HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.5%

Time: 12.7s

Alternatives: 14

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.5% accurate, 1.0× speedup

Math

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

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 99.6%

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

Alternative 2: 90.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.5:\\ \;\;\;\;1 + v \cdot 0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(v, u \cdot \mathsf{expm1}\left(\frac{2}{v}\right), -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.5) (+ 1.0 (* v 0.0)) (fma v (* u (expm1 (/ 2.0 v))) -1.0)))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.5f) {
		tmp = 1.0f + (v * 0.0f);
	} else {
		tmp = fmaf(v, (u * expm1f((2.0f / v))), -1.0f);
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.5))
		tmp = Float32(Float32(1.0) + Float32(v * Float32(0.0)));
	else
		tmp = fma(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 < 0.5

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-define 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-define 100.0%

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

   3. Simplified 100.0%

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

      1. fma-undefine 100.0%

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

      2. +-commutative 100.0%

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

      3. add-sqr-sqrt 99.9%

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

      4. log-prod 100.0%

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

      5. +-commutative 100.0%

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

      6. fma-undefine 100.0%

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

      7. +-commutative 100.0%

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

      8. fma-undefine 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. count-2 100.0%

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

   8. Simplified 100.0%

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

      1. fma-undefine 100.0%

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

      2. add-log-exp 100.0%

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

      3. *-commutative 100.0%

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

      4. fma-define 100.0%

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

      5. exp-to-pow 100.0%

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

      6. pow2 100.0%

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

      7. add-sqr-sqrt 100.0%

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

      8. *-commutative 100.0%

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

      9. fma-define 100.0%

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

   10. Applied egg-rr 100.0%

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

   11. Taylor expanded in v around inf 92.1%

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

   if 0.5 < v

   1. Initial program 92.4%

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

      1. +-commutative 92.4%

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

      2. fma-define 92.0%

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

      3. +-commutative 92.0%

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

      4. fma-define 91.6%

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

   3. Simplified 91.6%

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

   5. Taylor expanded in u around 0 78.8%

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

      1. *-commutative 78.8%

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

      2. associate-*l* 78.8%

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

      3. *-commutative 78.8%

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

      4. fmm-def 78.8%

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

      5. rec-exp 78.8%

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

      6. expm1-define 78.8%

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

      7. distribute-neg-frac 78.8%

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

      8. metadata-eval 78.8%

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

      9. metadata-eval 78.8%

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

   7. Simplified 78.8%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.5:\\ \;\;\;\;1 + v \cdot 0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(v, u \cdot \mathsf{expm1}\left(\frac{2}{v}\right), -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.2% 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 99.6%

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

3. Taylor expanded in u around 0 96.6%

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

Alternative 4: 90.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.5:\\ \;\;\;\;1 + v \cdot 0\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(v \cdot \mathsf{expm1}\left(\frac{2}{v}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.5) (+ 1.0 (* v 0.0)) (+ -1.0 (* u (* v (expm1 (/ 2.0 v)))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.5f) {
		tmp = 1.0f + (v * 0.0f);
	} else {
		tmp = -1.0f + (u * (v * expm1f((2.0f / v))));
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.5))
		tmp = Float32(Float32(1.0) + Float32(v * Float32(0.0)));
	else
		tmp = Float32(Float32(-1.0) + Float32(u * Float32(v * expm1(Float32(Float32(2.0) / v)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.5

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-define 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-define 100.0%

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

   3. Simplified 100.0%

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

      1. fma-undefine 100.0%

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

      2. +-commutative 100.0%

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

      3. add-sqr-sqrt 99.9%

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

      4. log-prod 100.0%

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

      5. +-commutative 100.0%

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

      6. fma-undefine 100.0%

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

      7. +-commutative 100.0%

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

      8. fma-undefine 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. count-2 100.0%

