HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.5%

Time: 17.9s

Alternatives: 14

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

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

FPCore

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

C

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

Julia

function code(u, v)
	return Float32(Float32(1.0) + Float32(v * log(Float32(u + Float32(Float32(Float32(1.0) - u) * (Float32(exp(1)) ^ Float32(Float32(-2.0) / v)))))))
end

MATLAB

function tmp = code(u, v)
	tmp = single(1.0) + (v * log((u + ((single(1.0) - u) * (single(2.71828182845904523536) ^ (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. Add Preprocessing
3. Step-by-step derivation

   1. *-un-lft-identity 99.5%

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

   2. exp-prod 99.5%

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

4. Applied egg-rr 99.5%

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

   1. exp-1-e 99.5%

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

6. Simplified 99.5%

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

7. Final simplification 99.5%

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

Alternative 2: 97.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.30000001192092896f) {
		tmp = 1.0f + (v * logf((u * -expm1f((-2.0f / v)))));
	} else {
		tmp = 1.0f + ((u * (v * ((1.0f / expf((-2.0f / v))) + -1.0f))) - 2.0f);
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.30000001192092896))
		tmp = Float32(Float32(1.0) + Float32(v * log(Float32(u * Float32(-expm1(Float32(Float32(-2.0) / v)))))));
	else
		tmp = Float32(Float32(1.0) + Float32(Float32(u * Float32(v * Float32(Float32(Float32(1.0) / exp(Float32(Float32(-2.0) / v))) + Float32(-1.0)))) - Float32(2.0)));
	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 99.9%

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

      3. +-commutative 99.9%

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

      4. fma-def 99.9%

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

   3. Simplified 99.9%

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

      1. fma-udef 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in u around inf 99.2%

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

      1. distribute-rgt-in 99.2%

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

      2. metadata-eval 99.2%

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

      3. distribute-lft-in 99.2%

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

      4. +-commutative 99.2%

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

      5. metadata-eval 99.2%

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

      6. sub-neg 99.2%

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

      7. distribute-rgt-in 99.2%

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

      8. +-commutative 99.2%

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

   9. Simplified 99.2%

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

   if 0.300000012 < v

   1. Initial program 94.0%

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

   3. Taylor expanded in u around 0 78.9%

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

4. Final simplification 97.7%

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

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 99.5%

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

3. Final simplification 99.5%

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

Alternative 4: 90.8% accurate, 1.8× speedup

Math

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

FPCore

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

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.30000001192092896f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + ((u * (v * ((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 <= 0.30000001192092896e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 + ((u * (v * ((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(0.30000001192092896))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + Float32(Float32(u * Float32(v * 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(0.30000001192092896))
		tmp = single(1.0);
	else
		tmp = single(1.0) + ((u * (v * ((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 < 0.300000012

   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 99.9%

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

      3. +-commutative 99.9%

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

      4. fma-def 99.9%

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

   3. Simplified 99.9%

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

      1. fma-udef 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in v around 0 91.8%

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

   if 0.300000012 < v

   1. Initial program 94.0%

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

   3. Taylor expanded in u around 0 78.9%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.30000001192092896:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(u \cdot \left(v \cdot \left(\frac{1}{e^{\frac{-2}{v}}} + -1\right)\right) - 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 90.8% accurate, 1.8× speedup

Math

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

FPCore

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

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.30000001192092896f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + ((expm1f((2.0f / v)) * (v * u)) - 2.0f);
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.30000001192092896))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + Float32(Float32(expm1(Float32(Float32(2.0) / v)) * Float32(v * u)) - Float32(2.0)));
	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 99.9%

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

      3. +-commutative 99.9%

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

      4. fma-def 99.9%

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

   3. Simplified 99.9%

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

      1. fma-udef 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in v around 0 91.8%

