HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.4%

Time: 16.7s

Alternatives: 21

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: 99.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.4%

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

   1. *-commutative 99.4%

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

   2. add-log-exp 99.3%

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

   3. exp-to-pow 99.4%

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

   4. +-commutative 99.4%

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

   5. fma-undefine 99.4%

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

4. Applied egg-rr 99.4%

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

Alternative 2: 99.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(u, v)
	tmp = single(1.0) + log(((u + ((single(1.0) - u) * exp((single(-2.0) / v)))) ^ v));
end

Derivation

1. Initial program 99.4%

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

   1. *-commutative 99.4%

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

   2. add-log-exp 99.3%

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

   3. exp-to-pow 99.4%

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

   4. +-commutative 99.4%

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

   5. fma-undefine 99.4%

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

4. Applied egg-rr 99.4%

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

   1. fma-undefine 99.4%

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

6. Applied egg-rr 99.4%

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

7. Final simplification 99.4%

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

Alternative 3: 99.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.4%

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

   1. +-commutative 99.4%

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

   2. fma-define 99.4%

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

   3. +-commutative 99.4%

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

   4. fma-define 99.4%

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

3. Simplified 99.4%

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

   1. fma-undefine 99.4%

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

6. Applied egg-rr 99.4%

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

7. Final simplification 99.4%

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

Alternative 4: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(u, v)
	tmp = single(1.0) + (v * log((u + ((single(1.0) - u) * exp((single(-2.0) / v))))));
end

Derivation

1. Initial program 99.4%

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

Alternative 5: 96.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float u, float v) {
	return 1.0f + (v * logf((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 + exp(((-2.0e0) / v)))))
end function

Julia

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

MATLAB

function tmp = code(u, v)
	tmp = single(1.0) + (v * log((u + exp((single(-2.0) / v)))));
end

Derivation

1. Initial program 99.4%

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

   1. expm1-log1p-u 99.3%

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

4. Applied egg-rr 99.3%

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

   1. expm1-undefine 99.3%

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

   2. sub-neg 99.3%

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

   3. log1p-undefine 99.3%

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

   4. rem-exp-log 99.3%

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

   5. associate-+r- 99.3%

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

   6. metadata-eval 99.3%

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

   7. metadata-eval 99.3%

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

6. Simplified 99.3%

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

7. Taylor expanded in u around 0 96.5%

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

Alternative 6: 90.8% accurate, 1.8× speedup

Math

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

FPCore

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

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f;
	} else {
		tmp = (u * (v * (expf((2.0f / v)) + -1.0f))) + -1.0f;
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.20000000298023224e0) then
        tmp = 1.0e0
    else
        tmp = (u * (v * (exp((2.0e0 / v)) + (-1.0e0)))) + (-1.0e0)
    end if
    code = tmp
end function

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   1. Initial program 99.9%

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

      1. add-sqr-sqrt 100.0%

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

      2. log-prod 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-undefine 100.0%

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

      5. +-commutative 100.0%

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

      6. fma-undefine 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. count-2 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in v around 0 93.0%

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

   if 0.200000003 < v

   1. Initial program 91.5%

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

      1. expm1-log1p-u 90.9%

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

   4. Applied egg-rr 90.9%

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

      1. expm1-undefine 90.9%

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

      2. sub-neg 90.9%

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

      3. log1p-undefine 90.8%

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

      4. rem-exp-log 90.8%

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

      5. associate-+r- 90.9%

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

      6. metadata-eval 90.9%

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

      7. metadata-eval 90.9%

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

   6. Simplified 90.9%

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

   7. Taylor expanded in u around 0 75.9%

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

      1. rec-exp 75.9%

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

      2. expm1-define 75.9%

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

      3. distribute-neg-frac 75.9%

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

      4. metadata-eval 75.9%

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

      5. associate-*r/ 75.9%

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

      6. metadata-eval 75.9%

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

   9. Simplified 75.9%

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

   10. Taylor expanded in v around 0 76.5%

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

4. Final simplification 91.9%

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

Alternative 7: 90.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\mathsf{expm1}\left(\frac{2}{v}\right) \cdot \left(u \cdot v\right) - 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224)
   1.0
   (+ 1.0 (- (* (expm1 (/ 2.0 v)) (* u v)) 2.0))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + ((expm1f((2.0f / v)) * (u * v)) - 2.0f);
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.20000000298023224))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + Float32(Float32(expm1(Float32(Float32(2.0) / v)) * Float32(u * v)) - Float32(2.0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   1. Initial program 99.9%

