HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.5%

Time: 17.0s

Alternatives: 11

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

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

FPCore

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

C

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

Julia

function code(u, v)
	return fma(v, log(Float32(u + Float32(exp(Float32(Float32(-2.0) / v)) / Float32(Float32(u + Float32(1.0)) / Float32(Float32(1.0) - (u ^ Float32(2.0))))))), 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. Add Preprocessing
5. Step-by-step derivation

   1. fma-udef 99.5%

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

6. Applied egg-rr 99.5%

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

   1. *-commutative 99.5%

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

   2. flip-- 99.5%

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

   3. associate-*r/ 99.5%

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

   4. metadata-eval 99.5%

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

   5. pow2 99.5%

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

8. Applied egg-rr 99.5%

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

   1. associate-/l* 99.5%

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

   2. +-commutative 99.5%

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

10. Simplified 99.5%

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

11. Final simplification 99.5%

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

Alternative 2: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{-2}{v}}\right)\right)\right) \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (+ 1.0 (* v (log (+ u (* (- 1.0 u) (expm1 (log1p (exp (/ -2.0 v))))))))))

C

float code(float u, float v) {
	return 1.0f + (v * logf((u + ((1.0f - u) * expm1f(log1pf(expf((-2.0f / v))))))));
}

Julia

function code(u, v)
	return Float32(Float32(1.0) + Float32(v * log(Float32(u + Float32(Float32(Float32(1.0) - u) * expm1(log1p(exp(Float32(Float32(-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. Add Preprocessing
3. Step-by-step derivation

   1. expm1-log1p-u 99.5%

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

4. Applied egg-rr 99.5%

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

5. Final simplification 99.5%

   \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{-2}{v}}\right)\right)\right) \]
6. Add Preprocessing

Alternative 3: 99.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, v)
	return fma(v, log(Float32(u + Float32(exp(Float32(Float32(-2.0) / v)) * Float32(Float32(1.0) - 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. Add Preprocessing
5. Step-by-step derivation

   1. fma-udef 99.5%

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

6. Applied egg-rr 99.5%

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

7. Final simplification 99.5%

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

Alternative 4: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(u, v)
	tmp = single(1.0) + (v * log((u + (exp((single(-2.0) / v)) * (single(1.0) - u)))));
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. Final simplification 99.5%

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

Alternative 5: 90.7% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.4000000059604645))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(u * Float32(v * Float32((Float32(exp(1)) ^ 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(0.4000000059604645))
		tmp = single(1.0);
	else
		tmp = (u * (v * ((single(2.71828182845904523536) ^ (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 < 0.400000006

   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. expm1-log1p-u 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in v around 0 93.1%

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

   if 0.400000006 < v

   1. Initial program 93.0%

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

   3. Taylor expanded in u around 0 66.8%

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

      1. sub-neg 66.8%

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

      2. rec-exp 66.8%

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

      3. metadata-eval 66.8%

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

      4. associate-*r/ 66.8%

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

      5. metadata-eval 66.8%

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

   5. Simplified 66.8%

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

   6. Taylor expanded in v around 0 67.4%

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

      1. *-un-lft-identity 67.4%

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

      2. exp-prod 67.4%

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

      3. e-exp-1 67.4%

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

   8. Applied egg-rr 67.4%

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

4. Final simplification 91.4%

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

Alternative 6: 90.7% accurate, 1.8× speedup

Math

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

FPCore

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

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.4000000059604645f) {
		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 <= 0.4000000059604645e0) 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(0.4000000059604645))
		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(0.4000000059604645))
		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 < 0.400000006

   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. expm1-log1p-u 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in v around 0 93.1%

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

   if 0.400000006 < v

   1. Initial program 93.0%

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

   3. Taylor expanded in u around 0 66.8%

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

      1. sub-neg 66.8%

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

      2. rec-exp 66.8%

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

      3. metadata-eval 66.8%

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

      4. associate-*r/ 66.8%

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

      5. metadata-eval 66.8%

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

   5. Simplified 66.8%

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

   6. Taylor expanded in v around 0 67.4%

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

4. Final simplification 91.4%

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

Alternative 7: 90.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.4000000059604645:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;u \cdot \left(v \cdot \mathsf{expm1}\left(\frac{2}{v}\right)\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.4000000059604645) 1.0 (+ (* u (* v (expm1 (/ 2.0 v)))) -1.0)))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.4000000059604645f) {
		tmp = 1.0f;
	} else {
		tmp = (u * (v * expm1f((2.0f / v)))) + -1.0f;
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.4000000059604645))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(u * Float32(v * expm1(Float32(Float32(2.0) / v)))) + Float32(-1.0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.400000006

