HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.5%

Time: 12.2s

Alternatives: 15

Speedup: 1.0×

Specification

\[\left(10^{-5} \leq u \land u \leq 1\right) \land \left(0 \leq v \land v \leq 109.746574\right)\]

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.5% accurate, 0.7× speedup

Math

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

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 99.5%

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

   1. +-commutative 99.5%

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

   2. fma-define 99.6%

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

   3. +-commutative 99.6%

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

   4. fma-define 99.5%

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

3. Simplified 99.5%

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

   1. fma-undefine 99.6%

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

6. Applied egg-rr 99.6%

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

7. Final simplification 99.6%

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

Alternative 2: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

Alternative 3: 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 99.5%

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

3. Taylor expanded in u around 0 96.6%

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

Alternative 4: 97.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.4000000059604645:\\ \;\;\;\;1 + v \cdot \log u\\ \mathbf{else}:\\ \;\;\;\;-1 - u \cdot \left(v \cdot \left(\frac{-1}{e^{\frac{-2}{v}}} - -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.4000000059604645)
   (+ 1.0 (* v (log u)))
   (- -1.0 (* u (* v (- (/ -1.0 (exp (/ -2.0 v))) -1.0))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.4000000059604645f) {
		tmp = 1.0f + (v * logf(u));
	} else {
		tmp = -1.0f - (u * (v * ((-1.0f / expf((-2.0f / v))) - -1.0f)));
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.4000000059604645e0) then
        tmp = 1.0e0 + (v * log(u))
    else
        tmp = (-1.0e0) - (u * (v * (((-1.0e0) / exp(((-2.0e0) / v))) - (-1.0e0))))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.4000000059604645))
		tmp = Float32(Float32(1.0) + Float32(v * log(u)));
	else
		tmp = Float32(Float32(-1.0) - Float32(u * Float32(v * Float32(Float32(Float32(-1.0) / exp(Float32(Float32(-2.0) / v))) - Float32(-1.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.4000000059604645))
		tmp = single(1.0) + (v * log(u));
	else
		tmp = single(-1.0) - (u * (v * ((single(-1.0) / exp((single(-2.0) / v))) - single(-1.0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.400000006

   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-exp-log 99.9%

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

      2. *-commutative 99.9%

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

      3. log-prod 99.9%

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

      4. add-log-exp 99.9%

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

      5. sub-neg 99.9%

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

      6. log1p-define 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in u around inf 99.2%

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

      1. mul-1-neg 99.2%

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

      2. distribute-rgt-neg-in 99.2%

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

      3. log-rec 99.2%

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

      4. remove-double-neg 99.2%

         \[\leadsto 1 + v \cdot \color{blue}{\log u} \]

   7. Simplified 99.2%

      \[\leadsto 1 + \color{blue}{v \cdot \log u} \]

   if 0.400000006 < v

   1. Initial program 93.6%

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

      1. +-commutative 93.6%

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

      2. fma-define 93.5%

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

      3. +-commutative 93.5%

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

      4. fma-define 92.7%

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

   3. Simplified 92.7%

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

   5. Taylor expanded in u around 0 82.4%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.4000000059604645:\\ \;\;\;\;1 + v \cdot \log u\\ \mathbf{else}:\\ \;\;\;\;-1 - u \cdot \left(v \cdot \left(\frac{-1}{e^{\frac{-2}{v}}} - -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.4000000059604645:\\ \;\;\;\;1 + v \cdot \log u\\ \mathbf{else}:\\ \;\;\;\;v \cdot \left(u \cdot \mathsf{expm1}\left(\frac{2}{v}\right)\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.4000000059604645)
   (+ 1.0 (* v (log u)))
   (+ (* v (* u (expm1 (/ 2.0 v)))) -1.0)))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.4000000059604645f) {
		tmp = 1.0f + (v * logf(u));
	} else {
		tmp = (v * (u * expm1f((2.0f / v)))) + -1.0f;
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.4000000059604645))
		tmp = Float32(Float32(1.0) + Float32(v * log(u)));
	else
		tmp = Float32(Float32(v * Float32(u * expm1(Float32(Float32(2.0) / v)))) + Float32(-1.0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.400000006

   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-exp-log 99.9%

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

      2. *-commutative 99.9%

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

      3. log-prod 99.9%

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

      4. add-log-exp 99.9%

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

      5. sub-neg 99.9%

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

      6. log1p-define 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in u around inf 99.2%

