HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.5%

Time: 9.4s

Alternatives: 9

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

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

FPCore

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

C

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

Julia

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

4. Applied rewrites 99.5%

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

   1. lift-fma.f32 N/A

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

   2. lower-+.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 99.5

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

6. Applied rewrites 99.5%

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

7. Final simplification 99.5%

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

Alternative 2: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, v)
	return fma(log(fma(Float32(Float32(1.0) - u), 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. Add Preprocessing

4. Applied rewrites 99.5%

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

Alternative 3: 95.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, v)
	return fma(log(Float32(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. Add Preprocessing

4. Applied rewrites 99.5%

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

   1. lift-fma.f32 N/A

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

   2. lower-+.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 99.5

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

6. Applied rewrites 99.5%

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

7. Taylor expanded in u around 0

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

   1. metadata-eval N/A

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

   2. distribute-neg-frac N/A

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

   3. metadata-eval N/A

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

   4. associate-*r/ N/A

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

   5. lower-exp.f32 N/A

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

   6. associate-*r/ N/A

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

   7. metadata-eval N/A

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

   8. distribute-neg-frac N/A

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

   9. metadata-eval N/A

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

   10. lower-/.f32 96.6

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

9. Applied rewrites 96.6%

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

Alternative 4: 91.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 - u, -2, 1\right) - \frac{\left(\mathsf{fma}\left(-4, 1 - u, 4\right) \cdot \left(1 - u\right)\right) \cdot -0.5 - \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-96, u, 192\right), u, -112\right), u, 16\right) \cdot u}{v}, 0.041666666666666664, \mathsf{fma}\left(\left(u - 1\right) \cdot \left(u - 1\right), \mathsf{fma}\left(16, 1 - u, -24\right), 8 \cdot \left(1 - u\right)\right) \cdot -0.16666666666666666\right)}{v}}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224)
   1.0
   (-
    (fma (- 1.0 u) -2.0 1.0)
    (/
     (-
      (* (* (fma -4.0 (- 1.0 u) 4.0) (- 1.0 u)) -0.5)
      (/
       (fma
        (/ (* (fma (fma (fma -96.0 u 192.0) u -112.0) u 16.0) u) v)
        0.041666666666666664
        (*
         (fma
          (* (- u 1.0) (- u 1.0))
          (fma 16.0 (- 1.0 u) -24.0)
          (* 8.0 (- 1.0 u)))
         -0.16666666666666666))
       v))
     v))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f;
	} else {
		tmp = fmaf((1.0f - u), -2.0f, 1.0f) - ((((fmaf(-4.0f, (1.0f - u), 4.0f) * (1.0f - u)) * -0.5f) - (fmaf(((fmaf(fmaf(fmaf(-96.0f, u, 192.0f), u, -112.0f), u, 16.0f) * u) / v), 0.041666666666666664f, (fmaf(((u - 1.0f) * (u - 1.0f)), fmaf(16.0f, (1.0f - u), -24.0f), (8.0f * (1.0f - u))) * -0.16666666666666666f)) / v)) / v);
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.20000000298023224))
		tmp = Float32(1.0);
	else
		tmp = Float32(fma(Float32(Float32(1.0) - u), Float32(-2.0), Float32(1.0)) - Float32(Float32(Float32(Float32(fma(Float32(-4.0), Float32(Float32(1.0) - u), Float32(4.0)) * Float32(Float32(1.0) - u)) * Float32(-0.5)) - Float32(fma(Float32(Float32(fma(fma(fma(Float32(-96.0), u, Float32(192.0)), u, Float32(-112.0)), u, Float32(16.0)) * u) / v), Float32(0.041666666666666664), Float32(fma(Float32(Float32(u - Float32(1.0)) * Float32(u - Float32(1.0))), fma(Float32(16.0), Float32(Float32(1.0) - u), Float32(-24.0)), Float32(Float32(8.0) * Float32(Float32(1.0) - u))) * Float32(-0.16666666666666666))) / v)) / v));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   1. Initial program 99.9%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 90.4%

