HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.5%

Time: 16.8s

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, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.2%

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

   1. +-commutative 99.2%

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

   2. fma-define 99.2%

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

   3. +-commutative 99.2%

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

   4. fma-define 99.3%

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

3. Simplified 99.3%

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

5. Final simplification 99.3%

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

Alternative 2: 99.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

float code(float u, float v) {
	return 1.0f + (v * logf((u + ((1.0f - u) * powf(expf(-2.0f), (1.0f / 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)) ** (1.0e0 / 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(-2.0)) ^ Float32(Float32(1.0) / v)))))))
end

MATLAB

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

Derivation

1. Initial program 99.2%

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

   1. div-inv 99.2%

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

   2. exp-prod 99.2%

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

4. Applied egg-rr 99.2%

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

5. Final simplification 99.2%

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

Alternative 3: 99.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ 1 + v \cdot \log \left(u + e^{\mathsf{log1p}\left(-u\right) - \frac{2}{v}}\right) \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (+ 1.0 (* v (log (+ u (exp (- (log1p (- u)) (/ 2.0 v))))))))

C

float code(float u, float v) {
	return 1.0f + (v * logf((u + expf((log1pf(-u) - (2.0f / v))))));
}

Julia

function code(u, v)
	return Float32(Float32(1.0) + Float32(v * log(Float32(u + exp(Float32(log1p(Float32(-u)) - Float32(Float32(2.0) / v)))))))
end

Derivation

1. Initial program 99.2%

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

   1. +-commutative 99.2%

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

   2. fma-define 99.2%

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

   3. +-commutative 99.2%

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

   4. fma-define 99.3%

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

3. Simplified 99.3%

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

   1. fma-undefine 99.2%

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

6. Applied egg-rr 99.2%

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

   1. add-exp-log 99.2%

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

   2. *-commutative 99.2%

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

   3. log-prod 99.2%

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

   4. add-log-exp 99.3%

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

   5. sub-neg 99.3%

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

   6. log1p-define 99.2%

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

8. Applied egg-rr 99.2%

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

9. Taylor expanded in v around 0 99.3%

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

    1. sub-neg 99.3%

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

    2. log1p-undefine 99.2%

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

    3. associate-*r/ 99.2%

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

    4. metadata-eval 99.2%

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

11. Simplified 99.2%

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

12. Final simplification 99.2%

    \[\leadsto 1 + v \cdot \log \left(u + e^{\mathsf{log1p}\left(-u\right) - \frac{2}{v}}\right) \]
13. 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 99.2%

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

3. Final simplification 99.2%

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

Alternative 5: 97.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.27000001072883606:\\ \;\;\;\;1 + v \cdot \log u\\ \mathbf{else}:\\ \;\;\;\;u \cdot \left(v \cdot \mathsf{expm1}\left(\frac{2}{v}\right) + \frac{-1}{u}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.27000001072883606)
   (+ 1.0 (* v (log u)))
   (* u (+ (* v (expm1 (/ 2.0 v))) (/ -1.0 u)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.27000001072883606f) {
		tmp = 1.0f + (v * logf(u));
	} else {
		tmp = u * ((v * expm1f((2.0f / v))) + (-1.0f / u));
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.27000001072883606))
		tmp = Float32(Float32(1.0) + Float32(v * log(u)));
	else
		tmp = Float32(u * Float32(Float32(v * expm1(Float32(Float32(2.0) / v))) + Float32(Float32(-1.0) / u)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.270000011

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. fma-define 99.9%

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

      3. +-commutative 99.9%

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

      4. fma-define 99.9%

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

   3. Simplified 99.9%

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

      1. fma-undefine 99.9%

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

   6. Applied egg-rr 99.9%

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

      1. add-exp-log 99.9%

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

      2. *-commutative 99.9%

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

      3. log-prod 99.9%

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

      4. add-log-exp 99.9%

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

      5. sub-neg 99.9%

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

      6. log1p-define 99.9%

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

   8. Applied egg-rr 99.9%

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

   9. Taylor expanded in u around inf 99.2%

      \[\leadsto \color{blue}{1 + -1 \cdot \left(v \cdot \log \left(\frac{1}{u}\right)\right)} \]
   10. 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%