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

      1. mul-1-neg 99.2%

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

      2. distribute-rgt-neg-in 99.2%

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

      3. log-rec 99.2%

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

      4. remove-double-neg 99.2%

         \[\leadsto 1 + v \cdot \color{blue}{\log u} \]

   7. Simplified 99.2%

      \[\leadsto 1 + \color{blue}{v \cdot \log u} \]

   if 0.400000006 < v

   1. Initial program 93.6%

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

   3. Taylor expanded in u around 0 81.5%

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

      1. +-commutative 81.5%

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

      2. associate-+l- 82.4%

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

      3. *-commutative 82.4%

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

      4. associate-*r* 82.4%

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

      5. rec-exp 82.4%

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

      6. expm1-define 82.4%

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

      7. distribute-neg-frac 82.4%

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

      8. metadata-eval 82.4%

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

      9. metadata-eval 82.4%

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

   5. Applied egg-rr 82.4%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.4000000059604645:\\ \;\;\;\;1 + v \cdot \log u\\ \mathbf{else}:\\ \;\;\;\;v \cdot \left(u \cdot \mathsf{expm1}\left(\frac{2}{v}\right)\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.4000000059604645:\\ \;\;\;\;1 + v \cdot \log u\\ \mathbf{else}:\\ \;\;\;\;-1 - \left(\frac{u \cdot -2 - \frac{0.6666666666666666 \cdot \frac{u}{v} + u \cdot 1.3333333333333333}{v}}{v} - u \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.4000000059604645)
   (+ 1.0 (* v (log u)))
   (-
    -1.0
    (-
     (/
      (-
       (* u -2.0)
       (/ (+ (* 0.6666666666666666 (/ u v)) (* u 1.3333333333333333)) v))
      v)
     (* u 2.0)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.4000000059604645f) {
		tmp = 1.0f + (v * logf(u));
	} else {
		tmp = -1.0f - ((((u * -2.0f) - (((0.6666666666666666f * (u / v)) + (u * 1.3333333333333333f)) / v)) / v) - (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 <= 0.4000000059604645e0) then
        tmp = 1.0e0 + (v * log(u))
    else
        tmp = (-1.0e0) - ((((u * (-2.0e0)) - (((0.6666666666666666e0 * (u / v)) + (u * 1.3333333333333333e0)) / v)) / v) - (u * 2.0e0))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.4000000059604645))
		tmp = Float32(Float32(1.0) + Float32(v * log(u)));
	else
		tmp = Float32(Float32(-1.0) - Float32(Float32(Float32(Float32(u * Float32(-2.0)) - Float32(Float32(Float32(Float32(0.6666666666666666) * Float32(u / v)) + Float32(u * Float32(1.3333333333333333))) / v)) / v) - Float32(u * Float32(2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.4000000059604645))
		tmp = single(1.0) + (v * log(u));
	else
		tmp = single(-1.0) - ((((u * single(-2.0)) - (((single(0.6666666666666666) * (u / v)) + (u * single(1.3333333333333333))) / v)) / v) - (u * single(2.0)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.400000006

   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-exp-log 99.9%

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

      2. *-commutative 99.9%

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

      3. log-prod 99.9%

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

      4. add-log-exp 99.9%

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

      5. sub-neg 99.9%

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

      6. log1p-define 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in u around inf 99.2%

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

      1. mul-1-neg 99.2%

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

      2. distribute-rgt-neg-in 99.2%

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

      3. log-rec 99.2%

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

      4. remove-double-neg 99.2%

         \[\leadsto 1 + v \cdot \color{blue}{\log u} \]

   7. Simplified 99.2%

      \[\leadsto 1 + \color{blue}{v \cdot \log u} \]

