HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.5%

Time: 11.8s

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: 42.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, v)
	return Float32(Float32(1.0) + Float32(log(fma(exp(Float32(Float32(-2.0) / v)), Float32(Float32(1.0) - u), u)) * 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. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 99.6

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

   4. lift-+.f32 N/A

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

   5. +-commutative N/A

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f32 93.7

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

4. Applied rewrites 93.9%

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

Alternative 3: 94.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, v)
	tmp = single(1.0) + (v * log((u * (single(1.0) - 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 v around inf

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

   1. associate-+r+ N/A

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

   2. fp-cancel-sign-sub-inv N/A

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

   3. metadata-eval N/A

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

   4. associate-*r/ N/A

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

   5. associate-+l- N/A

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

   6. fp-cancel-sub-sign-inv N/A

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

   7. metadata-eval N/A

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

   8. unpow2 N/A

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

   9. associate-/r* N/A

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

   10. associate-/l* N/A

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

   11. div-add N/A

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

   12. +-commutative N/A

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

   13. lower--.f32 N/A

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

5. Applied rewrites 43.3%

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

6. Taylor expanded in u around inf

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

   1. lower-*.f32 N/A

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

   2. lower-+.f32 N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f32 N/A

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

   5. lower-exp.f32 N/A

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

   6. lower-/.f32 95.3

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

8. Applied rewrites 95.3%

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

9. Final simplification 95.3%

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

Alternative 4: 86.6% 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 89.3%

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

Alternative 5: 6.0% 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 4.8%

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

Reproduce

herbie shell --seed 2024337 
(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))))))))