HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.5%

Time: 11.8s

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, 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 2: 91.4% accurate, 1.8× speedup

Math

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

FPCore

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

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + ((v * (u * ((1.0f / expf((-2.0f / v))) + -1.0f))) - 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 + ((v * (u * ((1.0e0 / exp(((-2.0e0) / v))) + (-1.0e0)))) - 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(Float32(v * Float32(u * Float32(Float32(Float32(1.0) / exp(Float32(Float32(-2.0) / v))) + Float32(-1.0)))) - 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) + ((v * (u * ((single(1.0) / exp((single(-2.0) / v))) + single(-1.0)))) - 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. log1p-expm1-u 100.0%

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

      2. expm1-udef 99.9%

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

      3. *-commutative 99.9%

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

      4. exp-to-pow 99.9%

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

      5. +-commutative 99.9%

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

      6. fma-udef 99.9%

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

   3. Applied egg-rr 99.9%

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

      1. pow-to-exp 99.9%

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

      2. expm1-def 100.0%

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

      3. log1p-expm1-u 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in v around 0 94.7%

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

   if 0.100000001 < v

   1. Initial program 94.9%

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

   2. Taylor expanded in u around 0 67.3%

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

4. Final simplification 92.1%

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

Alternative 3: 91.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f;
	} else {
		tmp = (v * (u * ((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 <= 0.10000000149011612e0) then
        tmp = 1.0e0
    else
        tmp = (v * (u * ((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(0.10000000149011612))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(v * Float32(u * 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(0.10000000149011612))
		tmp = single(1.0);
	else
		tmp = (v * (u * ((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 < 0.100000001

   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. log1p-expm1-u 100.0%

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

      2. expm1-udef 99.9%

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

      3. *-commutative 99.9%

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

      4. exp-to-pow 99.9%

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

      5. +-commutative 99.9%

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

      6. fma-udef 99.9%

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

   3. Applied egg-rr 99.9%

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

      1. pow-to-exp 99.9%

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

      2. expm1-def 100.0%

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

      3. log1p-expm1-u 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in v around 0 94.7%

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

   if 0.100000001 < v

   1. Initial program 94.9%

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

   2. Taylor expanded in u around 0 67.2%

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

4. Final simplification 92.1%

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

Alternative 4: 91.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f;
	} else {
		tmp = (v * (u * (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 <= 0.10000000149011612e0) then
        tmp = 1.0e0
    else
        tmp = (v * (u * (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(0.10000000149011612))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(v * Float32(u * 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(0.10000000149011612))
		tmp = single(1.0);
	else
		tmp = (v * (u * (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 < 0.100000001

   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. log1p-expm1-u 100.0%

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

      2. expm1-udef 99.9%

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

      3. *-commutative 99.9%

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

      4. exp-to-pow 99.9%

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

      5. +-commutative 99.9%

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

      6. fma-udef 99.9%

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

   3. Applied egg-rr 99.9%

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

      1. pow-to-exp 99.9%

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

      2. expm1-def 100.0%

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

      3. log1p-expm1-u 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in v around 0 94.7%

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

   if 0.100000001 < v

   1. Initial program 94.9%

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

      1. +-commutative 94.9%

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

      2. *-commutative 94.9%

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

      3. add-sqr-sqrt 93.6%

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

      4. associate-*r* 94.1%

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

      5. fma-def 93.7%

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

      6. +-commutative 93.7%

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

      7. fma-udef 93.8%

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

   3. Applied egg-rr 93.8%

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

   4. Taylor expanded in u around 0 76.1%

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

      1. associate--l+ 75.8%

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

      2. associate-*r* 75.8%

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

      3. rec-exp 76.0%

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

      4. expm1-def 76.0%

         \[\leadsto \left(v \cdot u\right) \cdot \color{blue}{\mathsf{expm1}\left(-\frac{-2}{v}\right)} + \left(-0.5 \cdot \frac{{\left(1 + -1 \cdot e^{\frac{-2}{v}}\right)}^{2} \cdot \left(v \cdot {u}^{2}\right)}{{\left(e^{\frac{-2}{v}}\right)}^{2}} - 1\right) \]

      5. distribute-neg-frac 76.0%

         \[\leadsto \left(v \cdot u\right) \cdot \mathsf{expm1}\left(\color{blue}{\frac{--2}{v}}\right) + \left(-0.5 \cdot \frac{{\left(1 + -1 \cdot e^{\frac{-2}{v}}\right)}^{2} \cdot \left(v \cdot {u}^{2}\right)}{{\left(e^{\frac{-2}{v}}\right)}^{2}} - 1\right) \]