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

   if 0.200000003 < v

   1. Initial program 92.3%

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

   3. Taylor expanded in v around -inf

      \[\leadsto \color{blue}{1 + \left(-2 \cdot \left(1 - u\right) + -1 \cdot \frac{-1 \cdot \frac{\frac{-1}{6} \cdot \left(-24 \cdot {\left(1 - u\right)}^{2} + \left(8 \cdot \left(1 - u\right) + 16 \cdot {\left(1 - u\right)}^{3}\right)\right) + \frac{1}{24} \cdot \frac{-96 \cdot {\left(1 - u\right)}^{4} + \left(-64 \cdot {\left(1 - u\right)}^{2} + \left(-48 \cdot {\left(1 - u\right)}^{2} + \left(16 \cdot \left(1 - u\right) + 192 \cdot {\left(1 - u\right)}^{3}\right)\right)\right)}{v}}{v} + \frac{-1}{2} \cdot \left(-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)\right)}{v}\right)} \]

   5. Applied rewrites 73.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1 - u, -2, 1\right) - \frac{-0.5 \cdot \left(\left(1 - u\right) \cdot \mathsf{fma}\left(-4, 1 - u, 4\right)\right) - \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left({\left(1 - u\right)}^{4}, -96, \mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), -112 + \left(1 - u\right) \cdot 192, 16 \cdot \left(1 - u\right)\right)\right)}{v}, 0.041666666666666664, -0.16666666666666666 \cdot \mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), \mathsf{fma}\left(16, 1 - u, -24\right), 8 \cdot \left(1 - u\right)\right)\right)}{v}}{v}} \]

   6. Taylor expanded in u around 0

      \[\leadsto \mathsf{fma}\left(1 - u, -2, 1\right) - \frac{\frac{-1}{2} \cdot \left(\left(1 - u\right) \cdot \mathsf{fma}\left(-4, 1 - u, 4\right)\right) - \frac{\mathsf{fma}\left(\frac{u \cdot \left(16 + u \cdot \left(u \cdot \left(192 + -96 \cdot u\right) - 112\right)\right)}{v}, \frac{1}{24}, \frac{-1}{6} \cdot \mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), \mathsf{fma}\left(16, 1 - u, -24\right), 8 \cdot \left(1 - u\right)\right)\right)}{v}}{v} \]
   7. Step-by-step derivation

   8. Applied rewrites 73.4%

      \[\leadsto \mathsf{fma}\left(1 - u, -2, 1\right) - \frac{-0.5 \cdot \left(\left(1 - u\right) \cdot \mathsf{fma}\left(-4, 1 - u, 4\right)\right) - \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-96, u, 192\right), u, -112\right), u, 16\right) \cdot u}{v}, 0.041666666666666664, -0.16666666666666666 \cdot \mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), \mathsf{fma}\left(16, 1 - u, -24\right), 8 \cdot \left(1 - u\right)\right)\right)}{v}}{v} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 - u, -2, 1\right) - \frac{\left(\mathsf{fma}\left(-4, 1 - u, 4\right) \cdot \left(1 - u\right)\right) \cdot -0.5 - \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-96, u, 192\right), u, -112\right), u, 16\right) \cdot u}{v}, 0.041666666666666664, \mathsf{fma}\left(\left(u - 1\right) \cdot \left(u - 1\right), \mathsf{fma}\left(16, 1 - u, -24\right), 8 \cdot \left(1 - u\right)\right) \cdot -0.16666666666666666\right)}{v}}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 - u, -2, 1\right) - \frac{\left(\mathsf{fma}\left(-4, 1 - u, 4\right) \cdot \left(1 - u\right)\right) \cdot -0.5 - \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(192, u, -112\right), u, 16\right) \cdot u}{v}, 0.041666666666666664, \mathsf{fma}\left(\left(u - 1\right) \cdot \left(u - 1\right), \mathsf{fma}\left(16, 1 - u, -24\right), 8 \cdot \left(1 - u\right)\right) \cdot -0.16666666666666666\right)}{v}}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224)
   1.0
   (-
    (fma (- 1.0 u) -2.0 1.0)
    (/
     (-
      (* (* (fma -4.0 (- 1.0 u) 4.0) (- 1.0 u)) -0.5)
      (/
       (fma
        (/ (* (fma (fma 192.0 u -112.0) u 16.0) u) v)
        0.041666666666666664
        (*
         (fma
          (* (- u 1.0) (- u 1.0))
          (fma 16.0 (- 1.0 u) -24.0)
          (* 8.0 (- 1.0 u)))
         -0.16666666666666666))
       v))
     v))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f;
	} else {
		tmp = fmaf((1.0f - u), -2.0f, 1.0f) - ((((fmaf(-4.0f, (1.0f - u), 4.0f) * (1.0f - u)) * -0.5f) - (fmaf(((fmaf(fmaf(192.0f, u, -112.0f), u, 16.0f) * u) / v), 0.041666666666666664f, (fmaf(((u - 1.0f) * (u - 1.0f)), fmaf(16.0f, (1.0f - u), -24.0f), (8.0f * (1.0f - u))) * -0.16666666666666666f)) / v)) / v);
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.20000000298023224))
		tmp = Float32(1.0);
	else
		tmp = Float32(fma(Float32(Float32(1.0) - u), Float32(-2.0), Float32(1.0)) - Float32(Float32(Float32(Float32(fma(Float32(-4.0), Float32(Float32(1.0) - u), Float32(4.0)) * Float32(Float32(1.0) - u)) * Float32(-0.5)) - Float32(fma(Float32(Float32(fma(fma(Float32(192.0), u, Float32(-112.0)), u, Float32(16.0)) * u) / v), Float32(0.041666666666666664), Float32(fma(Float32(Float32(u - Float32(1.0)) * Float32(u - Float32(1.0))), fma(Float32(16.0), Float32(Float32(1.0) - u), Float32(-24.0)), Float32(Float32(8.0) * Float32(Float32(1.0) - u))) * Float32(-0.16666666666666666))) / v)) / v));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   1. Initial program 99.9%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 90.4%