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

   8. Simplified 100.0%

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

      1. fma-undefine 100.0%

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

      2. add-log-exp 100.0%

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

      3. *-commutative 100.0%

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

      4. fma-define 100.0%

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

      5. exp-to-pow 100.0%

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

      6. pow2 100.0%

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

      7. add-sqr-sqrt 100.0%

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

      8. *-commutative 100.0%

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

      9. fma-define 100.0%

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

   10. Applied egg-rr 100.0%

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

   11. Taylor expanded in v around inf 92.1%

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

   if 0.5 < v

   1. Initial program 92.4%

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

   3. Taylor expanded in u around 0 78.8%

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

      1. +-commutative 78.8%

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

      2. associate-+l- 78.8%

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

      3. rec-exp 78.8%

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

      4. expm1-define 78.8%

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

      5. distribute-neg-frac 78.8%

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

      6. metadata-eval 78.8%

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

      7. metadata-eval 78.8%

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

   5. Applied egg-rr 78.8%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.5:\\ \;\;\;\;1 + v \cdot 0\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(v \cdot \mathsf{expm1}\left(\frac{2}{v}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 88.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.5:\\ \;\;\;\;\log e\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(\frac{\frac{0.6666666666666666 \cdot \frac{u}{v} + u \cdot 1.3333333333333333}{v} - u \cdot -2}{v} + u \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.5)
   (log E)
   (+
    -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 <= 0.5f) {
		tmp = logf(((float) M_E));
	} else {
		tmp = -1.0f + ((((((0.6666666666666666f * (u / v)) + (u * 1.3333333333333333f)) / v) - (u * -2.0f)) / v) + (u * 2.0f));
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.5))
		tmp = log(Float32(exp(1)));
	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(0.5))
		tmp = log(single(2.71828182845904523536));
	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 < 0.5

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-define 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-define 100.0%

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

   3. Simplified 100.0%

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

      1. fma-undefine 100.0%

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

      2. +-commutative 100.0%

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

      3. add-sqr-sqrt 99.9%

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

      4. log-prod 100.0%

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

      5. +-commutative 100.0%

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

      6. fma-undefine 100.0%

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

      7. +-commutative 100.0%

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

      8. fma-undefine 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. count-2 100.0%

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

   8. Simplified 100.0%

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

      1. add-log-exp 100.0%

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

      2. *-commutative 100.0%

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

      3. fma-define 100.0%

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

      4. exp-to-pow 99.9%

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

      5. pow2 99.9%

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

      6. add-sqr-sqrt 100.0%

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

      7. fma-undefine 100.0%

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

      8. add-log-exp 97.3%

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

      9. exp-sum 97.2%

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

   10. Applied egg-rr 97.2%

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

       1. exp-1-e 97.2%

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

   12. Simplified 97.2%

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

   13. Taylor expanded in v around 0 89.7%

       \[\leadsto \color{blue}{\log e} \]

   if 0.5 < v

   1. Initial program 92.4%

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

   3. Taylor expanded in u around 0 78.8%

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

   4. Taylor expanded in v around -inf 74.7%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.5:\\ \;\;\;\;\log e\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(\frac{\frac{0.6666666666666666 \cdot \frac{u}{v} + u \cdot 1.3333333333333333}{v} - u \cdot -2}{v} + u \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 90.8% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.5:\\ \;\;\;\;1 + v \cdot 0\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(\frac{\frac{0.6666666666666666 \cdot \frac{u}{v} + u \cdot 1.3333333333333333}{v} - u \cdot -2}{v} + u \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.5)
   (+ 1.0 (* v 0.0))
   (+
    -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 <= 0.5f) {
		tmp = 1.0f + (v * 0.0f);
	} 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 <= 0.5e0) then
        tmp = 1.0e0 + (v * 0.0e0)
    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(0.5))
		tmp = Float32(Float32(1.0) + Float32(v * Float32(0.0)));
	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(0.5))
		tmp = single(1.0) + (v * single(0.0));
	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 < 0.5

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-define 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-define 100.0%

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

   3. Simplified 100.0%

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

      1. fma-undefine 100.0%

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

      2. +-commutative 100.0%

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

      3. add-sqr-sqrt 99.9%

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

      4. log-prod 100.0%

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

      5. +-commutative 100.0%

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

      6. fma-undefine 100.0%

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

      7. +-commutative 100.0%

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

      8. fma-undefine 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. count-2 100.0%