      6. metadata-eval 76.0%

         \[\leadsto \left(v \cdot u\right) \cdot \mathsf{expm1}\left(\frac{\color{blue}{2}}{v}\right) + \left(-0.5 \cdot \frac{{\left(1 + -1 \cdot e^{\frac{-2}{v}}\right)}^{2} \cdot \left(v \cdot {u}^{2}\right)}{{\left(e^{\frac{-2}{v}}\right)}^{2}} - 1\right) \]

      7. fma-neg 76.0%

         \[\leadsto \left(v \cdot u\right) \cdot \mathsf{expm1}\left(\frac{2}{v}\right) + \color{blue}{\mathsf{fma}\left(-0.5, \frac{{\left(1 + -1 \cdot e^{\frac{-2}{v}}\right)}^{2} \cdot \left(v \cdot {u}^{2}\right)}{{\left(e^{\frac{-2}{v}}\right)}^{2}}, -1\right)} \]

   6. Simplified 76.0%

      \[\leadsto \color{blue}{\left(v \cdot u\right) \cdot \mathsf{expm1}\left(\frac{2}{v}\right) + \mathsf{fma}\left(-0.5, \frac{{\left(1 - e^{\frac{-2}{v}}\right)}^{2}}{\frac{{\left(e^{\frac{-2}{v}}\right)}^{2}}{v \cdot \left(u \cdot u\right)}}, -1\right)} \]

   7. Taylor expanded in u around 0 67.2%

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

4. Final simplification 92.1%

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

Alternative 5: 91.1% accurate, 11.1× speedup

Math

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

FPCore

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

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f;
	} else {
		tmp = ((-2.0f * (u * (u / v))) + (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 <= 0.10000000149011612e0) then
        tmp = 1.0e0
    else
        tmp = (((-2.0e0) * (u * (u / v))) + (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(0.10000000149011612))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(Float32(Float32(-2.0) * Float32(u * Float32(u / v))) + 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(0.10000000149011612))
		tmp = single(1.0);
	else
		tmp = ((single(-2.0) * (u * (u / v))) + (single(2.0) * (u + (u / v)))) + single(-1.0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.100000001

   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. log1p-expm1-u 100.0%

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

      2. expm1-udef 99.9%

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

      3. *-commutative 99.9%

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

      4. exp-to-pow 99.9%

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

      5. +-commutative 99.9%

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

      6. fma-udef 99.9%

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

   3. Applied egg-rr 99.9%

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

      1. pow-to-exp 99.9%

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

      2. expm1-def 100.0%

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

      3. log1p-expm1-u 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in v around 0 94.7%

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

   if 0.100000001 < v

   1. Initial program 94.9%

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

      1. +-commutative 94.9%

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

      2. *-commutative 94.9%

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

      3. add-sqr-sqrt 93.6%

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

      4. associate-*r* 94.1%

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

      5. fma-def 93.7%

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

      6. +-commutative 93.7%

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

      7. fma-udef 93.8%

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

   3. Applied egg-rr 93.8%

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

   4. Taylor expanded in u around 0 76.1%

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

      1. associate--l+ 75.8%

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

      2. associate-*r* 75.8%

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

      3. rec-exp 76.0%

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

      4. expm1-def 76.0%

         \[\leadsto \left(v \cdot u\right) \cdot \color{blue}{\mathsf{expm1}\left(-\frac{-2}{v}\right)} + \left(-0.5 \cdot \frac{{\left(1 + -1 \cdot e^{\frac{-2}{v}}\right)}^{2} \cdot \left(v \cdot {u}^{2}\right)}{{\left(e^{\frac{-2}{v}}\right)}^{2}} - 1\right) \]

      5. distribute-neg-frac 76.0%

         \[\leadsto \left(v \cdot u\right) \cdot \mathsf{expm1}\left(\color{blue}{\frac{--2}{v}}\right) + \left(-0.5 \cdot \frac{{\left(1 + -1 \cdot e^{\frac{-2}{v}}\right)}^{2} \cdot \left(v \cdot {u}^{2}\right)}{{\left(e^{\frac{-2}{v}}\right)}^{2}} - 1\right) \]

      6. metadata-eval 76.0%

         \[\leadsto \left(v \cdot u\right) \cdot \mathsf{expm1}\left(\frac{\color{blue}{2}}{v}\right) + \left(-0.5 \cdot \frac{{\left(1 + -1 \cdot e^{\frac{-2}{v}}\right)}^{2} \cdot \left(v \cdot {u}^{2}\right)}{{\left(e^{\frac{-2}{v}}\right)}^{2}} - 1\right) \]

      7. fma-neg 76.0%

         \[\leadsto \left(v \cdot u\right) \cdot \mathsf{expm1}\left(\frac{2}{v}\right) + \color{blue}{\mathsf{fma}\left(-0.5, \frac{{\left(1 + -1 \cdot e^{\frac{-2}{v}}\right)}^{2} \cdot \left(v \cdot {u}^{2}\right)}{{\left(e^{\frac{-2}{v}}\right)}^{2}}, -1\right)} \]

