HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.4%

Time: 20.1s

Alternatives: 13

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := u + e^{\frac{-2}{v}} \cdot \left(1 - u\right)\\ \mathsf{fma}\left(v, \log \left(\sqrt[3]{t\_0}\right) + \log \left(\sqrt[3]{{t\_0}^{2}}\right), 1\right) \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (let* ((t_0 (+ u (* (exp (/ -2.0 v)) (- 1.0 u)))))
   (fma v (+ (log (cbrt t_0)) (log (cbrt (pow t_0 2.0)))) 1.0)))

C

float code(float u, float v) {
	float t_0 = u + (expf((-2.0f / v)) * (1.0f - u));
	return fmaf(v, (logf(cbrtf(t_0)) + logf(cbrtf(powf(t_0, 2.0f)))), 1.0f);
}

Julia

function code(u, v)
	t_0 = Float32(u + Float32(exp(Float32(Float32(-2.0) / v)) * Float32(Float32(1.0) - u)))
	return fma(v, Float32(log(cbrt(t_0)) + log(cbrt((t_0 ^ Float32(2.0))))), Float32(1.0))
end

Derivation

1. Initial program 99.3%

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

   1. +-commutative 99.3%

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

   2. fma-define 99.3%

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

   3. +-commutative 99.3%

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

   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. *-un-lft-identity 99.3%

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

   2. exp-prod 99.2%

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

6. Applied egg-rr 99.2%

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

   1. exp-1-e 99.2%

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

8. Simplified 99.2%

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

   1. *-un-lft-identity 99.2%

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

   2. *-un-lft-identity 99.2%

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

   3. e-exp-1 99.2%

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

   4. pow-exp 99.3%

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

   5. *-un-lft-identity 99.3%

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

   6. add-cube-cbrt 99.1%

      \[\leadsto \mathsf{fma}\left(v, \log \color{blue}{\left(\left(\sqrt[3]{\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)} \cdot \sqrt[3]{\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)}\right)}, 1\right) \]

   7. log-prod 99.1%

      \[\leadsto \mathsf{fma}\left(v, \color{blue}{\log \left(\sqrt[3]{\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)} \cdot \sqrt[3]{\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)}\right) + \log \left(\sqrt[3]{\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)}\right)}, 1\right) \]

   8. cbrt-unprod 99.3%

      \[\leadsto \mathsf{fma}\left(v, \log \color{blue}{\left(\sqrt[3]{\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right) \cdot \mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)}\right)} + \log \left(\sqrt[3]{\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)}\right), 1\right) \]

   9. pow2 99.3%

      \[\leadsto \mathsf{fma}\left(v, \log \left(\sqrt[3]{\color{blue}{{\left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right)}^{2}}}\right) + \log \left(\sqrt[3]{\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)}\right), 1\right) \]

10. Applied egg-rr 99.3%

    \[\leadsto \mathsf{fma}\left(v, \color{blue}{\log \left(\sqrt[3]{{\left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right)}^{2}}\right) + \log \left(\sqrt[3]{\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)}\right)}, 1\right) \]
11. Step-by-step derivation

    1. +-commutative 99.3%

       \[\leadsto \mathsf{fma}\left(v, \color{blue}{\log \left(\sqrt[3]{\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)}\right) + \log \left(\sqrt[3]{{\left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right)}^{2}}\right)}, 1\right) \]

12. Simplified 99.3%

    \[\leadsto \mathsf{fma}\left(v, \color{blue}{\log \left(\sqrt[3]{\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)}\right) + \log \left(\sqrt[3]{{\left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right)}^{2}}\right)}, 1\right) \]

13. Taylor expanded in v around 0 99.4%

    \[\leadsto \mathsf{fma}\left(v, \log \left(\sqrt[3]{\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)}\right) + \color{blue}{\log \left(\sqrt[3]{{\left(u + e^{\frac{-2}{v}} \cdot \left(1 - u\right)\right)}^{2}}\right)}, 1\right) \]