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

   8. Simplified 100.0%

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

      1. fma-undefine 100.0%

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

      2. add-log-exp 100.0%

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

      3. *-commutative 100.0%

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

      4. fma-define 100.0%

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

      5. exp-to-pow 100.0%

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

      6. pow2 100.0%

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

      7. add-sqr-sqrt 100.0%

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

      8. *-commutative 100.0%

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

      9. fma-define 100.0%

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

   10. Applied egg-rr 100.0%

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

   11. Taylor expanded in v around inf 92.1%

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

   if 0.5 < v

   1. Initial program 92.4%

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

   3. Taylor expanded in u around 0 78.8%

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

   4. Taylor expanded in v around -inf 74.7%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.5:\\ \;\;\;\;1 + v \cdot 0\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(\frac{\frac{0.6666666666666666 \cdot \frac{u}{v} + u \cdot 1.3333333333333333}{v} - u \cdot -2}{v} + u \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 23.7% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.009999999776482582:\\ \;\;\;\;u \cdot 2\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(\frac{\frac{0.6666666666666666 \cdot \frac{u}{v} + u \cdot 1.3333333333333333}{v} - u \cdot -2}{v} + u \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.009999999776482582)
   (* u 2.0)
   (+
    -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 <= 0.009999999776482582f) {
		tmp = u * 2.0f;
	} 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 <= 0.009999999776482582e0) then
        tmp = u * 2.0e0
    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(0.009999999776482582))
		tmp = Float32(u * Float32(2.0));
	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(0.009999999776482582))
		tmp = u * single(2.0);
	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 < 0.00999999978

   1. Initial program 100.0%

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

   3. Taylor expanded in u around 0 6.2%

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

   4. Taylor expanded in v around inf 5.2%

      \[\leadsto \color{blue}{2 \cdot u - 1} \]

   5. Taylor expanded in u around inf 21.8%

      \[\leadsto \color{blue}{2 \cdot u} \]
   6. Step-by-step derivation

      1. *-commutative 21.8%

         \[\leadsto \color{blue}{u \cdot 2} \]

   7. Simplified 21.8%

      \[\leadsto \color{blue}{u \cdot 2} \]

   if 0.00999999978 < v

   1. Initial program 95.1%

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

   3. Taylor expanded in u around 0 52.6%

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

   4. Taylor expanded in v around -inf 53.9%

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

4. Final simplification 24.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.009999999776482582:\\ \;\;\;\;u \cdot 2\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(\frac{\frac{0.6666666666666666 \cdot \frac{u}{v} + u \cdot 1.3333333333333333}{v} - u \cdot -2}{v} + u \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 23.7% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.009999999776482582:\\ \;\;\;\;u \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(1 + u \cdot \left(2 + \frac{2 + \frac{1.3333333333333333 - 0.6666666666666666 \cdot \frac{-1}{v}}{v}}{v}\right)\right) - 2\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.009999999776482582)
   (* u 2.0)
   (-
    (+
     1.0
     (*
      u
      (+
       2.0
       (/
        (+ 2.0 (/ (- 1.3333333333333333 (* 0.6666666666666666 (/ -1.0 v))) v))
        v))))
    2.0)))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.009999999776482582f) {
		tmp = u * 2.0f;
	} else {
		tmp = (1.0f + (u * (2.0f + ((2.0f + ((1.3333333333333333f - (0.6666666666666666f * (-1.0f / v))) / v)) / v)))) - 2.0f;
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.009999999776482582e0) then
        tmp = u * 2.0e0
    else
        tmp = (1.0e0 + (u * (2.0e0 + ((2.0e0 + ((1.3333333333333333e0 - (0.6666666666666666e0 * ((-1.0e0) / v))) / v)) / v)))) - 2.0e0
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.009999999776482582))
		tmp = Float32(u * Float32(2.0));
	else
		tmp = Float32(Float32(Float32(1.0) + Float32(u * Float32(Float32(2.0) + Float32(Float32(Float32(2.0) + Float32(Float32(Float32(1.3333333333333333) - Float32(Float32(0.6666666666666666) * Float32(Float32(-1.0) / v))) / v)) / v)))) - Float32(2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.009999999776482582))
		tmp = u * single(2.0);
	else
		tmp = (single(1.0) + (u * (single(2.0) + ((single(2.0) + ((single(1.3333333333333333) - (single(0.6666666666666666) * (single(-1.0) / v))) / v)) / v)))) - single(2.0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.00999999978