   6. Simplified 76.0%

      \[\leadsto \color{blue}{\left(v \cdot u\right) \cdot \mathsf{expm1}\left(\frac{2}{v}\right) + \mathsf{fma}\left(-0.5, \frac{{\left(1 - e^{\frac{-2}{v}}\right)}^{2}}{\frac{{\left(e^{\frac{-2}{v}}\right)}^{2}}{v \cdot \left(u \cdot u\right)}}, -1\right)} \]

   7. Taylor expanded in v around inf 60.3%

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

      1. sub-neg 60.3%

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

      2. fma-def 60.3%

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

      3. unpow2 60.3%

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

      4. distribute-lft-out 60.3%

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

      5. metadata-eval 60.3%

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

   9. Simplified 60.3%

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

       1. fma-udef 60.3%

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

       2. associate-/l* 60.3%

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

       3. div-inv 60.3%

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

       4. clear-num 60.3%

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

   11. Applied egg-rr 60.3%

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

4. Final simplification 91.5%

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

Alternative 6: 91.0% accurate, 14.1× speedup

Math

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

FPCore

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

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + ((2.0f * (u / v)) - (2.0f + (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 + ((2.0e0 * (u / v)) - (2.0e0 + (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(Float32(Float32(2.0) * Float32(u / v)) - Float32(Float32(2.0) + 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) + ((single(2.0) * (u / v)) - (single(2.0) + (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. log1p-expm1-u 100.0%

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

      2. expm1-udef 99.9%

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

      3. *-commutative 99.9%

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

      4. exp-to-pow 99.9%

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

      5. +-commutative 99.9%

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

      6. fma-udef 99.9%

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

   3. Applied egg-rr 99.9%

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

      1. pow-to-exp 99.9%

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

      2. expm1-def 100.0%

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

      3. log1p-expm1-u 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in v around 0 94.7%

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

   if 0.100000001 < v

   1. Initial program 94.9%

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

   2. Taylor expanded in u around 0 67.1%

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

   3. Taylor expanded in v around -inf 58.2%

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

4. Final simplification 91.3%

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

Alternative 7: 91.0% accurate, 16.2× speedup

Math

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

FPCore

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

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f;
	} else {
		tmp = (2.0f * (u * (1.0f + (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 <= 0.10000000149011612e0) then
        tmp = 1.0e0
    else
        tmp = (2.0e0 * (u * (1.0e0 + (1.0e0 / v)))) + (-1.0e0)
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.10000000149011612))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(Float32(2.0) * Float32(u * Float32(Float32(1.0) + 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(0.10000000149011612))
		tmp = single(1.0);
	else
		tmp = (single(2.0) * (u * (single(1.0) + (single(1.0) / v)))) + single(-1.0);
	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. log1p-expm1-u 100.0%

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

      2. expm1-udef 99.9%

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

      3. *-commutative 99.9%

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

      4. exp-to-pow 99.9%

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

      5. +-commutative 99.9%

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

      6. fma-udef 99.9%

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

   3. Applied egg-rr 99.9%

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

      1. pow-to-exp 99.9%

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

      2. expm1-def 100.0%

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

      3. log1p-expm1-u 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in v around 0 94.7%

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

   if 0.100000001 < v

   1. Initial program 94.9%

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

      1. +-commutative 94.9%

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

      2. *-commutative 94.9%

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

      3. add-sqr-sqrt 93.6%

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

      4. associate-*r* 94.1%

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

      5. fma-def 93.7%

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

      6. +-commutative 93.7%

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

      7. fma-udef 93.8%

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

   3. Applied egg-rr 93.8%

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

   4. Taylor expanded in u around 0 76.1%

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

      1. associate--l+ 75.8%

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

      2. associate-*r* 75.8%

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

      3. rec-exp 76.0%

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

      4. expm1-def 76.0%

         \[\leadsto \left(v \cdot u\right) \cdot \color{blue}{\mathsf{expm1}\left(-\frac{-2}{v}\right)} + \left(-0.5 \cdot \frac{{\left(1 + -1 \cdot e^{\frac{-2}{v}}\right)}^{2} \cdot \left(v \cdot {u}^{2}\right)}{{\left(e^{\frac{-2}{v}}\right)}^{2}} - 1\right) \]

      5. distribute-neg-frac 76.0%

         \[\leadsto \left(v \cdot u\right) \cdot \mathsf{expm1}\left(\color{blue}{\frac{--2}{v}}\right) + \left(-0.5 \cdot \frac{{\left(1 + -1 \cdot e^{\frac{-2}{v}}\right)}^{2} \cdot \left(v \cdot {u}^{2}\right)}{{\left(e^{\frac{-2}{v}}\right)}^{2}} - 1\right) \]

