HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.5%

Time: 16.1s

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, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{-2}{v}}\\ \mathsf{fma}\left(v \cdot {\left(\sqrt[3]{\log \left(\mathsf{fma}\left(1 - u, t_0, u\right)\right)}\right)}^{2}, \sqrt[3]{\log \left(u + \left(1 - u\right) \cdot t_0\right)}, 1\right) \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (let* ((t_0 (exp (/ -2.0 v))))
   (fma
    (* v (pow (cbrt (log (fma (- 1.0 u) t_0 u))) 2.0))
    (cbrt (log (+ u (* (- 1.0 u) t_0))))
    1.0)))

C

float code(float u, float v) {
	float t_0 = expf((-2.0f / v));
	return fmaf((v * powf(cbrtf(logf(fmaf((1.0f - u), t_0, u))), 2.0f)), cbrtf(logf((u + ((1.0f - u) * t_0)))), 1.0f);
}

Julia

function code(u, v)
	t_0 = exp(Float32(Float32(-2.0) / v))
	return fma(Float32(v * (cbrt(log(fma(Float32(Float32(1.0) - u), t_0, u))) ^ Float32(2.0))), cbrt(log(Float32(u + Float32(Float32(Float32(1.0) - u) * t_0)))), Float32(1.0))
end

Derivation

1. Initial program 99.4%

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

   1. +-commutative 99.4%

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

   2. +-commutative 99.4%

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

   3. fma-udef 99.4%

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

   4. fma-udef 99.4%

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

   5. +-commutative 99.4%

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

   6. add-cube-cbrt 99.4%

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

   7. associate-*r* 99.4%

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

   8. fma-def 99.4%

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

3. Applied egg-rr 99.4%

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

4. Taylor expanded in v around 0 99.5%

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

5. Final simplification 99.5%

   \[\leadsto \mathsf{fma}\left(v \cdot {\left(\sqrt[3]{\log \left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right)}\right)}^{2}, \sqrt[3]{\log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right)}, 1\right) \]

Alternative 2: 99.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.4%

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

   1. +-commutative 99.4%

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

   2. fma-def 99.4%

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

   3. +-commutative 99.4%

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

   4. fma-def 99.4%

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

3. Simplified 99.4%

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

4. Taylor expanded in v around 0 99.4%

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

5. Final simplification 99.4%

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

Alternative 3: 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 99.4%

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

2. Final simplification 99.4%

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

Alternative 4: 91.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.30000001192092896:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;u \cdot \left(v \cdot \mathsf{expm1}\left(\frac{2}{v}\right)\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.30000001192092896) 1.0 (+ (* u (* v (expm1 (/ 2.0 v)))) -1.0)))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.30000001192092896f) {
		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(0.30000001192092896))
		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 < 0.300000012

   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. Taylor expanded in v around 0 100.0%

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

   5. Taylor expanded in v around 0 92.9%

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

   if 0.300000012 < v

   1. Initial program 93.5%

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

   2. Taylor expanded in u around 0 69.4%

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

      1. *-commutative 69.4%

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

      2. associate-*l* 69.4%

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

      3. fma-neg 69.4%

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

      4. rec-exp 69.4%

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

      5. expm1-def 69.4%

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

      6. distribute-neg-frac 69.4%

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

      7. metadata-eval 69.4%

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

      8. metadata-eval 69.4%

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

   4. Simplified 69.4%

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

      1. fma-udef 69.4%

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

      2. *-commutative 69.4%

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

   6. Applied egg-rr 69.4%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.30000001192092896:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;u \cdot \left(v \cdot \mathsf{expm1}\left(\frac{2}{v}\right)\right) + -1\\ \end{array} \]

Alternative 5: 90.5% accurate, 16.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(-2 + 2 \cdot \left(u + \frac{u}{v}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.10000000149011612) 1.0 (+ 1.0 (+ -2.0 (* 2.0 (+ u (/ u v)))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + (-2.0f + (2.0f * (u + (u / v))));
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.10000000149011612e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 + ((-2.0e0) + (2.0e0 * (u + (u / v))))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.10000000149011612))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + Float32(Float32(-2.0) + Float32(Float32(2.0) * Float32(u + Float32(u / v)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.10000000149011612))
		tmp = single(1.0);
	else
		tmp = single(1.0) + (single(-2.0) + (single(2.0) * (u + (u / v))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.100000001

