HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.5%

Time: 12.5s

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, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

   1. fma-undefine 99.4%

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

6. Applied egg-rr 99.4%

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

7. Final simplification 99.4%

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

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 99.4%

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

Alternative 3: 91.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.25:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;u \cdot \left(v \cdot \left(\frac{1}{e^{\frac{-2}{v}}} + -1\right)\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.25) 1.0 (+ (* u (* v (+ (/ 1.0 (exp (/ -2.0 v))) -1.0))) -1.0)))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.25f) {
		tmp = 1.0f;
	} else {
		tmp = (u * (v * ((1.0f / 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 <= 0.25e0) then
        tmp = 1.0e0
    else
        tmp = (u * (v * ((1.0e0 / 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(0.25))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(u * Float32(v * Float32(Float32(Float32(1.0) / 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(0.25))
		tmp = single(1.0);
	else
		tmp = (u * (v * ((single(1.0) / 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 < 0.25

   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-define 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-define 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. Add Preprocessing
   5. Step-by-step derivation

      1. fma-undefine 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in v around 0 92.5%

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

   if 0.25 < v

   1. Initial program 92.5%

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

   3. Taylor expanded in u around 0 69.9%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.25:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;u \cdot \left(v \cdot \left(\frac{1}{e^{\frac{-2}{v}}} + -1\right)\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.0% accurate, 9.7× speedup

Math

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

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224)
   1.0
   (+ (* u (+ 2.0 (+ (* -2.0 (/ u v)) (* 2.0 (/ 1.0 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 * (u / v)) + (2.0f * (1.0f / 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) * (u / v)) + (2.0e0 * (1.0e0 / 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(Float32(-2.0) * Float32(u / v)) + Float32(Float32(2.0) * Float32(Float32(1.0) / 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) * (u / v)) + (single(2.0) * (single(1.0) / 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. 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-define 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-define 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. Add Preprocessing
   5. Step-by-step derivation

      1. fma-undefine 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in v around 0 92.9%

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

   if 0.200000003 < v

   1. Initial program 92.9%

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

   3. Taylor expanded in v around inf 63.0%

      \[\leadsto 1 + v \cdot \color{blue}{\frac{-2 \cdot \left(1 - u\right) + 0.5 \cdot \frac{-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)}{v}}{v}} \]

   4. Taylor expanded in u around 0 63.0%

      \[\leadsto 1 + v \cdot \frac{-2 \cdot \left(1 - u\right) + 0.5 \cdot \frac{\color{blue}{u \cdot \left(4 + -4 \cdot u\right)}}{v}}{v} \]
   5. Step-by-step derivation

      1. *-commutative 63.0%

         \[\leadsto 1 + v \cdot \frac{-2 \cdot \left(1 - u\right) + 0.5 \cdot \frac{u \cdot \left(4 + \color{blue}{u \cdot -4}\right)}{v}}{v} \]

   6. Simplified 63.0%

      \[\leadsto 1 + v \cdot \frac{-2 \cdot \left(1 - u\right) + 0.5 \cdot \frac{\color{blue}{u \cdot \left(4 + u \cdot -4\right)}}{v}}{v} \]

   7. Taylor expanded in u around 0 63.3%

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

4. Final simplification 90.6%

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

Alternative 5: 90.8% accurate, 13.3× speedup

Math

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

FPCore

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

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 / u)));
	}
	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) / u)))
    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(u * Float32(Float32(2.0) + Float32(Float32(Float32(2.0) / v) + Float32(Float32(-1.0) / u))));
	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) / u)));
	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. 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-define 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-define 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. Add Preprocessing
   5. Step-by-step derivation

      1. fma-undefine 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in v around 0 92.9%

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

   if 0.200000003 < v

   1. Initial program 92.9%

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

   3. Taylor expanded in v around inf 63.0%

      \[\leadsto 1 + v \cdot \color{blue}{\frac{-2 \cdot \left(1 - u\right) + 0.5 \cdot \frac{-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)}{v}}{v}} \]

   4. Taylor expanded in u around 0 58.5%

      \[\leadsto 1 + v \cdot \frac{-2 \cdot \left(1 - u\right) + 0.5 \cdot \frac{\color{blue}{4 \cdot u}}{v}}{v} \]
   5. Step-by-step derivation

      1. *-commutative 58.5%

         \[\leadsto 1 + v \cdot \frac{-2 \cdot \left(1 - u\right) + 0.5 \cdot \frac{\color{blue}{u \cdot 4}}{v}}{v} \]

   6. Simplified 58.5%

      \[\leadsto 1 + v \cdot \frac{-2 \cdot \left(1 - u\right) + 0.5 \cdot \frac{\color{blue}{u \cdot 4}}{v}}{v} \]

   7. Taylor expanded in u around inf 58.7%

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

      1. associate--l+ 58.7%

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

      2. associate-*r/ 58.7%

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

      3. metadata-eval 58.7%

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

   9. Simplified 58.7%

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

4. Final simplification 90.2%

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

Alternative 6: 90.2% accurate, 17.7× speedup

Math

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

FPCore

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

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f;
	} else {
		tmp = u * (2.0f + (-1.0f / u));
	}
	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 + ((-1.0e0) / u))
    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(u * Float32(Float32(2.0) + Float32(Float32(-1.0) / u)));
	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(-1.0) / u));
	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. 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-define 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-define 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. Add Preprocessing
   5. Step-by-step derivation

      1. fma-undefine 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in v around 0 92.9%

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

   if 0.200000003 < v

   1. Initial program 92.9%

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

      1. +-commutative 92.9%

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

      2. fma-define 92.6%

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

      3. +-commutative 92.6%

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

      4. fma-define 92.5%

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

   3. Simplified 92.5%

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

      1. fma-undefine 93.0%

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

   6. Applied egg-rr 93.0%

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

   7. Taylor expanded in v around inf 52.7%

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

   8. Taylor expanded in u around inf 52.7%

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

4. Final simplification 89.8%

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

Alternative 7: 90.2% accurate, 21.2× speedup

Math

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

FPCore

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

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f;
	} else {
		tmp = (u * 2.0f) + -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 <= 0.20000000298023224e0) then
        tmp = 1.0e0
    else
        tmp = (u * 2.0e0) + (-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(2.0)) + 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(-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. 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-define 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-define 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. Add Preprocessing
   5. Step-by-step derivation

      1. fma-undefine 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in v around 0 92.9%

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

   if 0.200000003 < v

   1. Initial program 92.9%

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

      1. +-commutative 92.9%

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

      2. fma-define 92.6%

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

      3. +-commutative 92.6%

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

      4. fma-define 92.5%

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

   3. Simplified 92.5%

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

      1. fma-undefine 93.0%

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

   6. Applied egg-rr 93.0%

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

   7. Taylor expanded in v around inf 52.7%

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

   8. Taylor expanded in u around 0 52.7%

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

4. Final simplification 89.8%

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

Alternative 8: 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 99.4%

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

   1. fma-undefine 99.4%

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

6. Applied egg-rr 99.4%

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

7. Taylor expanded in v around 0 86.1%

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

Alternative 9: 5.9% 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. Add Preprocessing

3. Taylor expanded in u around 0 6.4%

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

Reproduce

herbie shell --seed 2024096 
(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))))))))