HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.5%

Time: 11.7s

Alternatives: 6

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

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

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 99.6%

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

Alternative 2: 94.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ 1 + \log \left(\mathsf{fma}\left(1 - \frac{2 - \frac{2 - \left|\frac{1.3333333333333333}{v}\right|}{v}}{v}, 1 - u, u\right)\right) \cdot v \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (+
  1.0
  (*
   (log
    (fma
     (- 1.0 (/ (- 2.0 (/ (- 2.0 (fabs (/ 1.3333333333333333 v))) v)) v))
     (- 1.0 u)
     u))
   v)))

C

float code(float u, float v) {
	return 1.0f + (logf(fmaf((1.0f - ((2.0f - ((2.0f - fabsf((1.3333333333333333f / v))) / v)) / v)), (1.0f - u), u)) * v);
}

Julia

function code(u, v)
	return Float32(Float32(1.0) + Float32(log(fma(Float32(Float32(1.0) - Float32(Float32(Float32(2.0) - Float32(Float32(Float32(2.0) - abs(Float32(Float32(1.3333333333333333) / v))) / v)) / v)), Float32(Float32(1.0) - u), u)) * v))
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

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

   1. fp-cancel-sign-sub-inv N/A

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

   2. metadata-eval N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \left(1 - \color{blue}{1} \cdot \frac{2 + -1 \cdot \frac{2 - \frac{4}{3} \cdot \frac{1}{v}}{v}}{v}\right)\right) \]

   3. *-lft-identity N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \left(1 - \color{blue}{\frac{2 + -1 \cdot \frac{2 - \frac{4}{3} \cdot \frac{1}{v}}{v}}{v}}\right)\right) \]

   4. lower--.f32 N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \color{blue}{\left(1 - \frac{2 + -1 \cdot \frac{2 - \frac{4}{3} \cdot \frac{1}{v}}{v}}{v}\right)}\right) \]

   5. lower-/.f32 N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \left(1 - \color{blue}{\frac{2 + -1 \cdot \frac{2 - \frac{4}{3} \cdot \frac{1}{v}}{v}}{v}}\right)\right) \]

   6. fp-cancel-sign-sub-inv N/A

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

   7. metadata-eval N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \left(1 - \frac{2 - \color{blue}{1} \cdot \frac{2 - \frac{4}{3} \cdot \frac{1}{v}}{v}}{v}\right)\right) \]

   8. *-lft-identity N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \left(1 - \frac{2 - \color{blue}{\frac{2 - \frac{4}{3} \cdot \frac{1}{v}}{v}}}{v}\right)\right) \]

   9. lower--.f32 N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \left(1 - \frac{\color{blue}{2 - \frac{2 - \frac{4}{3} \cdot \frac{1}{v}}{v}}}{v}\right)\right) \]

   10. lower-/.f32 N/A

       \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \left(1 - \frac{2 - \color{blue}{\frac{2 - \frac{4}{3} \cdot \frac{1}{v}}{v}}}{v}\right)\right) \]

   11. lower--.f32 N/A

       \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \left(1 - \frac{2 - \frac{\color{blue}{2 - \frac{4}{3} \cdot \frac{1}{v}}}{v}}{v}\right)\right) \]

   12. associate-*r/ N/A

       \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \left(1 - \frac{2 - \frac{2 - \color{blue}{\frac{\frac{4}{3} \cdot 1}{v}}}{v}}{v}\right)\right) \]

   13. metadata-eval N/A

       \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \left(1 - \frac{2 - \frac{2 - \frac{\color{blue}{\frac{4}{3}}}{v}}{v}}{v}\right)\right) \]

   14. lower-/.f32 2.6

       \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \left(1 - \frac{2 - \frac{2 - \color{blue}{\frac{1.3333333333333333}{v}}}{v}}{v}\right)\right) \]

5. Applied rewrites 2.6%

   \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \color{blue}{\left(1 - \frac{2 - \frac{2 - \frac{1.3333333333333333}{v}}{v}}{v}\right)}\right) \]
6. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto 1 + \color{blue}{v \cdot \log \left(u + \left(1 - u\right) \cdot \left(1 - \frac{2 - \frac{2 - \frac{\frac{4}{3}}{v}}{v}}{v}\right)\right)} \]

   2. *-commutative N/A

      \[\leadsto 1 + \color{blue}{\log \left(u + \left(1 - u\right) \cdot \left(1 - \frac{2 - \frac{2 - \frac{\frac{4}{3}}{v}}{v}}{v}\right)\right) \cdot v} \]

   3. lower-*.f32 2.6

      \[\leadsto 1 + \color{blue}{\log \left(u + \left(1 - u\right) \cdot \left(1 - \frac{2 - \frac{2 - \frac{1.3333333333333333}{v}}{v}}{v}\right)\right) \cdot v} \]