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

   11. Simplified 99.2%

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

   if 0.270000011 < v

   1. Initial program 93.2%

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

      1. div-inv 93.2%

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

      2. exp-prod 93.4%

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

   4. Applied egg-rr 93.4%

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

   5. Taylor expanded in u around 0 68.9%

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

      1. sub-neg 68.9%

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

      2. metadata-eval 68.9%

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

      3. rec-exp 68.9%

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

      4. associate-*r/ 68.9%

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

      5. metadata-eval 68.9%

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

   7. Simplified 68.9%

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

   8. Taylor expanded in u around inf 69.6%

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

      1. expm1-define 69.6%

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

   10. Simplified 69.6%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.27000001072883606:\\ \;\;\;\;1 + v \cdot \log u\\ \mathbf{else}:\\ \;\;\;\;u \cdot \left(v \cdot \mathsf{expm1}\left(\frac{2}{v}\right) + \frac{-1}{u}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.27000001072883606:\\ \;\;\;\;1 + v \cdot \log u\\ \mathbf{else}:\\ \;\;\;\;u \cdot \left(v \cdot \left(e^{\frac{2}{v}} + -1\right)\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.27000001072883606)
   (+ 1.0 (* v (log u)))
   (+ (* u (* v (+ (exp (/ 2.0 v)) -1.0))) -1.0)))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.27000001072883606f) {
		tmp = 1.0f + (v * logf(u));
	} 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 <= 0.27000001072883606e0) then
        tmp = 1.0e0 + (v * log(u))
    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(0.27000001072883606))
		tmp = Float32(Float32(1.0) + Float32(v * log(u)));
	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(0.27000001072883606))
		tmp = single(1.0) + (v * log(u));
	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 < 0.270000011

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. fma-define 99.9%

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

      3. +-commutative 99.9%

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

      4. fma-define 99.9%

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

   3. Simplified 99.9%

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

      1. fma-undefine 99.9%

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

   6. Applied egg-rr 99.9%

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

      1. add-exp-log 99.9%

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

      2. *-commutative 99.9%

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

      3. log-prod 99.9%

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

      4. add-log-exp 99.9%

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

      5. sub-neg 99.9%

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

      6. log1p-define 99.9%

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

   8. Applied egg-rr 99.9%

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

   9. Taylor expanded in u around inf 99.2%

      \[\leadsto \color{blue}{1 + -1 \cdot \left(v \cdot \log \left(\frac{1}{u}\right)\right)} \]
   10. 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%