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

      1. add-sqr-sqrt 100.0%

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

      2. log-prod 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-undefine 100.0%

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

      5. +-commutative 100.0%

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

      6. fma-undefine 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. count-2 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in v around 0 93.0%

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

   if 0.200000003 < v

   1. Initial program 91.5%

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

   3. Taylor expanded in u around 0 75.9%

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

      1. associate-*r* 75.9%

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

      2. sub-neg 75.9%

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

      3. metadata-eval 75.9%

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

      4. distribute-lft-in 75.9%

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

      5. rec-exp 75.9%

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

      6. distribute-neg-frac 75.9%

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

      7. metadata-eval 75.9%

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

   5. Applied egg-rr 75.9%

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

      1. distribute-lft-out 75.9%

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

      2. metadata-eval 75.9%

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

      3. sub-neg 75.9%

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

      4. expm1-undefine 75.9%

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

      5. *-commutative 75.9%

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

   7. Simplified 75.9%

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

Alternative 8: 90.7% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + v \cdot \left(\frac{\frac{u \cdot 2 - \frac{u \cdot -1.3333333333333333 + -0.6666666666666666 \cdot \frac{u}{v}}{v}}{v} - u \cdot -2}{v} - \frac{2}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224)
   1.0
   (+
    1.0
    (*
     v
     (-
      (/
       (-
        (/
         (-
          (* u 2.0)
          (/ (+ (* u -1.3333333333333333) (* -0.6666666666666666 (/ u v))) v))
         v)
        (* u -2.0))
       v)
      (/ 2.0 v))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + (v * ((((((u * 2.0f) - (((u * -1.3333333333333333f) + (-0.6666666666666666f * (u / v))) / v)) / v) - (u * -2.0f)) / v) - (2.0f / 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 <= 0.20000000298023224e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 + (v * ((((((u * 2.0e0) - (((u * (-1.3333333333333333e0)) + ((-0.6666666666666666e0) * (u / v))) / v)) / v) - (u * (-2.0e0))) / v) - (2.0e0 / v)))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.20000000298023224))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + Float32(v * Float32(Float32(Float32(Float32(Float32(Float32(u * Float32(2.0)) - Float32(Float32(Float32(u * Float32(-1.3333333333333333)) + Float32(Float32(-0.6666666666666666) * Float32(u / v))) / v)) / v) - Float32(u * Float32(-2.0))) / v) - Float32(Float32(2.0) / v))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.20000000298023224))
		tmp = single(1.0);
	else
		tmp = single(1.0) + (v * ((((((u * single(2.0)) - (((u * single(-1.3333333333333333)) + (single(-0.6666666666666666) * (u / v))) / v)) / v) - (u * single(-2.0))) / v) - (single(2.0) / v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   1. Initial program 99.9%

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

      1. add-sqr-sqrt 100.0%

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

      2. log-prod 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-undefine 100.0%

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

      5. +-commutative 100.0%

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

      6. fma-undefine 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. count-2 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in v around 0 93.0%