      6. metadata-eval 76.0%

         \[\leadsto \left(v \cdot u\right) \cdot \mathsf{expm1}\left(\frac{\color{blue}{2}}{v}\right) + \left(-0.5 \cdot \frac{{\left(1 + -1 \cdot e^{\frac{-2}{v}}\right)}^{2} \cdot \left(v \cdot {u}^{2}\right)}{{\left(e^{\frac{-2}{v}}\right)}^{2}} - 1\right) \]

      7. fma-neg 76.0%

         \[\leadsto \left(v \cdot u\right) \cdot \mathsf{expm1}\left(\frac{2}{v}\right) + \color{blue}{\mathsf{fma}\left(-0.5, \frac{{\left(1 + -1 \cdot e^{\frac{-2}{v}}\right)}^{2} \cdot \left(v \cdot {u}^{2}\right)}{{\left(e^{\frac{-2}{v}}\right)}^{2}}, -1\right)} \]

   6. Simplified 76.0%

      \[\leadsto \color{blue}{\left(v \cdot u\right) \cdot \mathsf{expm1}\left(\frac{2}{v}\right) + \mathsf{fma}\left(-0.5, \frac{{\left(1 - e^{\frac{-2}{v}}\right)}^{2}}{\frac{{\left(e^{\frac{-2}{v}}\right)}^{2}}{v \cdot \left(u \cdot u\right)}}, -1\right)} \]

   7. Taylor expanded in v around inf 60.3%

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

      1. sub-neg 60.3%

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

      2. fma-def 60.3%

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

      3. unpow2 60.3%

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

      4. distribute-lft-out 60.3%

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

      5. metadata-eval 60.3%

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

   9. Simplified 60.3%

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

   10. Taylor expanded in u around 0 58.1%

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

4. Final simplification 91.3%

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

Alternative 8: 91.0% accurate, 19.1× speedup

Math

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

FPCore

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

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		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 <= 0.10000000149011612e0) 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(0.10000000149011612))
		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(0.10000000149011612))
		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 < 0.100000001

   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. log1p-expm1-u 100.0%

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

      2. expm1-udef 99.9%

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

      3. *-commutative 99.9%

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

      4. exp-to-pow 99.9%

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

      5. +-commutative 99.9%

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

      6. fma-udef 99.9%

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

   3. Applied egg-rr 99.9%

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

      1. pow-to-exp 99.9%

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

      2. expm1-def 100.0%

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

      3. log1p-expm1-u 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in v around 0 94.7%

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

   if 0.100000001 < v

   1. Initial program 94.9%

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

   2. Taylor expanded in u around 0 67.1%

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

   3. Taylor expanded in v around inf 58.1%

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

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

   5. 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 91.3%

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

Alternative 9: 90.4% accurate, 29.9× speedup

Math

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

FPCore

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

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		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 <= 0.10000000149011612e0) 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(0.10000000149011612))
		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(0.10000000149011612))
		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 < 0.100000001

   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. log1p-expm1-u 100.0%

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

      2. expm1-udef 99.9%

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

      3. *-commutative 99.9%

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

      4. exp-to-pow 99.9%

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

      5. +-commutative 99.9%

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

      6. fma-udef 99.9%

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

   3. Applied egg-rr 99.9%

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

      1. pow-to-exp 99.9%

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

      2. expm1-def 100.0%

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

      3. log1p-expm1-u 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in v around 0 94.7%

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

   if 0.100000001 < v

   1. Initial program 94.9%

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

   2. Taylor expanded in v around inf 51.2%

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

      1. *-commutative 51.2%

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

   4. Simplified 51.2%

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

   5. Taylor expanded in u around 0 51.2%

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

4. Final simplification 90.6%

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

Alternative 10: 89.7% accurate, 68.2× speedup

Math

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

FPCore

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

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f;
	} else {
		tmp = -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 <= 0.10000000149011612e0) then
        tmp = 1.0e0
    else
        tmp = -1.0e0
    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(-1.0);
	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);
	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. log1p-expm1-u 100.0%

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

      2. expm1-udef 99.9%

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

      3. *-commutative 99.9%

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

      4. exp-to-pow 99.9%

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

      5. +-commutative 99.9%

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

      6. fma-udef 99.9%

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

   3. Applied egg-rr 99.9%

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

      1. pow-to-exp 99.9%

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

      2. expm1-def 100.0%

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

      3. log1p-expm1-u 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in v around 0 94.7%

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

   if 0.100000001 < v

   1. Initial program 94.9%

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

   2. Taylor expanded in u around 0 41.7%

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

4. Final simplification 89.7%

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

Alternative 11: 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 6.8%

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

3. Final simplification 6.8%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023240 
(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))))))))