HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.5%

Time: 14.2s

Alternatives: 7

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, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.4%

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

   1. +-commutative 99.4%

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

   2. fma-def 99.5%

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

   3. +-commutative 99.5%

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

   4. fma-def 99.6%

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

3. Simplified 99.6%

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

4. Final simplification 99.6%

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

Alternative 2: 99.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.4%

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

2. Taylor expanded in v around 0 99.4%

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

   1. +-commutative 99.4%

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

   2. *-commutative 99.4%

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

   3. fma-def 99.6%

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

4. Simplified 99.6%

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

5. Final simplification 99.6%

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

Alternative 3: 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 99.4%

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

2. Final simplification 99.4%

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

Alternative 4: 94.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(v, \log u, 1\right) \end{array} \]

FPCore

(FPCore (u v) :precision binary32 (fma v (log u) 1.0))

C

float code(float u, float v) {
	return fmaf(v, logf(u), 1.0f);
}

Julia

function code(u, v)
	return fma(v, log(u), Float32(1.0))
end

Derivation

1. Initial program 99.4%

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

   1. +-commutative 99.4%

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

   2. fma-def 99.5%

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

   3. +-commutative 99.5%

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

   4. fma-def 99.6%

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

3. Simplified 99.6%

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

   1. fma-udef 99.5%

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

5. Applied egg-rr 99.5%

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

   1. add-exp-log 99.5%

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

   2. *-commutative 99.5%

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

   3. log-prod 99.5%

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

   4. add-log-exp 99.5%

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

   5. sub-neg 99.5%

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

   6. log1p-def 99.5%

      \[\leadsto \mathsf{fma}\left(v, \log \left(e^{\frac{-2}{v} + \color{blue}{\mathsf{log1p}\left(-u\right)}} + u\right), 1\right) \]

7. Applied egg-rr 99.5%

   \[\leadsto \mathsf{fma}\left(v, \log \left(\color{blue}{e^{\frac{-2}{v} + \mathsf{log1p}\left(-u\right)}} + u\right), 1\right) \]

8. Taylor expanded in u around inf 95.1%

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

   1. mul-1-neg 95.1%

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

   2. log-rec 95.1%

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

   3. remove-double-neg 95.1%

      \[\leadsto \mathsf{fma}\left(v, \color{blue}{\log u}, 1\right) \]

10. Simplified 95.1%

    \[\leadsto \mathsf{fma}\left(v, \color{blue}{\log u}, 1\right) \]

11. Final simplification 95.1%

    \[\leadsto \mathsf{fma}\left(v, \log u, 1\right) \]

Alternative 5: 94.6% 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 99.4%

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

   1. +-commutative 99.4%

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

   2. fma-def 99.5%

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

   3. +-commutative 99.5%

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

   4. fma-def 99.6%

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

3. Simplified 99.6%

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

   1. fma-udef 99.5%

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

5. Applied egg-rr 99.5%

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

   1. add-exp-log 99.5%

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

   2. *-commutative 99.5%

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

   3. log-prod 99.5%

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

   4. add-log-exp 99.5%

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

   5. sub-neg 99.5%

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

   6. log1p-def 99.5%

      \[\leadsto \mathsf{fma}\left(v, \log \left(e^{\frac{-2}{v} + \color{blue}{\mathsf{log1p}\left(-u\right)}} + u\right), 1\right) \]

7. Applied egg-rr 99.5%

   \[\leadsto \mathsf{fma}\left(v, \log \left(\color{blue}{e^{\frac{-2}{v} + \mathsf{log1p}\left(-u\right)}} + u\right), 1\right) \]

8. Taylor expanded in u around inf 95.1%

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

   1. mul-1-neg 95.1%

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

   2. distribute-rgt-neg-in 95.1%

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

   3. log-rec 95.1%

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

   4. remove-double-neg 95.1%

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

10. Simplified 95.1%

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

11. Final simplification 95.1%

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

Alternative 6: 5.9% accurate, 213.0× 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 99.4%

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

2. Taylor expanded in u around 0 5.0%

   \[\leadsto \color{blue}{-1} \]

3. Final simplification 5.0%

   \[\leadsto -1 \]

Alternative 7: 86.7% accurate, 213.0× 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 99.4%

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

2. Taylor expanded in v around 0 99.4%

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

   1. +-commutative 99.4%

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

   2. *-commutative 99.4%

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

   3. fma-def 99.6%

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

4. Simplified 99.6%

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

5. Taylor expanded in v around 0 88.0%

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

6. Final simplification 88.0%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023331 
(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))))))))