14. Taylor expanded in v around 0 99.4%

    \[\leadsto \mathsf{fma}\left(v, \color{blue}{\log \left(\sqrt[3]{u + e^{\frac{-2}{v}} \cdot \left(1 - u\right)}\right) + \log \left(\sqrt[3]{{\left(u + e^{\frac{-2}{v}} \cdot \left(1 - u\right)\right)}^{2}}\right)}, 1\right) \]

15. Final simplification 99.4%

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

Alternative 2: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := u + e^{\frac{-2}{v}} \cdot \left(1 - u\right)\\ 1 + v \cdot \left(\log \left(\sqrt[3]{t\_0}\right) + \log \left(\sqrt[3]{{t\_0}^{2}}\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (let* ((t_0 (+ u (* (exp (/ -2.0 v)) (- 1.0 u)))))
   (+ 1.0 (* v (+ (log (cbrt t_0)) (log (cbrt (pow t_0 2.0))))))))

C

float code(float u, float v) {
	float t_0 = u + (expf((-2.0f / v)) * (1.0f - u));
	return 1.0f + (v * (logf(cbrtf(t_0)) + logf(cbrtf(powf(t_0, 2.0f)))));
}

Julia

function code(u, v)
	t_0 = Float32(u + Float32(exp(Float32(Float32(-2.0) / v)) * Float32(Float32(1.0) - u)))
	return Float32(Float32(1.0) + Float32(v * Float32(log(cbrt(t_0)) + log(cbrt((t_0 ^ Float32(2.0)))))))
end

Derivation

1. Initial program 99.3%

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

   1. +-commutative 99.3%

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

   2. fma-define 99.3%

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

   3. +-commutative 99.3%

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

   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. *-un-lft-identity 99.3%

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

   2. exp-prod 99.2%

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

6. Applied egg-rr 99.2%

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

   1. exp-1-e 99.2%

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

8. Simplified 99.2%

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

   1. *-un-lft-identity 99.2%

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

   2. *-un-lft-identity 99.2%

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

   3. e-exp-1 99.2%

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

   4. pow-exp 99.3%

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

   5. *-un-lft-identity 99.3%

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

   6. add-cube-cbrt 99.1%

      \[\leadsto \mathsf{fma}\left(v, \log \color{blue}{\left(\left(\sqrt[3]{\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)} \cdot \sqrt[3]{\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)}\right)}, 1\right) \]

   7. log-prod 99.1%

      \[\leadsto \mathsf{fma}\left(v, \color{blue}{\log \left(\sqrt[3]{\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)} \cdot \sqrt[3]{\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)}\right) + \log \left(\sqrt[3]{\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)}\right)}, 1\right) \]

   8. cbrt-unprod 99.3%

      \[\leadsto \mathsf{fma}\left(v, \log \color{blue}{\left(\sqrt[3]{\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right) \cdot \mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)}\right)} + \log \left(\sqrt[3]{\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)}\right), 1\right) \]

   9. pow2 99.3%

      \[\leadsto \mathsf{fma}\left(v, \log \left(\sqrt[3]{\color{blue}{{\left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right)}^{2}}}\right) + \log \left(\sqrt[3]{\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)}\right), 1\right) \]

10. Applied egg-rr 99.3%

    \[\leadsto \mathsf{fma}\left(v, \color{blue}{\log \left(\sqrt[3]{{\left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right)}^{2}}\right) + \log \left(\sqrt[3]{\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)}\right)}, 1\right) \]
11. Step-by-step derivation

    1. +-commutative 99.3%

       \[\leadsto \mathsf{fma}\left(v, \color{blue}{\log \left(\sqrt[3]{\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)}\right) + \log \left(\sqrt[3]{{\left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right)}^{2}}\right)}, 1\right) \]