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

   11. Simplified 99.2%

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

   if 0.270000011 < v

   1. Initial program 93.2%

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

   3. Taylor expanded in u around 0 69.4%

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

      1. fma-neg 69.4%

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

      2. rec-exp 69.4%

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

      3. expm1-define 69.4%

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

      4. distribute-neg-frac 69.4%

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

      5. metadata-eval 69.4%

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

      6. metadata-eval 69.4%

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

   5. Simplified 69.4%

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

   6. Taylor expanded in u around 0 69.8%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.27000001072883606:\\ \;\;\;\;1 + v \cdot \log u\\ \mathbf{else}:\\ \;\;\;\;u \cdot \left(v \cdot \left(e^{\frac{2}{v}} + -1\right)\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.27000001072883606:\\ \;\;\;\;1 + v \cdot \log u\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(\frac{\frac{0.6666666666666666 \cdot \frac{u}{v} + u \cdot 1.3333333333333333}{v} - u \cdot -2}{v} + u \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.27000001072883606)
   (+ 1.0 (* v (log u)))
   (+
    -1.0
    (+
     (/
      (-
       (/ (+ (* 0.6666666666666666 (/ u v)) (* u 1.3333333333333333)) v)
       (* u -2.0))
      v)
     (* u 2.0)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.27000001072883606f) {
		tmp = 1.0f + (v * logf(u));
	} else {
		tmp = -1.0f + ((((((0.6666666666666666f * (u / v)) + (u * 1.3333333333333333f)) / v) - (u * -2.0f)) / 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 <= 0.27000001072883606e0) then
        tmp = 1.0e0 + (v * log(u))
    else
        tmp = (-1.0e0) + ((((((0.6666666666666666e0 * (u / v)) + (u * 1.3333333333333333e0)) / v) - (u * (-2.0e0))) / v) + (u * 2.0e0))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.27000001072883606))
		tmp = Float32(Float32(1.0) + Float32(v * log(u)));
	else
		tmp = Float32(Float32(-1.0) + Float32(Float32(Float32(Float32(Float32(Float32(Float32(0.6666666666666666) * Float32(u / v)) + Float32(u * Float32(1.3333333333333333))) / v) - Float32(u * Float32(-2.0))) / v) + Float32(u * Float32(2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.27000001072883606))
		tmp = single(1.0) + (v * log(u));
	else
		tmp = single(-1.0) + ((((((single(0.6666666666666666) * (u / v)) + (u * single(1.3333333333333333))) / v) - (u * single(-2.0))) / v) + (u * single(2.0)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.270000011

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. fma-define 99.9%

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

      3. +-commutative 99.9%

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

      4. fma-define 99.9%

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

   3. Simplified 99.9%

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

      1. fma-undefine 99.9%

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

   6. Applied egg-rr 99.9%

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

      1. add-exp-log 99.9%

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

      2. *-commutative 99.9%

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

      3. log-prod 99.9%

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

      4. add-log-exp 99.9%

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

      5. sub-neg 99.9%

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

      6. log1p-define 99.9%

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

   8. Applied egg-rr 99.9%

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

   9. Taylor expanded in u around inf 99.2%

      \[\leadsto \color{blue}{1 + -1 \cdot \left(v \cdot \log \left(\frac{1}{u}\right)\right)} \]
   10. 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%

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

   11. Simplified 99.2%

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

   if 0.270000011 < v

   1. Initial program 93.2%

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

   3. Taylor expanded in u around 0 69.4%

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

      1. fma-neg 69.4%

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

      2. rec-exp 69.4%

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

      3. expm1-define 69.4%

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

      4. distribute-neg-frac 69.4%

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

      5. metadata-eval 69.4%

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

      6. metadata-eval 69.4%

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

   5. Simplified 69.4%

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

   6. Taylor expanded in v around -inf 67.0%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.27000001072883606:\\ \;\;\;\;1 + v \cdot \log u\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(\frac{\frac{0.6666666666666666 \cdot \frac{u}{v} + u \cdot 1.3333333333333333}{v} - u \cdot -2}{v} + u \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 91.2% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.15000000596046448:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(u \cdot 2 + \frac{\frac{u \cdot \left(1.3333333333333333 + \frac{0.6666666666666666}{v}\right)}{v} - u \cdot -2}{v}\right) - 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.15000000596046448)
   1.0
   (+
    1.0
    (-
     (+
      (* u 2.0)
      (/
       (-
        (/ (* u (+ 1.3333333333333333 (/ 0.6666666666666666 v))) v)
        (* u -2.0))
       v))
     2.0))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.15000000596046448f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + (((u * 2.0f) + ((((u * (1.3333333333333333f + (0.6666666666666666f / v))) / v) - (u * -2.0f)) / v)) - 2.0f);
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.15000000596046448e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 + (((u * 2.0e0) + ((((u * (1.3333333333333333e0 + (0.6666666666666666e0 / v))) / v) - (u * (-2.0e0))) / v)) - 2.0e0)
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.15000000596046448))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + Float32(Float32(Float32(u * Float32(2.0)) + Float32(Float32(Float32(Float32(u * Float32(Float32(1.3333333333333333) + Float32(Float32(0.6666666666666666) / v))) / v) - Float32(u * Float32(-2.0))) / v)) - Float32(2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.15000000596046448))
		tmp = single(1.0);
	else
		tmp = single(1.0) + (((u * single(2.0)) + ((((u * (single(1.3333333333333333) + (single(0.6666666666666666) / v))) / v) - (u * single(-2.0))) / v)) - single(2.0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.150000006