   if 0.400000006 < v

   1. Initial program 93.6%

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

   3. Taylor expanded in u around 0 81.5%

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

   4. Taylor expanded in v around -inf 78.9%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.4000000059604645:\\ \;\;\;\;1 + v \cdot \log u\\ \mathbf{else}:\\ \;\;\;\;-1 - \left(\frac{u \cdot -2 - \frac{0.6666666666666666 \cdot \frac{u}{v} + u \cdot 1.3333333333333333}{v}}{v} - u \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 90.8% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.46000000834465027:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 - \left(\frac{u \cdot -2 - \frac{0.6666666666666666 \cdot \frac{u}{v} + u \cdot 1.3333333333333333}{v}}{v} - u \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.46000000834465027)
   1.0
   (-
    -1.0
    (-
     (/
      (-
       (* u -2.0)
       (/ (+ (* 0.6666666666666666 (/ u v)) (* u 1.3333333333333333)) v))
      v)
     (* u 2.0)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.46000000834465027f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f - ((((u * -2.0f) - (((0.6666666666666666f * (u / v)) + (u * 1.3333333333333333f)) / v)) / v) - (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 <= 0.46000000834465027e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) - ((((u * (-2.0e0)) - (((0.6666666666666666e0 * (u / v)) + (u * 1.3333333333333333e0)) / v)) / v) - (u * 2.0e0))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.46000000834465027))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) - Float32(Float32(Float32(Float32(u * Float32(-2.0)) - Float32(Float32(Float32(Float32(0.6666666666666666) * Float32(u / v)) + Float32(u * Float32(1.3333333333333333))) / v)) / v) - Float32(u * Float32(2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.46000000834465027))
		tmp = single(1.0);
	else
		tmp = single(-1.0) - ((((u * single(-2.0)) - (((single(0.6666666666666666) * (u / v)) + (u * single(1.3333333333333333))) / v)) / v) - (u * single(2.0)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.460000008

   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-exp-log 99.9%

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

      2. *-commutative 99.9%

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

      3. log-prod 99.9%

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

      4. add-log-exp 99.9%

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

      5. sub-neg 99.9%

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

      6. log1p-define 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in v around 0 91.9%

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

   if 0.460000008 < v

   1. Initial program 94.0%

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

   3. Taylor expanded in u around 0 86.0%

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

   4. Taylor expanded in v around -inf 83.1%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.46000000834465027:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 - \left(\frac{u \cdot -2 - \frac{0.6666666666666666 \cdot \frac{u}{v} + u \cdot 1.3333333333333333}{v}}{v} - u \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 90.7% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.46000000834465027:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(u \cdot \left(2 - \frac{-2 + \frac{-1.3333333333333333 + \frac{-0.6666666666666666}{v}}{v}}{v}\right) - 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.46000000834465027)
   1.0
   (+
    1.0
    (-
     (*
      u
      (-
       2.0
       (/ (+ -2.0 (/ (+ -1.3333333333333333 (/ -0.6666666666666666 v)) v)) v)))
     2.0))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.46000000834465027f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + ((u * (2.0f - ((-2.0f + ((-1.3333333333333333f + (-0.6666666666666666f / 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 <= 0.46000000834465027e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 + ((u * (2.0e0 - (((-2.0e0) + (((-1.3333333333333333e0) + ((-0.6666666666666666e0) / v)) / v)) / v))) - 2.0e0)
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.46000000834465027))
		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) / v)) / v)) / v))) - Float32(2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.46000000834465027))
		tmp = single(1.0);
	else
		tmp = single(1.0) + ((u * (single(2.0) - ((single(-2.0) + ((single(-1.3333333333333333) + (single(-0.6666666666666666) / v)) / v)) / v))) - single(2.0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.460000008

   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-exp-log 99.9%

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

      2. *-commutative 99.9%

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

      3. log-prod 99.9%

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

      4. add-log-exp 99.9%

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

      5. sub-neg 99.9%

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

      6. log1p-define 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in v around 0 91.9%

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

   if 0.460000008 < v

   1. Initial program 94.0%

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

   3. Taylor expanded in u around 0 86.0%

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

   4. Taylor expanded in v around -inf 81.9%

      \[\leadsto 1 + \left(u \cdot \color{blue}{\left(2 + -1 \cdot \frac{-1 \cdot \frac{1.3333333333333333 + 0.6666666666666666 \cdot \frac{1}{v}}{v} - 2}{v}\right)} - 2\right) \]
   5. Step-by-step derivation

      1. mul-1-neg 81.9%

         \[\leadsto 1 + \left(u \cdot \left(2 + \color{blue}{\left(-\frac{-1 \cdot \frac{1.3333333333333333 + 0.6666666666666666 \cdot \frac{1}{v}}{v} - 2}{v}\right)}\right) - 2\right) \]

      2. unsub-neg 81.9%

         \[\leadsto 1 + \left(u \cdot \color{blue}{\left(2 - \frac{-1 \cdot \frac{1.3333333333333333 + 0.6666666666666666 \cdot \frac{1}{v}}{v} - 2}{v}\right)} - 2\right) \]