12. Simplified 99.3%

    \[\leadsto \mathsf{fma}\left(v, \color{blue}{\log \left(\sqrt[3]{\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)}\right) + \log \left(\sqrt[3]{{\left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right)}^{2}}\right)}, 1\right) \]

13. Taylor expanded in v around 0 99.4%

    \[\leadsto \color{blue}{1 + v \cdot \left(\log \left(\sqrt[3]{u + e^{\frac{-2}{v}} \cdot \left(1 - u\right)}\right) + \log \left(\sqrt[3]{{\left(u + e^{\frac{-2}{v}} \cdot \left(1 - u\right)\right)}^{2}}\right)\right)} \]

14. Final simplification 99.4%

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

Alternative 3: 99.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

   1. add-sqr-sqrt 99.2%

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

   2. log-prod 99.3%

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

   3. +-commutative 99.3%

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

   4. fma-undefine 99.3%

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

   5. +-commutative 99.3%

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

   6. fma-undefine 99.3%

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

4. Applied egg-rr 99.3%

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

   1. count-2 99.3%

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

6. Simplified 99.3%

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

7. Taylor expanded in v around 0 99.3%

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

8. Final simplification 99.3%

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

Alternative 4: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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)) * (1.0e0 - u)))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

3. Final simplification 99.3%

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

Alternative 5: 90.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(v \cdot u\right) \cdot \mathsf{expm1}\left(\frac{2}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.10000000149011612) 1.0 (+ -1.0 (* (* v u) (expm1 (/ 2.0 v))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + ((v * u) * expm1f((2.0f / v)));
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.10000000149011612))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(Float32(v * u) * expm1(Float32(Float32(2.0) / v))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.100000001

   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. Step-by-step derivation

      1. add-sqr-sqrt 100.0%

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

      2. log-prod 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-undefine 100.0%

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

      5. +-commutative 100.0%

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

      6. fma-undefine 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. count-2 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in v around 0 91.8%

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

   if 0.100000001 < v

   1. Initial program 91.5%

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

      1. +-commutative 91.5%

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

      2. fma-define 91.4%

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

      3. +-commutative 91.4%

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

      4. fma-define 91.4%

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

   3. Simplified 91.4%

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

      1. *-un-lft-identity 91.4%

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

      2. exp-prod 91.2%

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

   6. Applied egg-rr 91.2%

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

      1. exp-1-e 91.2%

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

   8. Simplified 91.2%

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

   9. Taylor expanded in u around 0 68.2%

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

       1. associate-+r+ 68.0%

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

       2. log-E 69.3%

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

       3. metadata-eval 69.3%

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

       4. metadata-eval 69.3%

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

       5. associate-*r* 69.3%

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

       6. rec-exp 69.3%

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

       7. log-E 69.3%

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

       8. metadata-eval 69.3%

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

       9. log-E 69.3%

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

       10. associate-*r/ 69.3%

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

       11. log-E 69.3%

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

       12. metadata-eval 69.3%

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

       13. metadata-eval 69.3%

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

   11. Simplified 69.3%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(v \cdot u\right) \cdot \mathsf{expm1}\left(\frac{2}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 90.7% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + u \cdot \left(\left(\frac{2 + \frac{1.3333333333333333}{v}}{v} - \frac{2}{u}\right) - -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.10000000149011612)
   1.0
   (+
    1.0
    (* u (- (- (/ (+ 2.0 (/ 1.3333333333333333 v)) v) (/ 2.0 u)) -2.0)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + (u * ((((2.0f + (1.3333333333333333f / v)) / 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 <= 0.10000000149011612e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 + (u * ((((2.0e0 + (1.3333333333333333e0 / v)) / v) - (2.0e0 / u)) - (-2.0e0)))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.10000000149011612))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + Float32(u * Float32(Float32(Float32(Float32(Float32(2.0) + Float32(Float32(1.3333333333333333) / v)) / v) - Float32(Float32(2.0) / u)) - Float32(-2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.10000000149011612))
		tmp = single(1.0);
	else
		tmp = single(1.0) + (u * ((((single(2.0) + (single(1.3333333333333333) / v)) / v) - (single(2.0) / u)) - single(-2.0)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.100000001