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. fma-define 99.9%

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

      3. +-commutative 99.9%

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

      4. fma-define 99.9%

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

   3. Simplified 99.9%

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

      1. fma-undefine 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in v around 0 91.3%

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

   if 0.150000006 < v

   1. Initial program 93.7%

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

   3. Taylor expanded in u around 0 63.1%

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

      1. fma-neg 63.1%

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

      2. rec-exp 63.1%

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

      3. expm1-define 63.1%

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

      4. distribute-neg-frac 63.1%

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

      5. metadata-eval 63.1%

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

      6. metadata-eval 63.1%

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

   5. Simplified 63.1%

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

   6. Taylor expanded in v around -inf 62.1%

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

   7. Taylor expanded in u around 0 62.1%

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

      1. associate-*r/ 62.1%

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

      2. metadata-eval 62.1%

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

   9. Simplified 62.1%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.15000000596046448:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(u \cdot 2 + \frac{\frac{u \cdot \left(1.3333333333333333 + \frac{0.6666666666666666}{v}\right)}{v} - u \cdot -2}{v}\right) - 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 91.2% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.15000000596046448:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(\frac{\frac{0.6666666666666666 \cdot \frac{u}{v} + u \cdot 1.3333333333333333}{v} - u \cdot -2}{v} + u \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.15000000596046448)
   1.0
   (+
    -1.0
    (+
     (/
      (-
       (/ (+ (* 0.6666666666666666 (/ u v)) (* u 1.3333333333333333)) v)
       (* u -2.0))
      v)
     (* u 2.0)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.15000000596046448f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + ((((((0.6666666666666666f * (u / v)) + (u * 1.3333333333333333f)) / v) - (u * -2.0f)) / 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 <= 0.15000000596046448e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + ((((((0.6666666666666666e0 * (u / v)) + (u * 1.3333333333333333e0)) / v) - (u * (-2.0e0))) / v) + (u * 2.0e0))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.15000000596046448))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(Float32(Float32(Float32(Float32(Float32(Float32(0.6666666666666666) * Float32(u / v)) + Float32(u * Float32(1.3333333333333333))) / v) - Float32(u * Float32(-2.0))) / v) + Float32(u * Float32(2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.15000000596046448))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + ((((((single(0.6666666666666666) * (u / v)) + (u * single(1.3333333333333333))) / v) - (u * single(-2.0))) / v) + (u * single(2.0)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.150000006

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. fma-define 99.9%

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

      3. +-commutative 99.9%

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

      4. fma-define 99.9%

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

   3. Simplified 99.9%

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

      1. fma-undefine 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in v around 0 91.3%

