HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.5%

Time: 7.2s

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

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

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 99.6%

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

3. Final simplification 99.6%

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

Alternative 2: 94.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(u, v)
	tmp = (log(((single(1.0) - exp((single(-2.0) / v))) * u)) * v) + 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 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. +-commutative N/A

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

5. Applied rewrites 88.7%

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

7. Applied rewrites 88.3%

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

8. Taylor expanded in u around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. mul-1-neg N/A

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

   4. unsub-neg N/A

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

   5. lower--.f32 N/A

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

   6. lower-exp.f32 N/A

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

   7. lower-/.f32 95.0

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

10. Applied rewrites 95.0%

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

11. Final simplification 95.0%

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

Alternative 3: 87.9% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(u, v)
	tmp = (log((single(2.0) * (u / v))) * v) + 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 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. +-commutative N/A

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

5. Applied rewrites 88.3%

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

7. Applied rewrites 88.7%

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

8. Taylor expanded in v around inf

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

10. Applied rewrites 88.7%

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

11. Taylor expanded in u around inf

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

13. Applied rewrites 89.1%

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

14. Final simplification 89.1%

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

Alternative 4: 87.3% 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 88.7%

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

Alternative 5: 5.7% 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 5.6%

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

Reproduce

herbie shell --seed 2024332 
(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))))))))