HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.5%

Time: 11.8s

Alternatives: 5

Speedup: 1.0×

• unsoundness detected in the egraph

Specification

\[\left(10^{-5} \leq u \land u \leq 1\right) \land \left(0 \leq v \land v \leq 109.746574\right)\]

Math

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

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)
use fmin_fmax_functions
    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

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

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)
use fmin_fmax_functions
    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: 96.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.5%

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

2. Taylor expanded in u around 0

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

4. Applied rewrites 96.3%

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

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. add-flip N/A

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

   4. metadata-eval N/A

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

   5. lower--.f32 96.3%

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

6. Applied rewrites 96.3%

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

Alternative 2: 96.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.5%

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

2. Taylor expanded in u around 0

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

4. Applied rewrites 96.3%

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

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 96.3%

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

   6. lift-+.f32 N/A

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

   7. +-commutative N/A

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

   8. add-flip N/A

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

   9. sub-flip N/A

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

   10. lift-*.f32 N/A

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

   11. remove-double-neg N/A

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

   12. lower-fma.f32 96.3%

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

6. Applied rewrites 96.3%

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

Alternative 3: 86.8% accurate, 9.9× speedup

Math

\[10 - 9 \]

FPCore

(FPCore (u v)
  :precision binary32
  (- 10.0 9.0))

C

float code(float u, float v) {
	return 10.0f - 9.0f;
}

Fortran

real(4) function code(u, v)
use fmin_fmax_functions
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    code = 10.0e0 - 9.0e0
end function

Julia

function code(u, v)
	return Float32(Float32(10.0) - Float32(9.0))
end

MATLAB

function tmp = code(u, v)
	tmp = single(10.0) - single(9.0);
end

Derivation

1. Initial program 99.5%

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

2. Taylor expanded in v around inf

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

   1. lower-*.f32 N/A

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

   2. lower--.f32 7.8%

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

4. Applied rewrites 7.8%

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

   1. +-rgt-identity N/A

      \[\leadsto \color{blue}{\left(1 + -2 \cdot \left(1 - u\right)\right) + 0} \]

   2. lift-+.f32 N/A

      \[\leadsto \color{blue}{\left(1 + -2 \cdot \left(1 - u\right)\right)} + 0 \]

   3. +-commutative N/A

      \[\leadsto \color{blue}{\left(-2 \cdot \left(1 - u\right) + 1\right)} + 0 \]

   4. +-commutative N/A

      \[\leadsto \color{blue}{\left(1 + -2 \cdot \left(1 - u\right)\right)} + 0 \]

   5. lift-+.f32 N/A

      \[\leadsto \color{blue}{\left(1 + -2 \cdot \left(1 - u\right)\right)} + 0 \]

   6. +-inverses N/A

      \[\leadsto \left(1 + -2 \cdot \left(1 - u\right)\right) + \color{blue}{\left(\left(\left(3 + 3\right) + 3\right) - \left(\left(3 + 3\right) + 3\right)\right)} \]

   7. associate-+r- N/A

      \[\leadsto \color{blue}{\left(\left(1 + -2 \cdot \left(1 - u\right)\right) + \left(\left(3 + 3\right) + 3\right)\right) - \left(\left(3 + 3\right) + 3\right)} \]

   8. lower--.f32 N/A

      \[\leadsto \color{blue}{\left(\left(1 + -2 \cdot \left(1 - u\right)\right) + \left(\left(3 + 3\right) + 3\right)\right) - \left(\left(3 + 3\right) + 3\right)} \]

6. Applied rewrites 7.8%

   \[\leadsto \color{blue}{\left(\left(-2 \cdot \left(1 - u\right) - -1\right) + 9\right) - 9} \]

7. Taylor expanded in v around 0

   \[\leadsto \color{blue}{10} - 9 \]
8. Step-by-step derivation

9. Applied rewrites 86.8%

   \[\leadsto \color{blue}{10} - 9 \]
10. Add Preprocessing

Alternative 4: 5.8% accurate, 35.6× speedup

Math

\[-1 \]

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)
use fmin_fmax_functions
    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 99.5%

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

2. Taylor expanded in u around 0

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

4. Applied rewrites 5.8%

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

Reproduce

herbie shell --seed 2025322 
(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))))))))