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

   if 0.200000003 < v

   1. Initial program 91.5%

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

      1. expm1-log1p-u 90.9%

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

   4. Applied egg-rr 90.9%

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

      1. expm1-undefine 90.9%

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

      2. sub-neg 90.9%

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

      3. log1p-undefine 90.8%

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

      4. rem-exp-log 90.8%

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

      5. associate-+r- 90.9%

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

      6. metadata-eval 90.9%

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

      7. metadata-eval 90.9%

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

   6. Simplified 90.9%

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

   7. Taylor expanded in u around 0 75.9%

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

      1. rec-exp 75.9%

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

      2. expm1-define 75.9%

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

      3. distribute-neg-frac 75.9%

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

      4. metadata-eval 75.9%

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

      5. associate-*r/ 75.9%

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

      6. metadata-eval 75.9%

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

   9. Simplified 75.9%

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

   10. Taylor expanded in v around -inf 72.3%

       \[\leadsto 1 + v \cdot \left(\color{blue}{-1 \cdot \frac{-2 \cdot u + -1 \cdot \frac{-1 \cdot \frac{-1.3333333333333333 \cdot u + -0.6666666666666666 \cdot \frac{u}{v}}{v} + 2 \cdot u}{v}}{v}} - \frac{2}{v}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + v \cdot \left(\frac{\frac{u \cdot 2 - \frac{u \cdot -1.3333333333333333 + -0.6666666666666666 \cdot \frac{u}{v}}{v}}{v} - u \cdot -2}{v} - \frac{2}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 90.7% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{\frac{\frac{u}{v} \cdot 0.6666666666666666 + u \cdot 1.3333333333333333}{v} - u \cdot -2}{v} - \left(2 + u \cdot -2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224)
   1.0
   (+
    1.0
    (-
     (/
      (-
       (/ (+ (* (/ u v) 0.6666666666666666) (* u 1.3333333333333333)) v)
       (* u -2.0))
      v)
     (+ 2.0 (* u -2.0))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + (((((((u / v) * 0.6666666666666666f) + (u * 1.3333333333333333f)) / v) - (u * -2.0f)) / 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 <= 0.20000000298023224e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 + (((((((u / v) * 0.6666666666666666e0) + (u * 1.3333333333333333e0)) / v) - (u * (-2.0e0))) / v) - (2.0e0 + (u * (-2.0e0))))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.20000000298023224))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + Float32(Float32(Float32(Float32(Float32(Float32(Float32(u / v) * Float32(0.6666666666666666)) + Float32(u * Float32(1.3333333333333333))) / v) - Float32(u * Float32(-2.0))) / 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(0.20000000298023224))
		tmp = single(1.0);
	else
		tmp = single(1.0) + (((((((u / v) * single(0.6666666666666666)) + (u * single(1.3333333333333333))) / v) - (u * single(-2.0))) / v) - (single(2.0) + (u * single(-2.0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   1. Initial program 99.9%

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

      1. add-sqr-sqrt 100.0%

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

      2. log-prod 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-undefine 100.0%

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

      5. +-commutative 100.0%

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

      6. fma-undefine 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. count-2 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in v around 0 93.0%

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

   if 0.200000003 < v

   1. Initial program 91.5%

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

      1. expm1-log1p-u 90.9%

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

   4. Applied egg-rr 90.9%

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

      1. expm1-undefine 90.9%

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

      2. sub-neg 90.9%

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

      3. log1p-undefine 90.8%

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

      4. rem-exp-log 90.8%

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

      5. associate-+r- 90.9%

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

      6. metadata-eval 90.9%

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

      7. metadata-eval 90.9%

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

   6. Simplified 90.9%

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

   7. Taylor expanded in u around 0 75.9%

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

      1. rec-exp 75.9%

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

      2. expm1-define 75.9%

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

      3. distribute-neg-frac 75.9%

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

      4. metadata-eval 75.9%

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

      5. associate-*r/ 75.9%

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

      6. metadata-eval 75.9%

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

   9. Simplified 75.9%

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

   10. Taylor expanded in v around -inf 72.3%

       \[\leadsto 1 + \color{blue}{\left(-1 \cdot \left(2 + -2 \cdot u\right) + -1 \cdot \frac{-2 \cdot u + -1 \cdot \frac{0.6666666666666666 \cdot \frac{u}{v} + 1.3333333333333333 \cdot u}{v}}{v}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{\frac{\frac{u}{v} \cdot 0.6666666666666666 + u \cdot 1.3333333333333333}{v} - u \cdot -2}{v} - \left(2 + u \cdot -2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 90.7% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(u \cdot \left(2 + \frac{2 + \frac{1.3333333333333333 + 0.6666666666666666 \cdot \frac{1}{v}}{v}}{v}\right) - 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224)
   1.0
   (+
    1.0
    (-
     (*
      u
      (+
       2.0
       (/
        (+ 2.0 (/ (+ 1.3333333333333333 (* 0.6666666666666666 (/ 1.0 v))) v))
        v)))
     2.0))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + ((u * (2.0f + ((2.0f + ((1.3333333333333333f + (0.6666666666666666f * (1.0f / v))) / v)) / 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 <= 0.20000000298023224e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 + ((u * (2.0e0 + ((2.0e0 + ((1.3333333333333333e0 + (0.6666666666666666e0 * (1.0e0 / v))) / v)) / v))) - 2.0e0)
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.20000000298023224))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + Float32(Float32(u * Float32(Float32(2.0) + Float32(Float32(Float32(2.0) + Float32(Float32(Float32(1.3333333333333333) + Float32(Float32(0.6666666666666666) * Float32(Float32(1.0) / v))) / v)) / v))) - Float32(2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.20000000298023224))
		tmp = single(1.0);
	else
		tmp = single(1.0) + ((u * (single(2.0) + ((single(2.0) + ((single(1.3333333333333333) + (single(0.6666666666666666) * (single(1.0) / v))) / v)) / v))) - single(2.0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   1. Initial program 99.9%