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

   if 0.150000006 < v

   1. Initial program 93.7%

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

   3. Taylor expanded in u around 0 63.1%

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

      1. fma-neg 63.1%

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

      2. rec-exp 63.1%

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

      3. expm1-define 63.1%

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

      4. distribute-neg-frac 63.1%

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

      5. metadata-eval 63.1%

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

      6. metadata-eval 63.1%

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

   5. Simplified 63.1%

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

   6. Taylor expanded in v around -inf 62.5%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.15000000596046448:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(\frac{\frac{0.6666666666666666 \cdot \frac{u}{v} + u \cdot 1.3333333333333333}{v} - u \cdot -2}{v} + u \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 91.1% accurate, 10.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.15000000596046448:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;u \cdot \left(\left(\frac{2 + \frac{1.3333333333333333}{v}}{v} + \frac{-1}{u}\right) - -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.15000000596046448)
   1.0
   (* u (- (+ (/ (+ 2.0 (/ 1.3333333333333333 v)) v) (/ -1.0 u)) -2.0))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.15000000596046448f) {
		tmp = 1.0f;
	} else {
		tmp = u * ((((2.0f + (1.3333333333333333f / v)) / v) + (-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 <= 0.15000000596046448e0) then
        tmp = 1.0e0
    else
        tmp = u * ((((2.0e0 + (1.3333333333333333e0 / v)) / v) + ((-1.0e0) / u)) - (-2.0e0))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.15000000596046448))
		tmp = Float32(1.0);
	else
		tmp = Float32(u * Float32(Float32(Float32(Float32(Float32(2.0) + Float32(Float32(1.3333333333333333) / v)) / v) + Float32(Float32(-1.0) / u)) - Float32(-2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.15000000596046448))
		tmp = single(1.0);
	else
		tmp = u * ((((single(2.0) + (single(1.3333333333333333) / v)) / v) + (single(-1.0) / u)) - single(-2.0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.150000006

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. fma-define 99.9%

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

      3. +-commutative 99.9%

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

      4. fma-define 99.9%

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

   3. Simplified 99.9%

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

      1. fma-undefine 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in v around 0 91.3%

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

   if 0.150000006 < v

   1. Initial program 93.7%

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

      1. div-inv 93.7%

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

      2. exp-prod 93.8%

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

   4. Applied egg-rr 93.8%

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

   5. Taylor expanded in u around 0 62.7%

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

      1. sub-neg 62.7%

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

      2. metadata-eval 62.7%

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

      3. rec-exp 62.7%

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

      4. associate-*r/ 62.7%

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

      5. metadata-eval 62.7%

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

   7. Simplified 62.7%

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

   8. Taylor expanded in v around -inf 60.9%

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

   9. Taylor expanded in u around -inf 61.3%

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

       1. mul-1-neg 61.3%

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

       2. *-commutative 61.3%

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

       3. distribute-rgt-neg-in 61.3%

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

       4. sub-neg 61.3%

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

       5. +-commutative 61.3%

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

       6. mul-1-neg 61.3%

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

       7. unsub-neg 61.3%

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

       8. associate-*r/ 61.3%

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

       9. metadata-eval 61.3%

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

       10. metadata-eval 61.3%

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

   11. Simplified 61.3%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.15000000596046448:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;u \cdot \left(\left(\frac{2 + \frac{1.3333333333333333}{v}}{v} + \frac{-1}{u}\right) - -2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 90.8% accurate, 11.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.15000000596046448:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(2 \cdot \frac{u}{v} - \left(2 + u \cdot -2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.15000000596046448)
   1.0
   (+ 1.0 (- (* 2.0 (/ u v)) (+ 2.0 (* u -2.0))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.15000000596046448f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + ((2.0f * (u / v)) - (2.0f + (u * -2.0f)));
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.15000000596046448e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 + ((2.0e0 * (u / v)) - (2.0e0 + (u * (-2.0e0))))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.15000000596046448))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + Float32(Float32(Float32(2.0) * Float32(u / v)) - Float32(Float32(2.0) + Float32(u * Float32(-2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.15000000596046448))
		tmp = single(1.0);
	else
		tmp = single(1.0) + ((single(2.0) * (u / v)) - (single(2.0) + (u * single(-2.0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.150000006

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. fma-define 99.9%

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

      3. +-commutative 99.9%

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

      4. fma-define 99.9%

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

   3. Simplified 99.9%

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

      1. fma-undefine 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in v around 0 91.3%

