HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.5%

Time: 24.2s

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 99.5%

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

   1. +-commutative 99.5%

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

   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 99.5%

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

3. Simplified 99.5%

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

4. Final simplification 99.5%

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

Alternative 2: 99.5% accurate, 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 99.5%

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

2. Final simplification 99.5%

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

Alternative 3: 90.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.11999999731779099:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;u \cdot \left(2 + \left(\left(\frac{0.6666666666666666}{{v}^{3}} + \frac{1.3333333333333333}{v \cdot v}\right) + \frac{2}{v}\right)\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.11999999731779099)
   1.0
   (+
    (*
     u
     (+
      2.0
      (+
       (+ (/ 0.6666666666666666 (pow v 3.0)) (/ 1.3333333333333333 (* v v)))
       (/ 2.0 v))))
    -1.0)))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.11999999731779099f) {
		tmp = 1.0f;
	} else {
		tmp = (u * (2.0f + (((0.6666666666666666f / powf(v, 3.0f)) + (1.3333333333333333f / (v * v))) + (2.0f / v)))) + -1.0f;
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.11999999731779099e0) then
        tmp = 1.0e0
    else
        tmp = (u * (2.0e0 + (((0.6666666666666666e0 / (v ** 3.0e0)) + (1.3333333333333333e0 / (v * v))) + (2.0e0 / v)))) + (-1.0e0)
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.11999999731779099))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(u * Float32(Float32(2.0) + Float32(Float32(Float32(Float32(0.6666666666666666) / (v ^ Float32(3.0))) + Float32(Float32(1.3333333333333333) / Float32(v * v))) + Float32(Float32(2.0) / v)))) + Float32(-1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.11999999731779099))
		tmp = single(1.0);
	else
		tmp = (u * (single(2.0) + (((single(0.6666666666666666) / (v ^ single(3.0))) + (single(1.3333333333333333) / (v * v))) + (single(2.0) / v)))) + single(-1.0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.119999997

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-def 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-def 100.0%

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

   3. Simplified 100.0%

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

      1. fma-udef 100.0%

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

      2. fma-udef 100.0%

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

      3. +-commutative 100.0%

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

      4. add-log-exp 99.9%

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

      5. *-commutative 99.9%

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

      6. exp-to-pow 99.9%

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

      7. +-commutative 99.9%

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

      8. fma-udef 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in v around 0 93.8%

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

   if 0.119999997 < v

   1. Initial program 92.3%

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

   2. Taylor expanded in u around 0 65.3%

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

      1. expm1-log1p-u 65.3%

         \[\leadsto 1 + v \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(u \cdot \left(\frac{1}{e^{\frac{-2}{v}}} - 1\right)\right)\right)} - 2 \cdot \frac{1}{v}\right) \]

      2. expm1-udef 63.1%

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

      3. rec-exp 63.1%

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

      4. expm1-def 63.1%

         \[\leadsto 1 + v \cdot \left(\left(e^{\mathsf{log1p}\left(u \cdot \color{blue}{\mathsf{expm1}\left(-\frac{-2}{v}\right)}\right)} - 1\right) - 2 \cdot \frac{1}{v}\right) \]

   4. Applied egg-rr 63.1%

      \[\leadsto 1 + v \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left(u \cdot \mathsf{expm1}\left(-\frac{-2}{v}\right)\right)} - 1\right)} - 2 \cdot \frac{1}{v}\right) \]
   5. Step-by-step derivation

      1. expm1-def 65.3%

         \[\leadsto 1 + v \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(u \cdot \mathsf{expm1}\left(-\frac{-2}{v}\right)\right)\right)} - 2 \cdot \frac{1}{v}\right) \]

      2. expm1-log1p 65.3%

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

      3. distribute-neg-frac 65.3%

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

      4. metadata-eval 65.3%

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

   6. Simplified 65.3%

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

   7. Taylor expanded in v around 0 65.4%

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

   8. Taylor expanded in v around inf 68.3%

      \[\leadsto u \cdot \color{blue}{\left(2 + \left(0.6666666666666666 \cdot \frac{1}{{v}^{3}} + \left(1.3333333333333333 \cdot \frac{1}{{v}^{2}} + 2 \cdot \frac{1}{v}\right)\right)\right)} - 1 \]
   9. Step-by-step derivation

      1. associate-+r+ 68.3%

         \[\leadsto u \cdot \left(2 + \color{blue}{\left(\left(0.6666666666666666 \cdot \frac{1}{{v}^{3}} + 1.3333333333333333 \cdot \frac{1}{{v}^{2}}\right) + 2 \cdot \frac{1}{v}\right)}\right) - 1 \]

      2. associate-*r/ 68.3%

         \[\leadsto u \cdot \left(2 + \left(\left(\color{blue}{\frac{0.6666666666666666 \cdot 1}{{v}^{3}}} + 1.3333333333333333 \cdot \frac{1}{{v}^{2}}\right) + 2 \cdot \frac{1}{v}\right)\right) - 1 \]