   1. Initial program 100.0%

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

   3. Taylor expanded in u around 0 6.2%

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

   4. Taylor expanded in v around inf 5.2%

      \[\leadsto \color{blue}{2 \cdot u - 1} \]

   5. Taylor expanded in u around inf 21.8%

      \[\leadsto \color{blue}{2 \cdot u} \]
   6. Step-by-step derivation

      1. *-commutative 21.8%

         \[\leadsto \color{blue}{u \cdot 2} \]

   7. Simplified 21.8%

      \[\leadsto \color{blue}{u \cdot 2} \]

   if 0.00999999978 < v

   1. Initial program 95.1%

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

   3. Taylor expanded in u around 0 52.6%

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

      1. associate-+r- 52.6%

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

      2. rec-exp 52.6%

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

      3. expm1-define 52.6%

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

      4. distribute-neg-frac 52.6%

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

      5. metadata-eval 52.6%

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

   5. Applied egg-rr 52.6%

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

   6. Taylor expanded in v around -inf 53.8%

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

4. Final simplification 24.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.009999999776482582:\\ \;\;\;\;u \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(1 + u \cdot \left(2 + \frac{2 + \frac{1.3333333333333333 - 0.6666666666666666 \cdot \frac{-1}{v}}{v}}{v}\right)\right) - 2\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 23.7% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.009999999776482582:\\ \;\;\;\;u \cdot 2\\ \mathbf{else}:\\ \;\;\;\;1 + \left(u \cdot \left(2 + \frac{2 + \frac{1.3333333333333333 - 0.6666666666666666 \cdot \frac{-1}{v}}{v}}{v}\right) - 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.009999999776482582)
   (* u 2.0)
   (+
    1.0
    (-
     (*
      u
      (+
       2.0
       (/
        (+ 2.0 (/ (- 1.3333333333333333 (* 0.6666666666666666 (/ -1.0 v))) v))
        v)))
     2.0))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.009999999776482582f) {
		tmp = u * 2.0f;
	} else {
		tmp = 1.0f + ((u * (2.0f + ((2.0f + ((1.3333333333333333f - (0.6666666666666666f * (-1.0f / v))) / v)) / v))) - 2.0f);
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.009999999776482582e0) then
        tmp = u * 2.0e0
    else
        tmp = 1.0e0 + ((u * (2.0e0 + ((2.0e0 + ((1.3333333333333333e0 - (0.6666666666666666e0 * ((-1.0e0) / v))) / v)) / v))) - 2.0e0)
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.009999999776482582))
		tmp = Float32(u * Float32(2.0));
	else
		tmp = Float32(Float32(1.0) + Float32(Float32(u * Float32(Float32(2.0) + Float32(Float32(Float32(2.0) + Float32(Float32(Float32(1.3333333333333333) - Float32(Float32(0.6666666666666666) * Float32(Float32(-1.0) / v))) / v)) / v))) - Float32(2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.009999999776482582))
		tmp = u * single(2.0);
	else
		tmp = single(1.0) + ((u * (single(2.0) + ((single(2.0) + ((single(1.3333333333333333) - (single(0.6666666666666666) * (single(-1.0) / v))) / v)) / v))) - single(2.0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.00999999978

   1. Initial program 100.0%

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

   3. Taylor expanded in u around 0 6.2%

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

   4. Taylor expanded in v around inf 5.2%

      \[\leadsto \color{blue}{2 \cdot u - 1} \]

   5. Taylor expanded in u around inf 21.8%

      \[\leadsto \color{blue}{2 \cdot u} \]
   6. Step-by-step derivation

      1. *-commutative 21.8%

         \[\leadsto \color{blue}{u \cdot 2} \]

