{2} + {\left(e^{\frac{-2}{v}} \cdot \left(1 - u\right)\right)}^{4}\right) + {u}^{4}\right) \cdot \left(u - e^{\frac{-2}{v}} \cdot \left(1 - u\right)\right)}\right)} \]

   2. lift-*.f32 N/A

      \[\leadsto 1 + v \cdot \log \left(\frac{{u}^{6} - {\left(e^{\frac{-2}{v}} \cdot \left(1 - u\right)\right)}^{6}}{\color{blue}{\left(\left({\left(\left(\left(1 - u\right) \cdot u\right) \cdot e^{\frac{-2}{v}}\right)}^{2} + {\left(e^{\frac{-2}{v}} \cdot \left(1 - u\right)\right)}^{4}\right) + {u}^{4}\right) \cdot \left(u - e^{\frac{-2}{v}} \cdot \left(1 - u\right)\right)}}\right) \]

   3. associate-/r* N/A

      \[\leadsto 1 + v \cdot \log \color{blue}{\left(\frac{\frac{{u}^{6} - {\left(e^{\frac{-2}{v}} \cdot \left(1 - u\right)\right)}^{6}}{\left({\left(\left(\left(1 - u\right) \cdot u\right) \cdot e^{\frac{-2}{v}}\right)}^{2} + {\left(e^{\frac{-2}{v}} \cdot \left(1 - u\right)\right)}^{4}\right) + {u}^{4}}}{u - e^{\frac{-2}{v}} \cdot \left(1 - u\right)}\right)} \]

6. Applied rewrites 93.9%

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

Alternative 4: 87.1% 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 99.7%

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

3. Taylor expanded in v around 0

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

5. Applied rewrites 87.7%

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

Alternative 5: 5.8% 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 99.7%

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

3. Taylor expanded in u around 0

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

5. Applied rewrites 5.4%

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

Reproduce

herbie shell --seed 2024339 
(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))))))))