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

   if 0.300000012 < v

   1. Initial program 94.0%

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

   3. Taylor expanded in u around 0 86.8%

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

   4. Taylor expanded in u around 0 78.9%

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

      1. associate-*r* 78.9%

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

      2. rec-exp 78.9%

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

      3. metadata-eval 78.9%

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

      4. associate-*r/ 78.9%

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

      5. log-E 78.9%

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

      6. rec-exp 78.9%

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

      7. rec-exp 78.9%

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

      8. log-E 78.9%

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

      9. distribute-lft-neg-in 78.9%

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

      10. metadata-eval 78.9%

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

      11. expm1-def 78.9%

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

      12. associate-*r/ 78.9%

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

      13. metadata-eval 78.9%

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

      14. metadata-eval 78.9%

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

      15. distribute-neg-frac 78.9%

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

   6. Simplified 78.9%

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

4. Final simplification 90.8%

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

Alternative 6: 90.4% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.05000000074505806:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(1 - u\right) \cdot -2 + 0.5 \cdot \frac{-4 \cdot \left(\left(1 - u\right) \cdot \left(1 - u\right)\right) + \left(1 - u\right) \cdot 4}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.05000000074505806)
   1.0
   (+
    1.0
    (+
     (* (- 1.0 u) -2.0)
     (* 0.5 (/ (+ (* -4.0 (* (- 1.0 u) (- 1.0 u))) (* (- 1.0 u) 4.0)) v))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.05000000074505806f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + (((1.0f - u) * -2.0f) + (0.5f * (((-4.0f * ((1.0f - u) * (1.0f - u))) + ((1.0f - u) * 4.0f)) / 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 <= 0.05000000074505806e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 + (((1.0e0 - u) * (-2.0e0)) + (0.5e0 * ((((-4.0e0) * ((1.0e0 - u) * (1.0e0 - u))) + ((1.0e0 - u) * 4.0e0)) / v)))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.05000000074505806))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + Float32(Float32(Float32(Float32(1.0) - u) * Float32(-2.0)) + Float32(Float32(0.5) * Float32(Float32(Float32(Float32(-4.0) * Float32(Float32(Float32(1.0) - u) * Float32(Float32(1.0) - u))) + Float32(Float32(Float32(1.0) - u) * Float32(4.0))) / v))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.05000000074505806))
		tmp = single(1.0);
	else
		tmp = single(1.0) + (((single(1.0) - u) * single(-2.0)) + (single(0.5) * (((single(-4.0) * ((single(1.0) - u) * (single(1.0) - u))) + ((single(1.0) - u) * single(4.0))) / v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.0500000007

   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. Add Preprocessing
   5. Step-by-step derivation

      1. fma-udef 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in v around 0 92.9%

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

   if 0.0500000007 < v

   1. Initial program 94.0%

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

   3. Taylor expanded in v around inf 63.4%

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

      1. unpow2 63.4%

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

   5. Applied egg-rr 63.4%

      \[\leadsto 1 + \left(-2 \cdot \left(1 - u\right) + 0.5 \cdot \frac{-4 \cdot \color{blue}{\left(\left(1 - u\right) \cdot \left(1 - u\right)\right)} + 4 \cdot \left(1 - u\right)}{v}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.05000000074505806:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(1 - u\right) \cdot -2 + 0.5 \cdot \frac{-4 \cdot \left(\left(1 - u\right) \cdot \left(1 - u\right)\right) + \left(1 - u\right) \cdot 4}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 90.3% accurate, 10.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.05000000074505806:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(1 - u\right) \cdot -2 + 0.5 \cdot \frac{u \cdot 4}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.05000000074505806)
   1.0
   (+ 1.0 (+ (* (- 1.0 u) -2.0) (* 0.5 (/ (* u 4.0) v))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.05000000074505806f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + (((1.0f - u) * -2.0f) + (0.5f * ((u * 4.0f) / 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 <= 0.05000000074505806e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 + (((1.0e0 - u) * (-2.0e0)) + (0.5e0 * ((u * 4.0e0) / v)))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.05000000074505806))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + Float32(Float32(Float32(Float32(1.0) - u) * Float32(-2.0)) + Float32(Float32(0.5) * Float32(Float32(u * Float32(4.0)) / v))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.05000000074505806))
		tmp = single(1.0);
	else
		tmp = single(1.0) + (((single(1.0) - u) * single(-2.0)) + (single(0.5) * ((u * single(4.0)) / v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.0500000007