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

      1. add-sqr-sqrt 100.0%

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

      2. log-prod 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-undefine 100.0%

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

      5. +-commutative 100.0%

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

      6. fma-undefine 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. count-2 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in v around 0 93.0%

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

   if 0.200000003 < v

   1. Initial program 91.5%

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

   3. Taylor expanded in u around 0 75.9%

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

   4. Taylor expanded in v around -inf 72.3%

      \[\leadsto 1 + \left(u \cdot \color{blue}{\left(2 + -1 \cdot \frac{-1 \cdot \frac{1.3333333333333333 + 0.6666666666666666 \cdot \frac{1}{v}}{v} - 2}{v}\right)} - 2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(u \cdot \left(2 + \frac{2 + \frac{1.3333333333333333 + 0.6666666666666666 \cdot \frac{1}{v}}{v}}{v}\right) - 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 90.6% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(u \cdot \left(v \cdot \left(\frac{2}{v} + \frac{\frac{2 + \frac{1.3333333333333333}{v}}{v}}{v}\right)\right) - 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224)
   1.0
   (+
    1.0
    (-
     (* u (* v (+ (/ 2.0 v) (/ (/ (+ 2.0 (/ 1.3333333333333333 v)) v) v))))
     2.0))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + ((u * (v * ((2.0f / v) + (((2.0f + (1.3333333333333333f / v)) / v) / 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 <= 0.20000000298023224e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 + ((u * (v * ((2.0e0 / v) + (((2.0e0 + (1.3333333333333333e0 / v)) / v) / v)))) - 2.0e0)
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.20000000298023224))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + Float32(Float32(u * Float32(v * Float32(Float32(Float32(2.0) / v) + Float32(Float32(Float32(Float32(2.0) + Float32(Float32(1.3333333333333333) / v)) / v) / v)))) - Float32(2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.20000000298023224))
		tmp = single(1.0);
	else
		tmp = single(1.0) + ((u * (v * ((single(2.0) / v) + (((single(2.0) + (single(1.3333333333333333) / v)) / v) / v)))) - single(2.0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   1. Initial program 99.9%

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

      1. add-sqr-sqrt 100.0%

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

      2. log-prod 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-undefine 100.0%

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

      5. +-commutative 100.0%

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

      6. fma-undefine 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. count-2 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in v around 0 93.0%

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

   if 0.200000003 < v

   1. Initial program 91.5%

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

   3. Taylor expanded in u around 0 75.9%

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

   4. Taylor expanded in v around -inf 69.2%

      \[\leadsto 1 + \left(u \cdot \left(v \cdot \color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{2 + 1.3333333333333333 \cdot \frac{1}{v}}{v} - 2}{v}\right)}\right) - 2\right) \]
   5. Step-by-step derivation

      1. div-sub 69.3%

         \[\leadsto 1 + \left(u \cdot \left(v \cdot \left(-1 \cdot \color{blue}{\left(\frac{-1 \cdot \frac{2 + 1.3333333333333333 \cdot \frac{1}{v}}{v}}{v} - \frac{2}{v}\right)}\right)\right) - 2\right) \]

      2. mul-1-neg 69.3%

         \[\leadsto 1 + \left(u \cdot \left(v \cdot \left(-1 \cdot \left(\frac{\color{blue}{-\frac{2 + 1.3333333333333333 \cdot \frac{1}{v}}{v}}}{v} - \frac{2}{v}\right)\right)\right) - 2\right) \]

      3. un-div-inv 69.3%

         \[\leadsto 1 + \left(u \cdot \left(v \cdot \left(-1 \cdot \left(\frac{-\frac{2 + \color{blue}{\frac{1.3333333333333333}{v}}}{v}}{v} - \frac{2}{v}\right)\right)\right) - 2\right) \]

   6. Applied egg-rr 69.3%

      \[\leadsto 1 + \left(u \cdot \left(v \cdot \left(-1 \cdot \color{blue}{\left(\frac{-\frac{2 + \frac{1.3333333333333333}{v}}{v}}{v} - \frac{2}{v}\right)}\right)\right) - 2\right) \]
   7. Step-by-step derivation