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

   if 0.200000003 < v

   1. Initial program 92.3%

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

   3. Taylor expanded in v around -inf

      \[\leadsto \color{blue}{1 + \left(-2 \cdot \left(1 - u\right) + -1 \cdot \frac{-1 \cdot \frac{\frac{-1}{6} \cdot \left(-24 \cdot {\left(1 - u\right)}^{2} + \left(8 \cdot \left(1 - u\right) + 16 \cdot {\left(1 - u\right)}^{3}\right)\right) + \frac{1}{24} \cdot \frac{-96 \cdot {\left(1 - u\right)}^{4} + \left(-64 \cdot {\left(1 - u\right)}^{2} + \left(-48 \cdot {\left(1 - u\right)}^{2} + \left(16 \cdot \left(1 - u\right) + 192 \cdot {\left(1 - u\right)}^{3}\right)\right)\right)}{v}}{v} + \frac{-1}{2} \cdot \left(-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)\right)}{v}\right)} \]

   5. Applied rewrites 73.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1 - u, -2, 1\right) - \frac{-0.5 \cdot \left(\left(1 - u\right) \cdot \mathsf{fma}\left(-4, 1 - u, 4\right)\right) - \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left({\left(1 - u\right)}^{4}, -96, \mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), -112 + \left(1 - u\right) \cdot 192, 16 \cdot \left(1 - u\right)\right)\right)}{v}, 0.041666666666666664, -0.16666666666666666 \cdot \mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), \mathsf{fma}\left(16, 1 - u, -24\right), 8 \cdot \left(1 - u\right)\right)\right)}{v}}{v}} \]

   6. Taylor expanded in u around 0

      \[\leadsto \mathsf{fma}\left(1 - u, -2, 1\right) - \frac{\frac{-1}{2} \cdot \left(\left(1 - u\right) \cdot \mathsf{fma}\left(-4, 1 - u, 4\right)\right) - \frac{\mathsf{fma}\left(\frac{u \cdot \left(16 + u \cdot \left(192 \cdot u - 112\right)\right)}{v}, \frac{1}{24}, \frac{-1}{6} \cdot \mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), \mathsf{fma}\left(16, 1 - u, -24\right), 8 \cdot \left(1 - u\right)\right)\right)}{v}}{v} \]
   7. Step-by-step derivation