   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. Taylor expanded in v around 0 100.0%

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

   5. Taylor expanded in v around 0 93.5%

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

   if 0.100000001 < v

   1. Initial program 93.5%

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

   2. Taylor expanded in u around 0 63.5%

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

      1. rec-exp 63.5%

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

      2. div-inv 63.5%

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

      3. distribute-lft-neg-in 63.5%

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

      4. metadata-eval 63.5%

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

      5. *-commutative 63.5%

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

      6. exp-lft-sqr 63.6%

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

   4. Applied egg-rr 63.6%

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

      1. unpow2 63.6%

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

      2. metadata-eval 63.6%

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

      3. exp-prod 63.5%

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

      4. *-commutative 63.5%

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

      5. metadata-eval 63.5%

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

      6. associate-*r/ 63.5%

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

      7. metadata-eval 63.5%

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

   6. Simplified 63.5%

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

   7. Taylor expanded in v around inf 56.7%

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

      1. sub-neg 56.7%

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

      2. distribute-lft-out 56.7%

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

      3. metadata-eval 56.7%

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

   9. Simplified 56.7%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(-2 + 2 \cdot \left(u + \frac{u}{v}\right)\right)\\ \end{array} \]

Alternative 6: 90.5% accurate, 19.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + 2 \cdot \left(u + \frac{u}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.10000000149011612) 1.0 (+ -1.0 (* 2.0 (+ u (/ u v))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + (2.0f * (u + (u / v)));
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.10000000149011612e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + (2.0e0 * (u + (u / v)))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.10000000149011612))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(Float32(2.0) * Float32(u + Float32(u / v))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.10000000149011612))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + (single(2.0) * (u + (u / v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.100000001

   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. Taylor expanded in v around 0 100.0%

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

   5. Taylor expanded in v around 0 93.5%

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

   if 0.100000001 < v

   1. Initial program 93.5%

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

   2. Taylor expanded in u around 0 64.1%

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

      1. *-commutative 64.1%

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

      2. associate-*l* 64.1%

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

      3. fma-neg 64.1%

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

      4. rec-exp 64.1%

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

      5. expm1-def 64.1%

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

      6. distribute-neg-frac 64.1%

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

      7. metadata-eval 64.1%

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

      8. metadata-eval 64.1%

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

   4. Simplified 64.1%

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

   5. Taylor expanded in v around inf 56.7%

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

      1. sub-neg 56.7%

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

      2. distribute-lft-out 56.7%

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

      3. metadata-eval 56.7%

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

   7. Simplified 56.7%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + 2 \cdot \left(u + \frac{u}{v}\right)\\ \end{array} \]

Alternative 7: 89.9% accurate, 29.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.10000000149011612) 1.0 (+ -1.0 (* u 2.0))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + (u * 2.0f);
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.10000000149011612e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + (u * 2.0e0)
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.10000000149011612))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(u * Float32(2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.10000000149011612))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + (u * single(2.0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.100000001

   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. Taylor expanded in v around 0 100.0%

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

   5. Taylor expanded in v around 0 93.5%

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

   if 0.100000001 < v

   1. Initial program 93.5%

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

   2. Taylor expanded in v around inf 48.4%

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

      1. *-commutative 48.4%

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

   4. Simplified 48.4%

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

   5. Taylor expanded in u around 0 48.4%

      \[\leadsto \color{blue}{2 \cdot u - 1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot 2\\ \end{array} \]

Alternative 8: 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 99.4%

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

2. Taylor expanded in u around 0 6.2%

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

3. Final simplification 6.2%

   \[\leadsto -1 \]

Alternative 9: 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 99.4%

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

   1. +-commutative 99.4%

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

   2. fma-def 99.4%

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

   3. +-commutative 99.4%

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

   4. fma-def 99.4%

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

3. Simplified 99.4%

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

4. Taylor expanded in v around 0 99.4%

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

5. Taylor expanded in v around 0 86.2%

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

6. Final simplification 86.2%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023174 
(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))))))))