      1. distribute-neg-frac2 69.3%

         \[\leadsto 1 + \left(u \cdot \left(v \cdot \left(-1 \cdot \left(\frac{\color{blue}{\frac{2 + \frac{1.3333333333333333}{v}}{-v}}}{v} - \frac{2}{v}\right)\right)\right) - 2\right) \]

   8. Simplified 69.3%

      \[\leadsto 1 + \left(u \cdot \left(v \cdot \left(-1 \cdot \color{blue}{\left(\frac{\frac{2 + \frac{1.3333333333333333}{v}}{-v}}{v} - \frac{2}{v}\right)}\right)\right) - 2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(u \cdot \left(v \cdot \left(\frac{2}{v} + \frac{\frac{2 + \frac{1.3333333333333333}{v}}{v}}{v}\right)\right) - 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 90.6% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\left(2 + u \cdot -2\right) + \frac{u \cdot -2 + -1.3333333333333333 \cdot \frac{u}{v}}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224)
   1.0
   (-
    1.0
    (+
     (+ 2.0 (* u -2.0))
     (/ (+ (* u -2.0) (* -1.3333333333333333 (/ u v))) v)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f - ((2.0f + (u * -2.0f)) + (((u * -2.0f) + (-1.3333333333333333f * (u / v))) / 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 <= 0.20000000298023224e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 - ((2.0e0 + (u * (-2.0e0))) + (((u * (-2.0e0)) + ((-1.3333333333333333e0) * (u / v))) / v))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.20000000298023224))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) - Float32(Float32(Float32(2.0) + Float32(u * Float32(-2.0))) + Float32(Float32(Float32(u * Float32(-2.0)) + Float32(Float32(-1.3333333333333333) * Float32(u / v))) / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.20000000298023224))
		tmp = single(1.0);
	else
		tmp = single(1.0) - ((single(2.0) + (u * single(-2.0))) + (((u * single(-2.0)) + (single(-1.3333333333333333) * (u / v))) / v));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   1. Initial program 99.9%

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

      1. add-sqr-sqrt 100.0%

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

      2. log-prod 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-undefine 100.0%

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

      5. +-commutative 100.0%

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

      6. fma-undefine 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. count-2 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in v around 0 93.0%

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

   if 0.200000003 < v

   1. Initial program 91.5%

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

      1. expm1-log1p-u 90.9%

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

   4. Applied egg-rr 90.9%

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

      1. expm1-undefine 90.9%

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

      2. sub-neg 90.9%

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

      3. log1p-undefine 90.8%

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

      4. rem-exp-log 90.8%

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

      5. associate-+r- 90.9%

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

      6. metadata-eval 90.9%

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

      7. metadata-eval 90.9%

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

   6. Simplified 90.9%

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

   7. Taylor expanded in u around 0 75.9%

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

      1. rec-exp 75.9%

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

      2. expm1-define 75.9%

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

      3. distribute-neg-frac 75.9%

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

      4. metadata-eval 75.9%

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

      5. associate-*r/ 75.9%

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

      6. metadata-eval 75.9%

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

   9. Simplified 75.9%

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

   10. Taylor expanded in v around -inf 69.3%

       \[\leadsto 1 + \color{blue}{\left(-1 \cdot \left(2 + -2 \cdot u\right) + -1 \cdot \frac{-2 \cdot u + -1.3333333333333333 \cdot \frac{u}{v}}{v}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\left(2 + u \cdot -2\right) + \frac{u \cdot -2 + -1.3333333333333333 \cdot \frac{u}{v}}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 90.6% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(u \cdot 2 - \frac{u \cdot -2 + -1.3333333333333333 \cdot \frac{u}{v}}{v}\right) - 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224)
   1.0
   (+
    1.0
    (-
     (- (* u 2.0) (/ (+ (* u -2.0) (* -1.3333333333333333 (/ u v))) v))
     2.0))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + (((u * 2.0f) - (((u * -2.0f) + (-1.3333333333333333f * (u / v))) / 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 <= 0.20000000298023224e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 + (((u * 2.0e0) - (((u * (-2.0e0)) + ((-1.3333333333333333e0) * (u / v))) / v)) - 2.0e0)
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.20000000298023224))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + Float32(Float32(Float32(u * Float32(2.0)) - Float32(Float32(Float32(u * Float32(-2.0)) + Float32(Float32(-1.3333333333333333) * Float32(u / v))) / v)) - Float32(2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.20000000298023224))
		tmp = single(1.0);
	else
		tmp = single(1.0) + (((u * single(2.0)) - (((u * single(-2.0)) + (single(-1.3333333333333333) * (u / v))) / v)) - single(2.0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   1. Initial program 99.9%

