HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.4% → 99.4%

Time: 9.0s

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: 99.4% 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: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \log \left(e^{-2 \cdot {\left({v}^{-0.5}\right)}^{2}} \cdot \left(1 - u\right) + u\right) \cdot v + 1 \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (+ (* (log (+ (* (exp (* -2.0 (pow (pow v -0.5) 2.0))) (- 1.0 u)) u)) v) 1.0))

C

float code(float u, float v) {
	return (logf(((expf((-2.0f * powf(powf(v, -0.5f), 2.0f))) * (1.0f - u)) + u)) * v) + 1.0f;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    code = (log(((exp(((-2.0e0) * ((v ** (-0.5e0)) ** 2.0e0))) * (1.0e0 - u)) + u)) * v) + 1.0e0
end function

Julia

function code(u, v)
	return Float32(Float32(log(Float32(Float32(exp(Float32(Float32(-2.0) * ((v ^ Float32(-0.5)) ^ Float32(2.0)))) * Float32(Float32(1.0) - u)) + u)) * v) + Float32(1.0))
end

MATLAB

function tmp = code(u, v)
	tmp = (log(((exp((single(-2.0) * ((v ^ single(-0.5)) ^ single(2.0)))) * (single(1.0) - u)) + u)) * 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. Step-by-step derivation

   1. lift-exp.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. clear-num N/A

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

   4. associate-/r/ N/A

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

   5. exp-prod N/A

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

   6. pow-to-exp N/A

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

   7. lower-exp.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. lower-log.f32 N/A

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

   10. lower-exp.f32 N/A

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

   11. lower-/.f32 99.6

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

4. Applied rewrites 99.6%

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

   1. lift-log.f32 N/A

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

   2. lift-exp.f32 N/A

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

   3. rem-log-exp 99.7

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

   4. lift-/.f32 N/A

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

   5. inv-pow N/A

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

   6. sqr-pow N/A

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

   7. pow2 N/A

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

   8. lower-pow.f32 N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. lower-pow.f32 N/A

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

   12. metadata-eval 99.7

       \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{{\left({v}^{\color{blue}{-0.5}}\right)}^{2} \cdot -2}\right) \]

6. Applied rewrites 99.7%

   \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\color{blue}{{\left({v}^{-0.5}\right)}^{2}} \cdot -2}\right) \]

7. Final simplification 99.7%

   \[\leadsto \log \left(e^{-2 \cdot {\left({v}^{-0.5}\right)}^{2}} \cdot \left(1 - u\right) + u\right) \cdot v + 1 \]
8. Add Preprocessing

Alternative 2: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(u, v)
	tmp = (log(((exp((single(-2.0) / v)) * (single(1.0) - u)) + u)) * 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. Final simplification 99.7%

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

Alternative 3: 44.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

4. Applied rewrites 95.9%

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

Alternative 4: 86.5% 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 90.1%

   \[\leadsto \color{blue}{1} \]
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 99.7%

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

3. Taylor expanded in u around 0

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

5. Applied rewrites 4.5%

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

Reproduce

herbie shell --seed 2024285 
(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))))))))