      3. metadata-eval 68.3%

         \[\leadsto u \cdot \left(2 + \left(\left(\frac{\color{blue}{0.6666666666666666}}{{v}^{3}} + 1.3333333333333333 \cdot \frac{1}{{v}^{2}}\right) + 2 \cdot \frac{1}{v}\right)\right) - 1 \]

      4. associate-*r/ 68.3%

         \[\leadsto u \cdot \left(2 + \left(\left(\frac{0.6666666666666666}{{v}^{3}} + \color{blue}{\frac{1.3333333333333333 \cdot 1}{{v}^{2}}}\right) + 2 \cdot \frac{1}{v}\right)\right) - 1 \]

      5. metadata-eval 68.3%

         \[\leadsto u \cdot \left(2 + \left(\left(\frac{0.6666666666666666}{{v}^{3}} + \frac{\color{blue}{1.3333333333333333}}{{v}^{2}}\right) + 2 \cdot \frac{1}{v}\right)\right) - 1 \]

      6. unpow2 68.3%

         \[\leadsto u \cdot \left(2 + \left(\left(\frac{0.6666666666666666}{{v}^{3}} + \frac{1.3333333333333333}{\color{blue}{v \cdot v}}\right) + 2 \cdot \frac{1}{v}\right)\right) - 1 \]

      7. associate-*r/ 68.3%

         \[\leadsto u \cdot \left(2 + \left(\left(\frac{0.6666666666666666}{{v}^{3}} + \frac{1.3333333333333333}{v \cdot v}\right) + \color{blue}{\frac{2 \cdot 1}{v}}\right)\right) - 1 \]

      8. metadata-eval 68.3%

         \[\leadsto u \cdot \left(2 + \left(\left(\frac{0.6666666666666666}{{v}^{3}} + \frac{1.3333333333333333}{v \cdot v}\right) + \frac{\color{blue}{2}}{v}\right)\right) - 1 \]

   10. Simplified 68.3%

       \[\leadsto u \cdot \color{blue}{\left(2 + \left(\left(\frac{0.6666666666666666}{{v}^{3}} + \frac{1.3333333333333333}{v \cdot v}\right) + \frac{2}{v}\right)\right)} - 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.11999999731779099:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;u \cdot \left(2 + \left(\left(\frac{0.6666666666666666}{{v}^{3}} + \frac{1.3333333333333333}{v \cdot v}\right) + \frac{2}{v}\right)\right) + -1\\ \end{array} \]

Alternative 4: 90.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.15000000596046448:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;u \cdot \left(v \cdot \left(e^{\frac{2}{v}} + -1\right)\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.15000000596046448)
   1.0
   (+ (* u (* v (+ (exp (/ 2.0 v)) -1.0))) -1.0)))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.15000000596046448f) {
		tmp = 1.0f;
	} else {
		tmp = (u * (v * (expf((2.0f / v)) + -1.0f))) + -1.0f;
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.15000000596046448e0) then
        tmp = 1.0e0
    else
        tmp = (u * (v * (exp((2.0e0 / v)) + (-1.0e0)))) + (-1.0e0)
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.15000000596046448))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(u * Float32(v * Float32(exp(Float32(Float32(2.0) / v)) + Float32(-1.0)))) + Float32(-1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.15000000596046448))
		tmp = single(1.0);
	else
		tmp = (u * (v * (exp((single(2.0) / v)) + single(-1.0)))) + single(-1.0);
	end
	tmp_2 = 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. Step-by-step derivation

      1. +-commutative 99.9%

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

      2. fma-def 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-def 100.0%

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

   3. Simplified 100.0%

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

      1. fma-udef 99.9%

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

      2. fma-udef 99.9%

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

      3. +-commutative 99.9%

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

      4. add-log-exp 99.9%

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

      5. *-commutative 99.9%

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

      6. exp-to-pow 99.9%

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

      7. +-commutative 99.9%

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

      8. fma-udef 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in v around 0 93.5%

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

   if 0.150000006 < v

   1. Initial program 92.0%

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

   2. Taylor expanded in u around 0 68.5%

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

      1. expm1-log1p-u 68.5%

         \[\leadsto 1 + v \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(u \cdot \left(\frac{1}{e^{\frac{-2}{v}}} - 1\right)\right)\right)} - 2 \cdot \frac{1}{v}\right) \]

      2. expm1-udef 66.2%

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

      3. rec-exp 66.2%

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

      4. expm1-def 66.2%

         \[\leadsto 1 + v \cdot \left(\left(e^{\mathsf{log1p}\left(u \cdot \color{blue}{\mathsf{expm1}\left(-\frac{-2}{v}\right)}\right)} - 1\right) - 2 \cdot \frac{1}{v}\right) \]

   4. Applied egg-rr 66.2%

      \[\leadsto 1 + v \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left(u \cdot \mathsf{expm1}\left(-\frac{-2}{v}\right)\right)} - 1\right)} - 2 \cdot \frac{1}{v}\right) \]
   5. Step-by-step derivation

      1. expm1-def 68.5%

         \[\leadsto 1 + v \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(u \cdot \mathsf{expm1}\left(-\frac{-2}{v}\right)\right)\right)} - 2 \cdot \frac{1}{v}\right) \]