   8. Applied rewrites 69.1%

      \[\leadsto \mathsf{fma}\left(1 - u, -2, 1\right) - \frac{-0.5 \cdot \left(\left(1 - u\right) \cdot \mathsf{fma}\left(-4, 1 - u, 4\right)\right) - \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(192, u, -112\right), u, 16\right) \cdot u}{v}, 0.041666666666666664, -0.16666666666666666 \cdot \mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), \mathsf{fma}\left(16, 1 - u, -24\right), 8 \cdot \left(1 - u\right)\right)\right)}{v}}{v} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 - u, -2, 1\right) - \frac{\left(\mathsf{fma}\left(-4, 1 - u, 4\right) \cdot \left(1 - u\right)\right) \cdot -0.5 - \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(192, u, -112\right), u, 16\right) \cdot u}{v}, 0.041666666666666664, \mathsf{fma}\left(\left(u - 1\right) \cdot \left(u - 1\right), \mathsf{fma}\left(16, 1 - u, -24\right), 8 \cdot \left(1 - u\right)\right) \cdot -0.16666666666666666\right)}{v}}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.5% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.15000000596046448:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, u, -1\right) - \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(u - 1\right) \cdot \left(u - 1\right), \mathsf{fma}\left(16, 1 - u, -24\right), 8 \cdot \left(1 - u\right)\right)}{v}, 0.16666666666666666, \left(\mathsf{fma}\left(-4, 1 - u, 4\right) \cdot \left(1 - u\right)\right) \cdot -0.5\right)}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.15000000596046448)
   1.0
   (-
    (fma 2.0 u -1.0)
    (/
     (fma
      (/
       (fma
        (* (- u 1.0) (- u 1.0))
        (fma 16.0 (- 1.0 u) -24.0)
        (* 8.0 (- 1.0 u)))
       v)
      0.16666666666666666
      (* (* (fma -4.0 (- 1.0 u) 4.0) (- 1.0 u)) -0.5))
     v))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.15000000596046448f) {
		tmp = 1.0f;
	} else {
		tmp = fmaf(2.0f, u, -1.0f) - (fmaf((fmaf(((u - 1.0f) * (u - 1.0f)), fmaf(16.0f, (1.0f - u), -24.0f), (8.0f * (1.0f - u))) / v), 0.16666666666666666f, ((fmaf(-4.0f, (1.0f - u), 4.0f) * (1.0f - u)) * -0.5f)) / v);
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.15000000596046448))
		tmp = Float32(1.0);
	else
		tmp = Float32(fma(Float32(2.0), u, Float32(-1.0)) - Float32(fma(Float32(fma(Float32(Float32(u - Float32(1.0)) * Float32(u - Float32(1.0))), fma(Float32(16.0), Float32(Float32(1.0) - u), Float32(-24.0)), Float32(Float32(8.0) * Float32(Float32(1.0) - u))) / v), Float32(0.16666666666666666), Float32(Float32(fma(Float32(-4.0), Float32(Float32(1.0) - u), Float32(4.0)) * Float32(Float32(1.0) - u)) * Float32(-0.5))) / v));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.150000006

   1. Initial program 99.9%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 90.6%

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

   if 0.150000006 < v

   1. Initial program 92.6%

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

   3. Taylor expanded in v around -inf

      \[\leadsto \color{blue}{1 + \left(-2 \cdot \left(1 - u\right) + -1 \cdot \frac{\frac{-1}{2} \cdot \left(-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)\right) + \frac{1}{6} \cdot \frac{-24 \cdot {\left(1 - u\right)}^{2} + \left(8 \cdot \left(1 - u\right) + 16 \cdot {\left(1 - u\right)}^{3}\right)}{v}}{v}\right)} \]
   4. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(1 + -2 \cdot \left(1 - u\right)\right) + -1 \cdot \frac{\frac{-1}{2} \cdot \left(-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)\right) + \frac{1}{6} \cdot \frac{-24 \cdot {\left(1 - u\right)}^{2} + \left(8 \cdot \left(1 - u\right) + 16 \cdot {\left(1 - u\right)}^{3}\right)}{v}}{v}} \]

      2. mul-1-neg N/A

         \[\leadsto \left(1 + -2 \cdot \left(1 - u\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{\frac{-1}{2} \cdot \left(-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)\right) + \frac{1}{6} \cdot \frac{-24 \cdot {\left(1 - u\right)}^{2} + \left(8 \cdot \left(1 - u\right) + 16 \cdot {\left(1 - u\right)}^{3}\right)}{v}}{v}\right)\right)} \]