   4. lift-+.f32 N/A

      \[\leadsto 1 + \log \color{blue}{\left(u + \left(1 - u\right) \cdot \left(1 - \frac{2 - \frac{2 - \frac{\frac{4}{3}}{v}}{v}}{v}\right)\right)} \cdot v \]

   5. +-commutative N/A

      \[\leadsto 1 + \log \color{blue}{\left(\left(1 - u\right) \cdot \left(1 - \frac{2 - \frac{2 - \frac{\frac{4}{3}}{v}}{v}}{v}\right) + u\right)} \cdot v \]

   6. lift-*.f32 N/A

      \[\leadsto 1 + \log \left(\color{blue}{\left(1 - u\right) \cdot \left(1 - \frac{2 - \frac{2 - \frac{\frac{4}{3}}{v}}{v}}{v}\right)} + u\right) \cdot v \]

   7. *-commutative N/A

      \[\leadsto 1 + \log \left(\color{blue}{\left(1 - \frac{2 - \frac{2 - \frac{\frac{4}{3}}{v}}{v}}{v}\right) \cdot \left(1 - u\right)} + u\right) \cdot v \]

   8. lower-fma.f32 46.7

      \[\leadsto 1 + \log \color{blue}{\left(\mathsf{fma}\left(1 - \frac{2 - \frac{2 - \frac{1.3333333333333333}{v}}{v}}{v}, 1 - u, u\right)\right)} \cdot v \]

7. Applied rewrites 44.1%

   \[\leadsto 1 + \color{blue}{\log \left(\mathsf{fma}\left(1 - \frac{2 - \frac{2 - \frac{1.3333333333333333}{v}}{v}}{v}, 1 - u, u\right)\right) \cdot v} \]
8. Step-by-step derivation

9. Applied rewrites 95.2%

   \[\leadsto 1 + \log \left(\mathsf{fma}\left(1 - \frac{2 - \frac{2 - \left|\frac{1.3333333333333333}{v}\right|}{v}}{v}, 1 - u, u\right)\right) \cdot v \]
10. Add Preprocessing

Alternative 3: 47.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.05000000074505806:\\ \;\;\;\;1 + \log \left(\mathsf{fma}\left(1 - \frac{2}{v}, 1 - u, u\right)\right) \cdot v\\ \mathbf{else}:\\ \;\;\;\;\left(u \cdot v\right) \cdot \left(e^{\frac{2}{v}} - 1\right) - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.05000000074505806)
   (+ 1.0 (* (log (fma (- 1.0 (/ 2.0 v)) (- 1.0 u) u)) v))
   (- (* (* u v) (- (exp (/ 2.0 v)) 1.0)) 1.0)))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.05000000074505806f) {
		tmp = 1.0f + (logf(fmaf((1.0f - (2.0f / v)), (1.0f - u), u)) * v);
	} else {
		tmp = ((u * v) * (expf((2.0f / v)) - 1.0f)) - 1.0f;
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.05000000074505806))
		tmp = Float32(Float32(1.0) + Float32(log(fma(Float32(Float32(1.0) - Float32(Float32(2.0) / v)), Float32(Float32(1.0) - u), u)) * v));
	else
		tmp = Float32(Float32(Float32(u * v) * Float32(exp(Float32(Float32(2.0) / v)) - Float32(1.0))) - Float32(1.0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.0500000007

   1. Initial program 100.0%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f32 100.0

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

      4. lift-+.f32 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f32 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f32 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in v around -inf

      \[\leadsto 1 + \log \left(\mathsf{fma}\left(\color{blue}{1 + -1 \cdot \frac{2 + -1 \cdot \frac{2 - \frac{4}{3} \cdot \frac{1}{v}}{v}}{v}}, 1 - u, u\right)\right) \cdot v \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 1 + \log \left(\mathsf{fma}\left(\color{blue}{-1 \cdot \frac{2 + -1 \cdot \frac{2 - \frac{4}{3} \cdot \frac{1}{v}}{v}}{v} + 1}, 1 - u, u\right)\right) \cdot v \]

      2. *-commutative N/A

         \[\leadsto 1 + \log \left(\mathsf{fma}\left(\color{blue}{\frac{2 + -1 \cdot \frac{2 - \frac{4}{3} \cdot \frac{1}{v}}{v}}{v} \cdot -1} + 1, 1 - u, u\right)\right) \cdot v \]

      3. lower-fma.f32 N/A

         \[\leadsto 1 + \log \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{2 + -1 \cdot \frac{2 - \frac{4}{3} \cdot \frac{1}{v}}{v}}{v}, -1, 1\right)}, 1 - u, u\right)\right) \cdot v \]

      4. lower-/.f32 N/A

         \[\leadsto 1 + \log \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{2 + -1 \cdot \frac{2 - \frac{4}{3} \cdot \frac{1}{v}}{v}}{v}}, -1, 1\right), 1 - u, u\right)\right) \cdot v \]