      2. expm1-log1p 68.5%

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

      3. distribute-neg-frac 68.5%

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

      4. metadata-eval 68.5%

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

   6. Simplified 68.5%

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

   7. Taylor expanded in v around 0 68.7%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.15000000596046448:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;u \cdot \left(v \cdot \left(e^{\frac{2}{v}} + -1\right)\right) + -1\\ \end{array} \]

Alternative 5: 90.5% accurate, 12.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.4000000059604645:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;u \cdot \left(2 + \left(\frac{1.3333333333333333}{v \cdot v} + \frac{2}{v}\right)\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.4000000059604645)
   1.0
   (+ (* u (+ 2.0 (+ (/ 1.3333333333333333 (* v v)) (/ 2.0 v)))) -1.0)))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.4000000059604645f) {
		tmp = 1.0f;
	} else {
		tmp = (u * (2.0f + ((1.3333333333333333f / (v * v)) + (2.0f / v)))) + -1.0f;
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.4000000059604645e0) then
        tmp = 1.0e0
    else
        tmp = (u * (2.0e0 + ((1.3333333333333333e0 / (v * v)) + (2.0e0 / v)))) + (-1.0e0)
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.4000000059604645))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(u * Float32(Float32(2.0) + Float32(Float32(Float32(1.3333333333333333) / Float32(v * v)) + Float32(Float32(2.0) / v)))) + Float32(-1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.4000000059604645))
		tmp = single(1.0);
	else
		tmp = (u * (single(2.0) + ((single(1.3333333333333333) / (v * v)) + (single(2.0) / v)))) + single(-1.0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.400000006

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. fma-def 99.9%

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

      3. +-commutative 99.9%

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

      4. fma-def 99.9%

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

   3. Simplified 99.9%

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

      1. fma-udef 99.9%

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

      2. fma-udef 99.9%

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

      3. +-commutative 99.9%

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

      4. add-log-exp 99.8%

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

      5. *-commutative 99.8%

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

      6. exp-to-pow 99.8%

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

      7. +-commutative 99.8%

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

      8. fma-udef 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in v around 0 92.6%

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

   if 0.400000006 < v

   1. Initial program 91.5%

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

   2. Taylor expanded in u around 0 79.7%

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

      1. expm1-log1p-u 79.7%

         \[\leadsto 1 + v \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(u \cdot \left(\frac{1}{e^{\frac{-2}{v}}} - 1\right)\right)\right)} - 2 \cdot \frac{1}{v}\right) \]

      2. expm1-udef 76.8%

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

      3. rec-exp 76.8%

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

      4. expm1-def 76.8%

         \[\leadsto 1 + v \cdot \left(\left(e^{\mathsf{log1p}\left(u \cdot \color{blue}{\mathsf{expm1}\left(-\frac{-2}{v}\right)}\right)} - 1\right) - 2 \cdot \frac{1}{v}\right) \]

   4. Applied egg-rr 76.8%

      \[\leadsto 1 + v \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left(u \cdot \mathsf{expm1}\left(-\frac{-2}{v}\right)\right)} - 1\right)} - 2 \cdot \frac{1}{v}\right) \]
   5. Step-by-step derivation

      1. expm1-def 79.7%

         \[\leadsto 1 + v \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(u \cdot \mathsf{expm1}\left(-\frac{-2}{v}\right)\right)\right)} - 2 \cdot \frac{1}{v}\right) \]

      2. expm1-log1p 79.7%

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

      3. distribute-neg-frac 79.7%

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

      4. metadata-eval 79.7%

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

   6. Simplified 79.7%

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

   7. Taylor expanded in v around 0 79.9%

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

   8. Taylor expanded in v around inf 77.1%

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

      1. +-commutative 77.1%

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

      2. associate-*r/ 77.1%

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

      3. metadata-eval 77.1%

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

      4. associate-*r/ 77.1%

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

      5. metadata-eval 77.1%

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

      6. unpow2 77.1%

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

   10. Simplified 77.1%

       \[\leadsto u \cdot \color{blue}{\left(2 + \left(\frac{2}{v} + \frac{1.3333333333333333}{v \cdot v}\right)\right)} - 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.4000000059604645:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;u \cdot \left(2 + \left(\frac{1.3333333333333333}{v \cdot v} + \frac{2}{v}\right)\right) + -1\\ \end{array} \]

Alternative 6: 5.8% 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 99.5%

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

2. Taylor expanded in u around 0 5.4%

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

3. Final simplification 5.4%

   \[\leadsto -1 \]

Alternative 7: 86.9% 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 99.5%

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

   1. +-commutative 99.5%

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

   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 99.5%

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

3. Simplified 99.5%

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

   1. fma-udef 99.5%

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

   2. fma-udef 99.5%

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

   3. +-commutative 99.5%

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

   4. add-log-exp 99.4%

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

   5. *-commutative 99.4%

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

   6. exp-to-pow 99.4%

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

   7. +-commutative 99.4%

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

   8. fma-udef 99.4%

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

5. Applied egg-rr 99.4%

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

6. Taylor expanded in v around 0 88.5%

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

7. Final simplification 88.5%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023278 
(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))))))))