   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. Add Preprocessing
   5. Step-by-step derivation

      1. fma-udef 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in v around 0 92.9%

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

   if 0.0500000007 < v

   1. Initial program 94.0%

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

   3. Taylor expanded in v around inf 63.4%

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

   4. Taylor expanded in u around 0 62.3%

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

      1. *-commutative 62.3%

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

   6. Simplified 62.3%

      \[\leadsto 1 + \left(-2 \cdot \left(1 - u\right) + 0.5 \cdot \frac{\color{blue}{u \cdot 4}}{v}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.05000000074505806:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(1 - u\right) \cdot -2 + 0.5 \cdot \frac{u \cdot 4}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 90.3% accurate, 11.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.05000000074505806:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(u \cdot \left(2 + 2 \cdot \frac{1}{v}\right) - 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.05000000074505806)
   1.0
   (+ 1.0 (- (* u (+ 2.0 (* 2.0 (/ 1.0 v)))) 2.0))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.05000000074505806f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + ((u * (2.0f + (2.0f * (1.0f / v)))) - 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 <= 0.05000000074505806e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 + ((u * (2.0e0 + (2.0e0 * (1.0e0 / v)))) - 2.0e0)
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.05000000074505806))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + Float32(Float32(u * Float32(Float32(2.0) + Float32(Float32(2.0) * Float32(Float32(1.0) / v)))) - Float32(2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.05000000074505806))
		tmp = single(1.0);
	else
		tmp = single(1.0) + ((u * (single(2.0) + (single(2.0) * (single(1.0) / v)))) - single(2.0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.0500000007

   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. Add Preprocessing
   5. Step-by-step derivation

      1. fma-udef 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in v around 0 92.9%

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

   if 0.0500000007 < v

   1. Initial program 94.0%

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

   3. Taylor expanded in v around inf 63.4%

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

   4. Taylor expanded in u around 0 62.3%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.05000000074505806:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(u \cdot \left(2 + 2 \cdot \frac{1}{v}\right) - 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 90.3% accurate, 13.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.05000000074505806:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(2 \cdot \left(u + \frac{u}{v}\right) - 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.05000000074505806) 1.0 (+ 1.0 (- (* 2.0 (+ u (/ u v))) 2.0))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.05000000074505806f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + ((2.0f * (u + (u / v))) - 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 <= 0.05000000074505806e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 + ((2.0e0 * (u + (u / v))) - 2.0e0)
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.05000000074505806))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + Float32(Float32(Float32(2.0) * Float32(u + Float32(u / v))) - Float32(2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.05000000074505806))
		tmp = single(1.0);
	else
		tmp = single(1.0) + ((single(2.0) * (u + (u / v))) - single(2.0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.0500000007

   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. Add Preprocessing
   5. Step-by-step derivation

      1. fma-udef 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in v around 0 92.9%

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

   if 0.0500000007 < v

   1. Initial program 94.0%

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

   3. Taylor expanded in u around 0 77.8%

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

   4. Taylor expanded in u around 0 68.6%

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

      1. associate-*r* 68.6%

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

      2. rec-exp 68.6%

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

      3. metadata-eval 68.6%

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

      4. associate-*r/ 68.6%

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

      5. log-E 68.6%

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

      6. rec-exp 68.6%

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

      7. rec-exp 68.6%

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

      8. log-E 68.6%

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

      9. distribute-lft-neg-in 68.6%

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

      10. metadata-eval 68.6%

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

      11. expm1-def 68.6%

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

      12. associate-*r/ 68.6%

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

      13. metadata-eval 68.6%

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

      14. metadata-eval 68.6%

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

      15. distribute-neg-frac 68.6%

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

   6. Simplified 68.6%

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

   7. Taylor expanded in v around inf 62.3%

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

      1. distribute-lft-out 62.3%

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

   9. Simplified 62.3%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.05000000074505806:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(2 \cdot \left(u + \frac{u}{v}\right) - 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 90.2% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.05000000074505806:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(u + \frac{u}{v}\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.05000000074505806) 1.0 (+ (* 2.0 (+ u (/ u v))) -1.0)))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.05000000074505806f) {
		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 <= 0.05000000074505806e0) 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(0.05000000074505806))
		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(0.05000000074505806))
		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 < 0.0500000007