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

   if 0.150000006 < v

   1. Initial program 93.7%

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

      1. div-inv 93.7%

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

      2. exp-prod 93.8%

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

   4. Applied egg-rr 93.8%

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

   5. Taylor expanded in u around 0 62.7%

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

      1. sub-neg 62.7%

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

      2. metadata-eval 62.7%

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

      3. rec-exp 62.7%

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

      4. associate-*r/ 62.7%

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

      5. metadata-eval 62.7%

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

   7. Simplified 62.7%

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

   8. Taylor expanded in v around -inf 58.6%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.15000000596046448:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(2 \cdot \frac{u}{v} - \left(2 + u \cdot -2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 90.8% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.15000000596046448:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + 2 \cdot \left(u + \frac{u}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.15000000596046448) 1.0 (+ -1.0 (* 2.0 (+ u (/ u v))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.15000000596046448f) {
		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 <= 0.15000000596046448e0) 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(0.15000000596046448))
		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(0.15000000596046448))
		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 < 0.150000006

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. fma-define 99.9%

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

      3. +-commutative 99.9%

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

      4. fma-define 99.9%

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

   3. Simplified 99.9%

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

      1. fma-undefine 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in v around 0 91.3%

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

   if 0.150000006 < v

   1. Initial program 93.7%

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

   3. Taylor expanded in u around 0 63.1%

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

      1. fma-neg 63.1%

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

      2. rec-exp 63.1%

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

      3. expm1-define 63.1%

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

      4. distribute-neg-frac 63.1%

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

      5. metadata-eval 63.1%

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

      6. metadata-eval 63.1%

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

   5. Simplified 63.1%

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

   6. Taylor expanded in v around inf 58.6%

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

      1. sub-neg 58.6%

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

      2. distribute-lft-out 58.6%

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

      3. metadata-eval 58.6%

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

   8. Simplified 58.6%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.15000000596046448:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + 2 \cdot \left(u + \frac{u}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 90.2% accurate, 21.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.15000000596046448:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.15000000596046448) 1.0 (+ -1.0 (* u 2.0))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.15000000596046448f) {
		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 <= 0.15000000596046448e0) 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(0.15000000596046448))
		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(0.15000000596046448))
		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 < 0.150000006

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. fma-define 99.9%

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

      3. +-commutative 99.9%

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

      4. fma-define 99.9%

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

   3. Simplified 99.9%

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

      1. fma-undefine 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in v around 0 91.3%

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

   if 0.150000006 < v

   1. Initial program 93.7%

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

   3. Taylor expanded in u around 0 63.1%

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

      1. fma-neg 63.1%

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

      2. rec-exp 63.1%

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

      3. expm1-define 63.1%

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

      4. distribute-neg-frac 63.1%

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

      5. metadata-eval 63.1%

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

      6. metadata-eval 63.1%

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

   5. Simplified 63.1%

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

   6. Taylor expanded in v around inf 49.0%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.15000000596046448:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 89.6% accurate, 35.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.15000000596046448:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v) :precision binary32 (if (<= v 0.15000000596046448) 1.0 -1.0))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.15000000596046448f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f;
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.15000000596046448e0) then
        tmp = 1.0e0
    else
        tmp = -1.0e0
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.15000000596046448))
		tmp = Float32(1.0);
	else
		tmp = Float32(-1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.15000000596046448))
		tmp = single(1.0);
	else
		tmp = single(-1.0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.150000006

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. fma-define 99.9%

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

      3. +-commutative 99.9%

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

      4. fma-define 99.9%

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

   3. Simplified 99.9%

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

      1. fma-undefine 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in v around 0 91.3%

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

   if 0.150000006 < v

   1. Initial program 93.7%

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

   3. Taylor expanded in u around 0 40.0%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.15000000596046448:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 5.9% accurate, 213.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.2%

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

3. Taylor expanded in u around 0 7.3%

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

4. Final simplification 7.3%

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

Reproduce

herbie shell --seed 2024076 
(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))))))))