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

      1. add-sqr-sqrt 100.0%

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

      2. log-prod 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-undefine 100.0%

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

      5. +-commutative 100.0%

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

      6. fma-undefine 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. count-2 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in v around 0 93.0%

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

   if 0.200000003 < v

   1. Initial program 91.5%

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

   3. Taylor expanded in u around 0 75.9%

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

   4. Taylor expanded in v around -inf 69.2%

      \[\leadsto 1 + \left(\color{blue}{\left(-1 \cdot \frac{-2 \cdot u + -1.3333333333333333 \cdot \frac{u}{v}}{v} + 2 \cdot u\right)} - 2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(u \cdot 2 - \frac{u \cdot -2 + -1.3333333333333333 \cdot \frac{u}{v}}{v}\right) - 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 90.6% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(u \cdot \left(2 + \frac{2 + 1.3333333333333333 \cdot \frac{1}{v}}{v}\right) - 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224)
   1.0
   (+
    1.0
    (- (* u (+ 2.0 (/ (+ 2.0 (* 1.3333333333333333 (/ 1.0 v))) v))) 2.0))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + ((u * (2.0f + ((2.0f + (1.3333333333333333f * (1.0f / v))) / 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 <= 0.20000000298023224e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 + ((u * (2.0e0 + ((2.0e0 + (1.3333333333333333e0 * (1.0e0 / v))) / v))) - 2.0e0)
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.20000000298023224))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + Float32(Float32(u * Float32(Float32(2.0) + Float32(Float32(Float32(2.0) + Float32(Float32(1.3333333333333333) * Float32(Float32(1.0) / v))) / v))) - Float32(2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.20000000298023224))
		tmp = single(1.0);
	else
		tmp = single(1.0) + ((u * (single(2.0) + ((single(2.0) + (single(1.3333333333333333) * (single(1.0) / v))) / v))) - single(2.0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   1. Initial program 99.9%

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

      1. add-sqr-sqrt 100.0%

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

      2. log-prod 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-undefine 100.0%

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

      5. +-commutative 100.0%

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

      6. fma-undefine 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. count-2 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in v around 0 93.0%

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

   if 0.200000003 < v

   1. Initial program 91.5%

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

   3. Taylor expanded in u around 0 75.9%

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

   4. Taylor expanded in v around -inf 69.2%

      \[\leadsto 1 + \left(u \cdot \left(v \cdot \color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{2 + 1.3333333333333333 \cdot \frac{1}{v}}{v} - 2}{v}\right)}\right) - 2\right) \]

   5. Taylor expanded in u around 0 69.2%

      \[\leadsto 1 + \left(\color{blue}{-1 \cdot \left(u \cdot \left(-1 \cdot \frac{2 + 1.3333333333333333 \cdot \frac{1}{v}}{v} - 2\right)\right)} - 2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(u \cdot \left(2 + \frac{2 + 1.3333333333333333 \cdot \frac{1}{v}}{v}\right) - 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 90.3% accurate, 13.3× speedup

Math

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

FPCore

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

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		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 <= 0.20000000298023224e0) 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(0.20000000298023224))
		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(0.20000000298023224))
		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 < 0.200000003

   1. Initial program 99.9%

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

      1. add-sqr-sqrt 100.0%

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

      2. log-prod 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-undefine 100.0%

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

      5. +-commutative 100.0%

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

      6. fma-undefine 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. count-2 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in v around 0 93.0%

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

   if 0.200000003 < v

   1. Initial program 91.5%

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

   3. Taylor expanded in u around 0 75.9%

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

   4. Taylor expanded in v around inf 65.1%

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

      1. distribute-lft-out 65.1%

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

   6. Simplified 65.1%

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

Alternative 16: 90.3% accurate, 15.2× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.20000000298023224e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + (2.0e0 * (u + (u / v)))
    end if
    code = tmp
end function

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   1. Initial program 99.9%

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

      1. add-sqr-sqrt 100.0%

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

      2. log-prod 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-undefine 100.0%