   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. expm1-log1p-u 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in v around 0 93.1%

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

   if 0.400000006 < v

   1. Initial program 93.0%

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

   3. Taylor expanded in u around 0 66.8%

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

      1. sub-neg 66.8%

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

      2. rec-exp 66.8%

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

      3. metadata-eval 66.8%

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

      4. associate-*r/ 66.8%

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

      5. metadata-eval 66.8%

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

   5. Simplified 66.8%

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

   6. Taylor expanded in v around 0 67.4%

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

      1. sub-neg 67.4%

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

      2. metadata-eval 67.4%

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

   8. Applied egg-rr 67.4%

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

      1. metadata-eval 67.4%

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

      2. sub-neg 67.4%

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

      3. expm1-def 67.4%

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

   10. Simplified 67.4%

       \[\leadsto u \cdot \left(v \cdot \color{blue}{\mathsf{expm1}\left(\frac{2}{v}\right)}\right) - 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.4000000059604645:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;u \cdot \left(v \cdot \mathsf{expm1}\left(\frac{2}{v}\right)\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 90.4% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(u + \frac{u}{v}\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224) 1.0 (+ (* 2.0 (+ u (/ u v))) -1.0)))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f;
	} else {
		tmp = (2.0f * (u + (u / 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 <= 0.20000000298023224e0) then
        tmp = 1.0e0
    else
        tmp = (2.0e0 * (u + (u / v))) + (-1.0e0)
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.20000000298023224))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(Float32(2.0) * Float32(u + Float32(u / v))) + Float32(-1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.20000000298023224))
		tmp = single(1.0);
	else
		tmp = (single(2.0) * (u + (u / v))) + single(-1.0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   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. expm1-log1p-u 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in v around 0 93.7%

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

   if 0.200000003 < v

   1. Initial program 93.6%

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

   3. Taylor expanded in u around 0 60.9%

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

      1. sub-neg 60.9%

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

      2. rec-exp 60.9%

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

      3. metadata-eval 60.9%

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

      4. associate-*r/ 60.9%

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

      5. metadata-eval 60.9%

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

   5. Simplified 60.9%

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

   6. Taylor expanded in v around inf 58.1%

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

      1. sub-neg 58.1%

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

      2. distribute-lft-out 58.1%

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

      3. metadata-eval 58.1%

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

   8. Simplified 58.1%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(u + \frac{u}{v}\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 90.4% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;u \cdot \left(2 + \frac{2}{v}\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224) 1.0 (+ (* u (+ 2.0 (/ 2.0 v))) -1.0)))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f;
	} else {
		tmp = (u * (2.0f + (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 <= 0.20000000298023224e0) then
        tmp = 1.0e0
    else
        tmp = (u * (2.0e0 + (2.0e0 / v))) + (-1.0e0)
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.20000000298023224))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(u * Float32(Float32(2.0) + 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(0.20000000298023224))
		tmp = single(1.0);
	else
		tmp = (u * (single(2.0) + (single(2.0) / v))) + single(-1.0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   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. expm1-log1p-u 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in v around 0 93.7%

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

   if 0.200000003 < v

   1. Initial program 93.6%

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

   3. Taylor expanded in u around 0 60.9%

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

      1. sub-neg 60.9%

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

      2. rec-exp 60.9%

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

      3. metadata-eval 60.9%

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

      4. associate-*r/ 60.9%

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

      5. metadata-eval 60.9%

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

   5. Simplified 60.9%

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

   6. Taylor expanded in v around 0 61.4%

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

   7. Taylor expanded in v around inf 58.1%

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

      1. associate-*r/ 58.1%

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

      2. metadata-eval 58.1%

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

   9. Simplified 58.1%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;u \cdot \left(2 + \frac{2}{v}\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 5.6% 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. Add Preprocessing

3. Taylor expanded in u around 0 5.8%

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

4. Final simplification 5.8%

   \[\leadsto -1 \]
5. Add Preprocessing

Alternative 11: 87.0% 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. Add Preprocessing
3. Step-by-step derivation

   1. expm1-log1p-u 99.5%

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

4. Applied egg-rr 99.5%

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

5. Taylor expanded in v around 0 87.4%

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

6. Final simplification 87.4%

   \[\leadsto 1 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024031 
(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))))))))