   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. Add Preprocessing
   5. Step-by-step derivation

      1. fma-udef 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in v around 0 92.9%

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

   if 0.0500000007 < v

   1. Initial program 94.0%

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

   3. Taylor expanded in u around 0 77.8%

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

   4. Taylor expanded in u around 0 68.6%

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

      1. associate-*r* 68.6%

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

      2. rec-exp 68.6%

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

      3. metadata-eval 68.6%

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

      4. associate-*r/ 68.6%

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

      5. log-E 68.6%

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

      6. rec-exp 68.6%

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

      7. rec-exp 68.6%

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

      8. log-E 68.6%

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

      9. distribute-lft-neg-in 68.6%

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

      10. metadata-eval 68.6%

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

      11. expm1-def 68.6%

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

      12. associate-*r/ 68.6%

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

      13. metadata-eval 68.6%

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

      14. metadata-eval 68.6%

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

      15. distribute-neg-frac 68.6%

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

   6. Simplified 68.6%

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

   7. Taylor expanded in v around inf 62.3%

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

      1. sub-neg 62.3%

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

      2. distribute-lft-out 62.3%

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

      3. metadata-eval 62.3%

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

   9. Simplified 62.3%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.05000000074505806:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(u + \frac{u}{v}\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 89.5% accurate, 17.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.05000000074505806:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(1 - u\right) \cdot -2\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.05000000074505806) 1.0 (+ 1.0 (* (- 1.0 u) -2.0))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.05000000074505806f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + ((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 <= 0.05000000074505806e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 + ((1.0e0 - u) * (-2.0e0))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.05000000074505806))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + Float32(Float32(Float32(1.0) - u) * Float32(-2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.05000000074505806))
		tmp = single(1.0);
	else
		tmp = single(1.0) + ((single(1.0) - u) * single(-2.0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.0500000007

   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. Add Preprocessing
   5. Step-by-step derivation

      1. fma-udef 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in v around 0 92.9%

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

   if 0.0500000007 < v

   1. Initial program 94.0%

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

   3. Taylor expanded in v around inf 53.8%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.05000000074505806:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(1 - u\right) \cdot -2\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 89.5% accurate, 21.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.05000000074505806:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.05000000074505806) 1.0 (+ -1.0 (* u 2.0))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.05000000074505806f) {
		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 <= 0.05000000074505806e0) 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(0.05000000074505806))
		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(0.05000000074505806))
		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 < 0.0500000007

   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. Add Preprocessing
   5. Step-by-step derivation

      1. fma-udef 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in v around 0 92.9%

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

   if 0.0500000007 < v

   1. Initial program 94.0%

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

   3. Taylor expanded in v around inf 53.8%

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

   4. Taylor expanded in u around 0 53.8%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.05000000074505806:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 88.9% accurate, 35.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.05000000074505806:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v) :precision binary32 (if (<= v 0.05000000074505806) 1.0 -1.0))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.05000000074505806f) {
		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 <= 0.05000000074505806e0) 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(0.05000000074505806))
		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(0.05000000074505806))
		tmp = single(1.0);
	else
		tmp = single(-1.0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.0500000007

   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. Add Preprocessing
   5. Step-by-step derivation

      1. fma-udef 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in v around 0 92.9%

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

   if 0.0500000007 < v

   1. Initial program 94.0%

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

   3. Taylor expanded in u around 0 45.0%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.05000000074505806:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 6.0% accurate, 213.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

(FPCore (u v) :precision binary32 -1.0)

C

float code(float u, float v) {
	return -1.0f;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    code = -1.0e0
end function

Julia

function code(u, v)
	return Float32(-1.0)
end

MATLAB

function tmp = code(u, v)
	tmp = single(-1.0);
end

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in u around 0 6.8%

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

4. Final simplification 6.8%

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

Reproduce

herbie shell --seed 2024020 
(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))))))))