      3. unsub-neg N/A

         \[\leadsto \color{blue}{\left(1 + -2 \cdot \left(1 - u\right)\right) - \frac{\frac{-1}{2} \cdot \left(-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)\right) + \frac{1}{6} \cdot \frac{-24 \cdot {\left(1 - u\right)}^{2} + \left(8 \cdot \left(1 - u\right) + 16 \cdot {\left(1 - u\right)}^{3}\right)}{v}}{v}} \]

      4. lower--.f32 N/A

         \[\leadsto \color{blue}{\left(1 + -2 \cdot \left(1 - u\right)\right) - \frac{\frac{-1}{2} \cdot \left(-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)\right) + \frac{1}{6} \cdot \frac{-24 \cdot {\left(1 - u\right)}^{2} + \left(8 \cdot \left(1 - u\right) + 16 \cdot {\left(1 - u\right)}^{3}\right)}{v}}{v}} \]

   5. Applied rewrites 64.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1 - u, -2, 1\right) - \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), \mathsf{fma}\left(16, 1 - u, -24\right), 8 \cdot \left(1 - u\right)\right)}{v}, 0.16666666666666666, -0.5 \cdot \left(\left(1 - u\right) \cdot \mathsf{fma}\left(-4, 1 - u, 4\right)\right)\right)}{v}} \]

   6. Taylor expanded in u around 0

      \[\leadsto \left(2 \cdot u - 1\right) - \frac{\color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), \mathsf{fma}\left(16, 1 - u, -24\right), 8 \cdot \left(1 - u\right)\right)}{v}, \frac{1}{6}, \frac{-1}{2} \cdot \left(\left(1 - u\right) \cdot \mathsf{fma}\left(-4, 1 - u, 4\right)\right)\right)}}{v} \]
   7. Step-by-step derivation

   8. Applied rewrites 64.6%

      \[\leadsto \mathsf{fma}\left(2, u, -1\right) - \frac{\color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), \mathsf{fma}\left(16, 1 - u, -24\right), 8 \cdot \left(1 - u\right)\right)}{v}, 0.16666666666666666, -0.5 \cdot \left(\left(1 - u\right) \cdot \mathsf{fma}\left(-4, 1 - u, 4\right)\right)\right)}}{v} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.15000000596046448:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, u, -1\right) - \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(u - 1\right) \cdot \left(u - 1\right), \mathsf{fma}\left(16, 1 - u, -24\right), 8 \cdot \left(1 - u\right)\right)}{v}, 0.16666666666666666, \left(\mathsf{fma}\left(-4, 1 - u, 4\right) \cdot \left(1 - u\right)\right) \cdot -0.5\right)}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.15000000596046448:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 - u, -2, 1\right) - \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(-16, u, 24\right), u, -8\right) \cdot u}{v}, 0.16666666666666666, \left(\mathsf{fma}\left(-4, 1 - u, 4\right) \cdot \left(1 - u\right)\right) \cdot -0.5\right)}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.15000000596046448)
   1.0
   (-
    (fma (- 1.0 u) -2.0 1.0)
    (/
     (fma
      (/ (* (fma (fma -16.0 u 24.0) u -8.0) u) v)
      0.16666666666666666
      (* (* (fma -4.0 (- 1.0 u) 4.0) (- 1.0 u)) -0.5))
     v))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.15000000596046448f) {
		tmp = 1.0f;
	} else {
		tmp = fmaf((1.0f - u), -2.0f, 1.0f) - (fmaf(((fmaf(fmaf(-16.0f, u, 24.0f), u, -8.0f) * u) / v), 0.16666666666666666f, ((fmaf(-4.0f, (1.0f - u), 4.0f) * (1.0f - u)) * -0.5f)) / v);
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.15000000596046448))
		tmp = Float32(1.0);
	else
		tmp = Float32(fma(Float32(Float32(1.0) - u), Float32(-2.0), Float32(1.0)) - Float32(fma(Float32(Float32(fma(fma(Float32(-16.0), u, Float32(24.0)), u, Float32(-8.0)) * u) / v), Float32(0.16666666666666666), Float32(Float32(fma(Float32(-4.0), Float32(Float32(1.0) - u), Float32(4.0)) * Float32(Float32(1.0) - u)) * Float32(-0.5))) / v));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.150000006