      3. sub-neg 81.9%

         \[\leadsto 1 + \left(u \cdot \left(2 - \frac{\color{blue}{-1 \cdot \frac{1.3333333333333333 + 0.6666666666666666 \cdot \frac{1}{v}}{v} + \left(-2\right)}}{v}\right) - 2\right) \]

      4. associate-*r/ 81.9%

         \[\leadsto 1 + \left(u \cdot \left(2 - \frac{\color{blue}{\frac{-1 \cdot \left(1.3333333333333333 + 0.6666666666666666 \cdot \frac{1}{v}\right)}{v}} + \left(-2\right)}{v}\right) - 2\right) \]

      5. distribute-lft-in 81.9%

         \[\leadsto 1 + \left(u \cdot \left(2 - \frac{\frac{\color{blue}{-1 \cdot 1.3333333333333333 + -1 \cdot \left(0.6666666666666666 \cdot \frac{1}{v}\right)}}{v} + \left(-2\right)}{v}\right) - 2\right) \]

      6. metadata-eval 81.9%

         \[\leadsto 1 + \left(u \cdot \left(2 - \frac{\frac{\color{blue}{-1.3333333333333333} + -1 \cdot \left(0.6666666666666666 \cdot \frac{1}{v}\right)}{v} + \left(-2\right)}{v}\right) - 2\right) \]

      7. neg-mul-1 81.9%

         \[\leadsto 1 + \left(u \cdot \left(2 - \frac{\frac{-1.3333333333333333 + \color{blue}{\left(-0.6666666666666666 \cdot \frac{1}{v}\right)}}{v} + \left(-2\right)}{v}\right) - 2\right) \]

      8. associate-*r/ 81.9%

         \[\leadsto 1 + \left(u \cdot \left(2 - \frac{\frac{-1.3333333333333333 + \left(-\color{blue}{\frac{0.6666666666666666 \cdot 1}{v}}\right)}{v} + \left(-2\right)}{v}\right) - 2\right) \]

      9. metadata-eval 81.9%

         \[\leadsto 1 + \left(u \cdot \left(2 - \frac{\frac{-1.3333333333333333 + \left(-\frac{\color{blue}{0.6666666666666666}}{v}\right)}{v} + \left(-2\right)}{v}\right) - 2\right) \]

      10. distribute-neg-frac 81.9%

          \[\leadsto 1 + \left(u \cdot \left(2 - \frac{\frac{-1.3333333333333333 + \color{blue}{\frac{-0.6666666666666666}{v}}}{v} + \left(-2\right)}{v}\right) - 2\right) \]

      11. metadata-eval 81.9%

          \[\leadsto 1 + \left(u \cdot \left(2 - \frac{\frac{-1.3333333333333333 + \frac{\color{blue}{-0.6666666666666666}}{v}}{v} + \left(-2\right)}{v}\right) - 2\right) \]

      12. metadata-eval 81.9%

          \[\leadsto 1 + \left(u \cdot \left(2 - \frac{\frac{-1.3333333333333333 + \frac{-0.6666666666666666}{v}}{v} + \color{blue}{-2}}{v}\right) - 2\right) \]

   6. Simplified 81.9%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.46000000834465027:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(u \cdot \left(2 - \frac{-2 + \frac{-1.3333333333333333 + \frac{-0.6666666666666666}{v}}{v}}{v}\right) - 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 90.7% accurate, 10.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.4000000059604645:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;u \cdot \left(\left(\frac{-1}{u} - \frac{-2 + \frac{-1.3333333333333333}{v}}{v}\right) - -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.4000000059604645)
   1.0
   (* u (- (- (/ -1.0 u) (/ (+ -2.0 (/ -1.3333333333333333 v)) v)) -2.0))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.4000000059604645f) {
		tmp = 1.0f;
	} else {
		tmp = u * (((-1.0f / u) - ((-2.0f + (-1.3333333333333333f / 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 <= 0.4000000059604645e0) then
        tmp = 1.0e0
    else
        tmp = u * ((((-1.0e0) / u) - (((-2.0e0) + ((-1.3333333333333333e0) / v)) / v)) - (-2.0e0))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.4000000059604645))
		tmp = Float32(1.0);
	else
		tmp = Float32(u * Float32(Float32(Float32(Float32(-1.0) / u) - Float32(Float32(Float32(-2.0) + Float32(Float32(-1.3333333333333333) / v)) / v)) - Float32(-2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.4000000059604645))
		tmp = single(1.0);
	else
		tmp = u * (((single(-1.0) / u) - ((single(-2.0) + (single(-1.3333333333333333) / v)) / v)) - single(-2.0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.400000006