   7. Simplified 21.8%

      \[\leadsto \color{blue}{u \cdot 2} \]

   if 0.00999999978 < v

   1. Initial program 95.1%

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

   3. Taylor expanded in u around 0 52.6%

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

   4. Taylor expanded in v around -inf 53.8%

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

4. Final simplification 24.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.009999999776482582:\\ \;\;\;\;u \cdot 2\\ \mathbf{else}:\\ \;\;\;\;1 + \left(u \cdot \left(2 + \frac{2 + \frac{1.3333333333333333 - 0.6666666666666666 \cdot \frac{-1}{v}}{v}}{v}\right) - 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 23.5% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.5:\\ \;\;\;\;u \cdot 2\\ \mathbf{else}:\\ \;\;\;\;1 - \left(2 - \left(u \cdot 2 - \frac{u \cdot -2 + \frac{u}{v} \cdot -1.3333333333333333}{v}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.5)
   (* u 2.0)
   (-
    1.0
    (-
     2.0
     (- (* u 2.0) (/ (+ (* u -2.0) (* (/ u v) -1.3333333333333333)) v))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.5f) {
		tmp = u * 2.0f;
	} else {
		tmp = 1.0f - (2.0f - ((u * 2.0f) - (((u * -2.0f) + ((u / v) * -1.3333333333333333f)) / 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 <= 0.5e0) then
        tmp = u * 2.0e0
    else
        tmp = 1.0e0 - (2.0e0 - ((u * 2.0e0) - (((u * (-2.0e0)) + ((u / v) * (-1.3333333333333333e0))) / v)))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.5))
		tmp = Float32(u * Float32(2.0));
	else
		tmp = Float32(Float32(1.0) - Float32(Float32(2.0) - Float32(Float32(u * Float32(2.0)) - Float32(Float32(Float32(u * Float32(-2.0)) + Float32(Float32(u / v) * Float32(-1.3333333333333333))) / v))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.5))
		tmp = u * single(2.0);
	else
		tmp = single(1.0) - (single(2.0) - ((u * single(2.0)) - (((u * single(-2.0)) + ((u / v) * single(-1.3333333333333333))) / v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.5

   1. Initial program 100.0%

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

   3. Taylor expanded in u around 0 6.5%

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

   4. Taylor expanded in v around inf 5.1%

      \[\leadsto \color{blue}{2 \cdot u - 1} \]

   5. Taylor expanded in u around inf 21.7%

      \[\leadsto \color{blue}{2 \cdot u} \]
   6. Step-by-step derivation

      1. *-commutative 21.7%

         \[\leadsto \color{blue}{u \cdot 2} \]

   7. Simplified 21.7%

      \[\leadsto \color{blue}{u \cdot 2} \]

   if 0.5 < v

   1. Initial program 92.4%

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

   3. Taylor expanded in u around 0 78.8%

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

   4. Taylor expanded in v around -inf 72.1%

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

4. Final simplification 24.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.5:\\ \;\;\;\;u \cdot 2\\ \mathbf{else}:\\ \;\;\;\;1 - \left(2 - \left(u \cdot 2 - \frac{u \cdot -2 + \frac{u}{v} \cdot -1.3333333333333333}{v}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 23.5% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.5:\\ \;\;\;\;u \cdot 2\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(u \cdot 2 - \frac{u \cdot -2 + \frac{u}{v} \cdot -1.3333333333333333}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.5)
   (* u 2.0)
   (+
    -1.0
    (- (* u 2.0) (/ (+ (* u -2.0) (* (/ u v) -1.3333333333333333)) v)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.5f) {
		tmp = u * 2.0f;
	} else {
		tmp = -1.0f + ((u * 2.0f) - (((u * -2.0f) + ((u / v) * -1.3333333333333333f)) / 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 <= 0.5e0) then
        tmp = u * 2.0e0
    else
        tmp = (-1.0e0) + ((u * 2.0e0) - (((u * (-2.0e0)) + ((u / v) * (-1.3333333333333333e0))) / v))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.5))
		tmp = Float32(u * Float32(2.0));
	else
		tmp = Float32(Float32(-1.0) + Float32(Float32(u * Float32(2.0)) - Float32(Float32(Float32(u * Float32(-2.0)) + Float32(Float32(u / v) * Float32(-1.3333333333333333))) / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.5))
		tmp = u * single(2.0);
	else
		tmp = single(-1.0) + ((u * single(2.0)) - (((u * single(-2.0)) + ((u / v) * single(-1.3333333333333333))) / v));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.5