   1. Initial program 99.9%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 90.6%

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

   if 0.150000006 < v

   1. Initial program 92.6%

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

   3. Taylor expanded in v around -inf

      \[\leadsto \color{blue}{1 + \left(-2 \cdot \left(1 - u\right) + -1 \cdot \frac{\frac{-1}{2} \cdot \left(-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)\right) + \frac{1}{6} \cdot \frac{-24 \cdot {\left(1 - u\right)}^{2} + \left(8 \cdot \left(1 - u\right) + 16 \cdot {\left(1 - u\right)}^{3}\right)}{v}}{v}\right)} \]
   4. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(1 + -2 \cdot \left(1 - u\right)\right) + -1 \cdot \frac{\frac{-1}{2} \cdot \left(-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)\right) + \frac{1}{6} \cdot \frac{-24 \cdot {\left(1 - u\right)}^{2} + \left(8 \cdot \left(1 - u\right) + 16 \cdot {\left(1 - u\right)}^{3}\right)}{v}}{v}} \]

      2. mul-1-neg N/A

         \[\leadsto \left(1 + -2 \cdot \left(1 - u\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{\frac{-1}{2} \cdot \left(-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)\right) + \frac{1}{6} \cdot \frac{-24 \cdot {\left(1 - u\right)}^{2} + \left(8 \cdot \left(1 - u\right) + 16 \cdot {\left(1 - u\right)}^{3}\right)}{v}}{v}\right)\right)} \]

      3. unsub-neg N/A

         \[\leadsto \color{blue}{\left(1 + -2 \cdot \left(1 - u\right)\right) - \frac{\frac{-1}{2} \cdot \left(-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)\right) + \frac{1}{6} \cdot \frac{-24 \cdot {\left(1 - u\right)}^{2} + \left(8 \cdot \left(1 - u\right) + 16 \cdot {\left(1 - u\right)}^{3}\right)}{v}}{v}} \]

      4. lower--.f32 N/A

         \[\leadsto \color{blue}{\left(1 + -2 \cdot \left(1 - u\right)\right) - \frac{\frac{-1}{2} \cdot \left(-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)\right) + \frac{1}{6} \cdot \frac{-24 \cdot {\left(1 - u\right)}^{2} + \left(8 \cdot \left(1 - u\right) + 16 \cdot {\left(1 - u\right)}^{3}\right)}{v}}{v}} \]

   5. Applied rewrites 64.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1 - u, -2, 1\right) - \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), \mathsf{fma}\left(16, 1 - u, -24\right), 8 \cdot \left(1 - u\right)\right)}{v}, 0.16666666666666666, -0.5 \cdot \left(\left(1 - u\right) \cdot \mathsf{fma}\left(-4, 1 - u, 4\right)\right)\right)}{v}} \]

   6. Taylor expanded in u around 0

      \[\leadsto \mathsf{fma}\left(1 - u, -2, 1\right) - \frac{\mathsf{fma}\left(\frac{u \cdot \left(u \cdot \left(24 + -16 \cdot u\right) - 8\right)}{v}, \frac{1}{6}, \frac{-1}{2} \cdot \left(\left(1 - u\right) \cdot \mathsf{fma}\left(-4, 1 - u, 4\right)\right)\right)}{v} \]
   7. Step-by-step derivation

   8. Applied rewrites 64.3%

      \[\leadsto \mathsf{fma}\left(1 - u, -2, 1\right) - \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(-16, u, 24\right), u, -8\right) \cdot u}{v}, 0.16666666666666666, -0.5 \cdot \left(\left(1 - u\right) \cdot \mathsf{fma}\left(-4, 1 - u, 4\right)\right)\right)}{v} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.15000000596046448:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 - u, -2, 1\right) - \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(-16, u, 24\right), u, -8\right) \cdot u}{v}, 0.16666666666666666, \left(\mathsf{fma}\left(-4, 1 - u, 4\right) \cdot \left(1 - u\right)\right) \cdot -0.5\right)}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 87.1% 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 99.5%

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

3. Taylor expanded in v around 0

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

5. Applied rewrites 86.0%

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

Alternative 9: 5.8% 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 99.5%

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

3. Taylor expanded in u around 0

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

5. Applied rewrites 4.8%

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

Reproduce

herbie shell --seed 2024235 
(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))))))))