   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-exp-log 99.9%

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

      2. *-commutative 99.9%

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

      3. log-prod 99.9%

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

      4. add-log-exp 99.9%

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

      5. sub-neg 99.9%

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

      6. log1p-define 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in v around 0 92.2%

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

   if 0.400000006 < v

   1. Initial program 93.6%

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

   3. Taylor expanded in u around 0 81.5%

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

   4. Taylor expanded in v around -inf 74.4%

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

      1. mul-1-neg 74.4%

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

      2. distribute-neg-frac2 74.4%

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

      3. sub-neg 74.4%

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

      4. associate-*r/ 74.4%

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

      5. distribute-lft-in 74.4%

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

      6. metadata-eval 74.4%

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

      7. neg-mul-1 74.4%

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

      8. associate-*r/ 74.4%

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

      9. metadata-eval 74.4%

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

      10. distribute-neg-frac 74.4%

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

      11. metadata-eval 74.4%

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

      12. metadata-eval 74.4%

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

   6. Simplified 74.4%

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

   7. Taylor expanded in u around -inf 74.4%

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

      1. mul-1-neg 74.4%

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

      2. distribute-rgt-neg-in 74.4%

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

      3. sub-neg 74.4%

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

      4. +-commutative 74.4%

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

      5. associate-*r/ 74.4%

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

      6. distribute-lft-in 74.4%

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

      7. metadata-eval 74.4%

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

      8. neg-mul-1 74.4%

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

      9. distribute-lft-neg-in 74.4%

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

      10. metadata-eval 74.4%

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

      11. associate-*r/ 74.4%

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

      12. metadata-eval 74.4%

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

      13. metadata-eval 74.4%

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

   9. Simplified 74.4%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.4000000059604645:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;u \cdot \left(\left(\frac{-1}{u} - \frac{-2 + \frac{-1.3333333333333333}{v}}{v}\right) - -2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 90.7% accurate, 10.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.4000000059604645:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(u \cdot \left(2 - \frac{-2 + \frac{-1.3333333333333333}{v}}{v}\right) - 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.4000000059604645)
   1.0
   (+ 1.0 (- (* u (- 2.0 (/ (+ -2.0 (/ -1.3333333333333333 v)) v))) 2.0))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.4000000059604645f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + ((u * (2.0f - ((-2.0f + (-1.3333333333333333f / 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 <= 0.4000000059604645e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 + ((u * (2.0e0 - (((-2.0e0) + ((-1.3333333333333333e0) / v)) / v))) - 2.0e0)
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.4000000059604645))
		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) / v)) / v))) - Float32(2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.4000000059604645))
		tmp = single(1.0);
	else
		tmp = single(1.0) + ((u * (single(2.0) - ((single(-2.0) + (single(-1.3333333333333333) / v)) / v))) - single(2.0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.400000006

   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-exp-log 99.9%

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

      2. *-commutative 99.9%

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

      3. log-prod 99.9%

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

      4. add-log-exp 99.9%

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

      5. sub-neg 99.9%

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

      6. log1p-define 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in v around 0 92.2%