      5. +-commutative N/A

         \[\leadsto 1 + \log \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{\color{blue}{-1 \cdot \frac{2 - \frac{4}{3} \cdot \frac{1}{v}}{v} + 2}}{v}, -1, 1\right), 1 - u, u\right)\right) \cdot v \]

      6. *-commutative N/A

         \[\leadsto 1 + \log \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{\color{blue}{\frac{2 - \frac{4}{3} \cdot \frac{1}{v}}{v} \cdot -1} + 2}{v}, -1, 1\right), 1 - u, u\right)\right) \cdot v \]

      7. lower-fma.f32 N/A

         \[\leadsto 1 + \log \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{\color{blue}{\mathsf{fma}\left(\frac{2 - \frac{4}{3} \cdot \frac{1}{v}}{v}, -1, 2\right)}}{v}, -1, 1\right), 1 - u, u\right)\right) \cdot v \]

      8. lower-/.f32 N/A

         \[\leadsto 1 + \log \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\color{blue}{\frac{2 - \frac{4}{3} \cdot \frac{1}{v}}{v}}, -1, 2\right)}{v}, -1, 1\right), 1 - u, u\right)\right) \cdot v \]

      9. lower--.f32 N/A

         \[\leadsto 1 + \log \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{\color{blue}{2 - \frac{4}{3} \cdot \frac{1}{v}}}{v}, -1, 2\right)}{v}, -1, 1\right), 1 - u, u\right)\right) \cdot v \]

      10. associate-*r/ N/A

          \[\leadsto 1 + \log \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{2 - \color{blue}{\frac{\frac{4}{3} \cdot 1}{v}}}{v}, -1, 2\right)}{v}, -1, 1\right), 1 - u, u\right)\right) \cdot v \]

      11. metadata-eval N/A

          \[\leadsto 1 + \log \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{2 - \frac{\color{blue}{\frac{4}{3}}}{v}}{v}, -1, 2\right)}{v}, -1, 1\right), 1 - u, u\right)\right) \cdot v \]

      12. lower-/.f32 49.6

          \[\leadsto 1 + \log \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{2 - \color{blue}{\frac{1.3333333333333333}{v}}}{v}, -1, 2\right)}{v}, -1, 1\right), 1 - u, u\right)\right) \cdot v \]

   7. Applied rewrites 50.1%

      \[\leadsto 1 + \log \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{2 - \frac{1.3333333333333333}{v}}{v}, -1, 2\right)}{v}, -1, 1\right)}, 1 - u, u\right)\right) \cdot v \]

   8. Taylor expanded in v around inf

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

   10. Applied rewrites 49.1%

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

   if 0.0500000007 < v

   1. Initial program 94.1%

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

   3. Taylor expanded in u around 0

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

      1. lower--.f32 N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. rec-exp N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. associate-*r/ N/A

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

      10. lower-expm1.f32 N/A

          \[\leadsto \left(u \cdot v\right) \cdot \color{blue}{\mathsf{expm1}\left(2 \cdot \frac{1}{v}\right)} - 1 \]

      11. associate-*r/ N/A

          \[\leadsto \left(u \cdot v\right) \cdot \mathsf{expm1}\left(\color{blue}{\frac{2 \cdot 1}{v}}\right) - 1 \]

      12. metadata-eval N/A

          \[\leadsto \left(u \cdot v\right) \cdot \mathsf{expm1}\left(\frac{\color{blue}{2}}{v}\right) - 1 \]

      13. lower-/.f32 38.0

          \[\leadsto \left(u \cdot v\right) \cdot \mathsf{expm1}\left(\color{blue}{\frac{2}{v}}\right) - 1 \]

   5. Applied rewrites 34.5%

      \[\leadsto \color{blue}{\left(u \cdot v\right) \cdot \mathsf{expm1}\left(\frac{2}{v}\right) - 1} \]
   6. Step-by-step derivation

   7. Applied rewrites 48.1%

      \[\leadsto \left(u \cdot v\right) \cdot \left(e^{\frac{2}{v}} - 1\right) - 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 87.3% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

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

   2. fp-cancel-sign-sub-inv N/A

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

   3. metadata-eval N/A

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

   4. associate-*r/ N/A

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

   5. associate-+l- N/A

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

   6. fp-cancel-sub-sign-inv N/A

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

   7. metadata-eval N/A

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

   8. unpow2 N/A

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

   9. associate-/r* N/A

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

   10. associate-/l* N/A

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

   11. div-add N/A

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

   12. +-commutative N/A

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

   13. lower--.f32 N/A

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

5. Applied rewrites 43.3%

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

6. Taylor expanded in v around inf

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

8. Applied rewrites 88.9%

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

9. Taylor expanded in u around inf

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

11. Applied rewrites 89.5%

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

Alternative 5: 86.6% 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 89.3%

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

Alternative 6: 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 99.6%

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

3. Taylor expanded in u around 0

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

5. Applied rewrites 4.8%

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

Reproduce

herbie shell --seed 2024337 
(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))))))))