   1. Initial program 100.0%

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

   3. Taylor expanded in u around 0 6.5%

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

   4. Taylor expanded in v around inf 5.1%

      \[\leadsto \color{blue}{2 \cdot u - 1} \]

   5. Taylor expanded in u around inf 21.7%

      \[\leadsto \color{blue}{2 \cdot u} \]
   6. Step-by-step derivation

      1. *-commutative 21.7%

         \[\leadsto \color{blue}{u \cdot 2} \]

   7. Simplified 21.7%

      \[\leadsto \color{blue}{u \cdot 2} \]

   if 0.5 < v

   1. Initial program 92.4%

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

   3. Taylor expanded in u around 0 78.8%

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

   4. Taylor expanded in v around -inf 72.0%

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

4. Final simplification 24.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.5:\\ \;\;\;\;u \cdot 2\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(u \cdot 2 - \frac{u \cdot -2 + \frac{u}{v} \cdot -1.3333333333333333}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 23.3% accurate, 15.2× speedup

Math

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

FPCore

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

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.5f) {
		tmp = u * 2.0f;
	} 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 <= 0.5e0) then
        tmp = u * 2.0e0
    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(0.5))
		tmp = Float32(u * Float32(2.0));
	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(0.5))
		tmp = u * single(2.0);
	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 < 0.5

   1. Initial program 100.0%

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

   3. Taylor expanded in u around 0 6.5%

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

   4. Taylor expanded in v around inf 5.1%

      \[\leadsto \color{blue}{2 \cdot u - 1} \]

   5. Taylor expanded in u around inf 21.7%

      \[\leadsto \color{blue}{2 \cdot u} \]
   6. Step-by-step derivation

      1. *-commutative 21.7%

         \[\leadsto \color{blue}{u \cdot 2} \]

   7. Simplified 21.7%

      \[\leadsto \color{blue}{u \cdot 2} \]

   if 0.5 < v

   1. Initial program 92.4%

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

   3. Taylor expanded in u around 0 78.8%

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

   4. Taylor expanded in v around inf 69.2%

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

      1. sub-neg 69.2%

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

      2. distribute-lft-out 69.2%

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

      3. metadata-eval 69.2%

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

   6. Simplified 69.2%

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

4. Final simplification 24.3%

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

Alternative 13: 20.1% accurate, 71.0× speedup

Math

\[\begin{array}{l} \\ u \cdot 2 \end{array} \]

FPCore

(FPCore (u v) :precision binary32 (* u 2.0))

C

float code(float u, float v) {
	return u * 2.0f;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    code = u * 2.0e0
end function

Julia

function code(u, v)
	return Float32(u * Float32(2.0))
end

MATLAB

function tmp = code(u, v)
	tmp = u * single(2.0);
end

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in u around 0 10.4%

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

4. Taylor expanded in v around inf 8.1%

   \[\leadsto \color{blue}{2 \cdot u - 1} \]

5. Taylor expanded in u around inf 20.7%

   \[\leadsto \color{blue}{2 \cdot u} \]
6. Step-by-step derivation

   1. *-commutative 20.7%

      \[\leadsto \color{blue}{u \cdot 2} \]

7. Simplified 20.7%

   \[\leadsto \color{blue}{u \cdot 2} \]
8. Add Preprocessing

Alternative 14: 5.6% 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 99.6%

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

   1. +-commutative 99.6%

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

   2. fma-define 99.5%

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

   3. +-commutative 99.5%

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

   4. fma-define 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 5.8%

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

Reproduce

herbie shell --seed 2024177 
(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))))))))