HairBSDF, sample_f, cosTheta

Percentage Accurate: 100.0% → 100.0%

Time: 14.9s

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: 100.0% 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: 100.0% 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 100.0%

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

2. Final simplification 100.0%

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

Alternative 2: 99.8% 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 100.0%

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

2. Taylor expanded in u around 0 99.9%

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

3. Final simplification 99.9%

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

Alternative 3: 99.7% 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 100.0%

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

2. Taylor expanded in u around 0 99.9%

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

3. Taylor expanded in v around inf 98.6%

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

4. Taylor expanded in u around inf 99.7%

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

   1. *-commutative 99.7%

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

   2. associate-*r* 99.7%

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

   3. mul-1-neg 99.7%

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

   4. log-rec 99.7%

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

   5. remove-double-neg 99.7%

      \[\leadsto 1 + \color{blue}{\log u} \cdot v \]

6. Simplified 99.7%

   \[\leadsto 1 + \color{blue}{\log u \cdot v} \]

7. Final simplification 99.7%

   \[\leadsto 1 + v \cdot \log u \]

Alternative 4: 99.3% accurate, 23.7× speedup

Math

\[\begin{array}{l} \\ 1 + \left(1 + \left(v \cdot u + -1\right)\right) \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in u around 0 99.9%

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

3. Taylor expanded in v around inf 98.6%

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

4. Taylor expanded in u around 0 98.6%

   \[\leadsto 1 + v \cdot \color{blue}{u} \]
5. Step-by-step derivation

   1. expm1-log1p-u 98.6%

      \[\leadsto 1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(v \cdot u\right)\right)} \]

   2. expm1-udef 98.6%

      \[\leadsto 1 + \color{blue}{\left(e^{\mathsf{log1p}\left(v \cdot u\right)} - 1\right)} \]

   3. log1p-udef 98.6%

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

   4. add-exp-log 98.6%

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

6. Applied egg-rr 98.6%

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

   1. associate--l+ 98.6%

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

8. Simplified 98.6%

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

9. Final simplification 98.6%

   \[\leadsto 1 + \left(1 + \left(v \cdot u + -1\right)\right) \]

Alternative 5: 99.3% accurate, 42.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in u around 0 99.9%

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

3. Taylor expanded in v around inf 98.6%

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

4. Taylor expanded in u around 0 98.6%

   \[\leadsto 1 + v \cdot \color{blue}{u} \]

5. Final simplification 98.6%

   \[\leadsto 1 + v \cdot u \]

Reproduce

herbie shell --seed 2023278 
(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))))))))