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

   if 0.400000006 < v

   1. Initial program 93.6%

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

   3. Taylor expanded in u around 0 81.5%

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

   4. Taylor expanded in v around -inf 74.4%

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

      1. mul-1-neg 74.4%

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

      2. distribute-neg-frac2 74.4%

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

      3. sub-neg 74.4%

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

      4. associate-*r/ 74.4%

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

      5. distribute-lft-in 74.4%

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

      6. metadata-eval 74.4%

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

      7. neg-mul-1 74.4%

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

      8. associate-*r/ 74.4%

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

      9. metadata-eval 74.4%

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

      10. distribute-neg-frac 74.4%

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

      11. metadata-eval 74.4%

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

      12. metadata-eval 74.4%

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

   6. Simplified 74.4%

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

   7. Taylor expanded in u around 0 74.4%

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

      1. mul-1-neg 74.4%

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

      2. distribute-rgt-neg-in 74.4%

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

      3. sub-neg 74.4%

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

      4. metadata-eval 74.4%

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

      5. distribute-neg-in 74.4%

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

      6. associate-*r/ 74.4%

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

      7. distribute-lft-in 74.4%

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

      8. metadata-eval 74.4%

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

      9. neg-mul-1 74.4%

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

      10. distribute-lft-neg-in 74.4%

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

      11. metadata-eval 74.4%

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

      12. associate-*r/ 74.4%

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

      13. metadata-eval 74.4%

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

      14. distribute-neg-in 74.4%

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

      15. +-commutative 74.4%

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

   9. Simplified 74.4%

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

       1. distribute-frac-neg2 74.4%

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

       2. unsub-neg 74.4%

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

   11. Applied egg-rr 74.4%

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

Alternative 11: 90.4% accurate, 13.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.46000000834465027:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(2 \cdot \left(u + \frac{u}{v}\right) - 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.46000000834465027) 1.0 (+ 1.0 (- (* 2.0 (+ u (/ u v))) 2.0))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.46000000834465027f) {
		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 <= 0.46000000834465027e0) 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(0.46000000834465027))
		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(0.46000000834465027))
		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 < 0.460000008

   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-exp-log 99.9%

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

      2. *-commutative 99.9%

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

      3. log-prod 99.9%

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

      4. add-log-exp 99.9%

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

      5. sub-neg 99.9%

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

      6. log1p-define 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in v around 0 91.9%

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

   if 0.460000008 < v

   1. Initial program 94.0%

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

   3. Taylor expanded in u around 0 86.0%

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

   4. Taylor expanded in v around inf 72.4%

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

      1. distribute-lft-out 72.4%

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

   6. Simplified 72.4%

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

Alternative 12: 90.4% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.46000000834465027:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + 2 \cdot \left(u + \frac{u}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.46000000834465027) 1.0 (+ -1.0 (* 2.0 (+ u (/ u v))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.46000000834465027f) {
		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 <= 0.46000000834465027e0) 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(0.46000000834465027))
		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(0.46000000834465027))
		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 < 0.460000008

   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-exp-log 99.9%

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

      2. *-commutative 99.9%

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

      3. log-prod 99.9%

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

      4. add-log-exp 99.9%

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

      5. sub-neg 99.9%

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

      6. log1p-define 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in v around 0 91.9%

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

   if 0.460000008 < v

   1. Initial program 94.0%

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

   3. Taylor expanded in u around 0 86.0%

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

   4. Taylor expanded in v around inf 72.3%

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

      1. sub-neg 72.3%

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

      2. distribute-lft-out 72.3%

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

      3. metadata-eval 72.3%

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

   6. Simplified 72.3%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.46000000834465027:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + 2 \cdot \left(u + \frac{u}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 89.9% accurate, 21.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.46000000834465027:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.46000000834465027) 1.0 (+ -1.0 (* u 2.0))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.46000000834465027f) {
		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 <= 0.46000000834465027e0) 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(0.46000000834465027))
		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(0.46000000834465027))
		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 < 0.460000008

   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-exp-log 99.9%

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

      2. *-commutative 99.9%

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

      3. log-prod 99.9%

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

      4. add-log-exp 99.9%

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

      5. sub-neg 99.9%

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

      6. log1p-define 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in v around 0 91.9%

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

   if 0.460000008 < v

   1. Initial program 94.0%

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

   3. Taylor expanded in u around 0 86.0%

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

   4. Taylor expanded in v around inf 63.5%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.46000000834465027:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 87.0% accurate, 213.0× speedup

Math

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

FPCore

(FPCore (u v) :precision binary32 1.0)

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

   1. add-exp-log 99.5%

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

   2. *-commutative 99.5%

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

   3. log-prod 99.5%

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

   4. add-log-exp 99.5%

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

   5. sub-neg 99.5%

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

   6. log1p-define 99.5%

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

4. Applied egg-rr 99.5%

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

5. Taylor expanded in v around 0 87.0%

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

Alternative 15: 5.8% accurate, 213.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

   1. +-commutative 99.5%

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

   2. fma-define 99.6%

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

   3. +-commutative 99.6%

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

   4. fma-define 99.5%

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

3. Simplified 99.5%

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

5. Taylor expanded in u around 0 6.0%

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

Reproduce

herbie shell --seed 2024135 
(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))))))))