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

      5. +-commutative 100.0%

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

      6. fma-undefine 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. count-2 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in v around 0 93.0%

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

   if 0.200000003 < v

   1. Initial program 91.5%

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

   3. Taylor expanded in u around 0 75.9%

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

   4. Taylor expanded in v around -inf 62.7%

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

      1. associate-*r/ 62.7%

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

      2. cancel-sign-sub-inv 62.7%

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

      3. metadata-eval 62.7%

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

      4. associate-*r/ 62.7%

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

      5. metadata-eval 62.7%

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

      6. distribute-lft-in 62.7%

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

      7. metadata-eval 62.7%

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

      8. neg-mul-1 62.7%

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

      9. distribute-neg-frac 62.7%

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

      10. metadata-eval 62.7%

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

   6. Simplified 62.7%

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

   7. Taylor expanded in v around inf 64.8%

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

      1. sub-neg 64.8%

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

      2. distribute-lft-out 64.8%

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

      3. metadata-eval 64.8%

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

   9. Simplified 64.8%

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

4. Final simplification 91.1%

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

Alternative 17: 89.8% accurate, 15.2× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.20000000298023224e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 + (u * (2.0e0 - (2.0e0 / u)))
    end if
    code = tmp
end function

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   1. Initial program 99.9%

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

      1. add-sqr-sqrt 100.0%

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

      2. log-prod 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-undefine 100.0%

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

      5. +-commutative 100.0%

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

      6. fma-undefine 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. count-2 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in v around 0 93.0%

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

   if 0.200000003 < v

   1. Initial program 91.5%

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

   3. Taylor expanded in u around 0 75.9%

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

   4. Taylor expanded in v around inf 56.9%

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

      1. *-commutative 56.9%

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

   6. Simplified 56.9%

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

   7. Taylor expanded in u around inf 56.9%

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

      1. associate-*r/ 56.9%

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

      2. metadata-eval 56.9%

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

   9. Simplified 56.9%

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

Alternative 18: 89.8% accurate, 17.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.20000000298023224e0) then
        tmp = 1.0e0
    else
        tmp = u * (2.0e0 + ((-1.0e0) / u))
    end if
    code = tmp
end function

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   1. Initial program 99.9%

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

      1. add-sqr-sqrt 100.0%

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

      2. log-prod 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-undefine 100.0%

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

      5. +-commutative 100.0%

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

      6. fma-undefine 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. count-2 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in v around 0 93.0%

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

   if 0.200000003 < v

   1. Initial program 91.5%

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

   3. Taylor expanded in u around 0 75.9%

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

   4. Taylor expanded in v around inf 56.9%

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

      1. *-commutative 56.9%

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

   6. Simplified 56.9%

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

   7. Taylor expanded in u around inf 56.9%

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

4. Final simplification 90.6%

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

Alternative 19: 89.8% accurate, 21.2× speedup

Math

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

FPCore

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

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		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 <= 0.20000000298023224e0) 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(0.20000000298023224))
		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(0.20000000298023224))
		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 < 0.200000003

   1. Initial program 99.9%

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

      1. add-sqr-sqrt 100.0%

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

      2. log-prod 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-undefine 100.0%

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

      5. +-commutative 100.0%

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

      6. fma-undefine 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. count-2 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in v around 0 93.0%

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

   if 0.200000003 < v

   1. Initial program 91.5%

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

   3. Taylor expanded in u around 0 75.9%

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

   4. Taylor expanded in v around inf 56.9%

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

      1. *-commutative 56.9%

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

   6. Simplified 56.9%

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

   7. Taylor expanded in u around 0 56.9%

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

4. Final simplification 90.6%

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

Alternative 20: 86.7% accurate, 213.0× speedup

Math

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

FPCore

(FPCore (u v) :precision binary32 1.0)

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

   1. add-sqr-sqrt 99.3%

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

   2. log-prod 99.4%

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

   3. +-commutative 99.4%

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

   4. fma-undefine 99.4%

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

   5. +-commutative 99.4%

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

   6. fma-undefine 99.4%

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

4. Applied egg-rr 99.4%

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

   1. count-2 99.4%

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

6. Simplified 99.4%

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

7. Taylor expanded in v around 0 87.0%

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

Alternative 21: 5.8% accurate, 213.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in u around 0 6.2%

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

Reproduce

herbie shell --seed 2024110 
(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))))))))