   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. Step-by-step derivation

      1. add-sqr-sqrt 100.0%

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

      2. log-prod 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-undefine 100.0%

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

      5. +-commutative 100.0%

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

      6. fma-undefine 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. count-2 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in v around 0 91.8%

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

   if 0.100000001 < v

   1. Initial program 91.5%

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

   3. Taylor expanded in u around 0 68.8%

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

   4. Taylor expanded in v around -inf 65.4%

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

   5. Taylor expanded in u around inf 65.1%

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

   6. Taylor expanded in u around -inf 65.1%

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

      1. mul-1-neg 65.1%

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

      2. *-commutative 65.1%

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

      3. distribute-rgt-neg-in 65.1%

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

      4. sub-neg 65.1%

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

      5. associate-*r/ 65.1%

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

      6. metadata-eval 65.1%

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

      7. +-commutative 65.1%

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

      8. mul-1-neg 65.1%

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

      9. associate-*r/ 65.1%

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

      10. metadata-eval 65.1%

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

      11. unsub-neg 65.1%

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

      12. metadata-eval 65.1%

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

   8. Simplified 65.1%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + u \cdot \left(\left(\frac{2 + \frac{1.3333333333333333}{v}}{v} - \frac{2}{u}\right) - -2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 90.7% accurate, 10.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 - \frac{1.3333333333333333 \cdot \frac{-1}{v} - 2}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.10000000149011612)
   1.0
   (+ -1.0 (* u (- 2.0 (/ (- (* 1.3333333333333333 (/ -1.0 v)) 2.0) v))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + (u * (2.0f - (((1.3333333333333333f * (-1.0f / v)) - 2.0f) / v)));
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.10000000149011612e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + (u * (2.0e0 - (((1.3333333333333333e0 * ((-1.0e0) / v)) - 2.0e0) / v)))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.10000000149011612))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(u * Float32(Float32(2.0) - Float32(Float32(Float32(Float32(1.3333333333333333) * Float32(Float32(-1.0) / v)) - Float32(2.0)) / v))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.10000000149011612))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + (u * (single(2.0) - (((single(1.3333333333333333) * (single(-1.0) / v)) - single(2.0)) / v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.100000001

   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. Step-by-step derivation

      1. add-sqr-sqrt 100.0%

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

      2. log-prod 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-undefine 100.0%

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

      5. +-commutative 100.0%

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

      6. fma-undefine 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. count-2 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in v around 0 91.8%

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

   if 0.100000001 < v

   1. Initial program 91.5%

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

   3. Taylor expanded in u around 0 68.8%

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

   4. Taylor expanded in v around -inf 65.4%

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

   5. Taylor expanded in u around 0 65.8%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 - \frac{1.3333333333333333 \cdot \frac{-1}{v} - 2}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 90.4% accurate, 13.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(-2 + 2 \cdot \left(u + \frac{u}{v}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.10000000149011612) 1.0 (+ 1.0 (+ -2.0 (* 2.0 (+ u (/ u v)))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + (-2.0f + (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 <= 0.10000000149011612e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 + ((-2.0e0) + (2.0e0 * (u + (u / v))))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.10000000149011612))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + Float32(Float32(-2.0) + 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(0.10000000149011612))
		tmp = single(1.0);
	else
		tmp = single(1.0) + (single(-2.0) + (single(2.0) * (u + (u / v))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.100000001

   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. Step-by-step derivation

      1. add-sqr-sqrt 100.0%

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

      2. log-prod 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-undefine 100.0%

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

      5. +-commutative 100.0%

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

      6. fma-undefine 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. count-2 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in v around 0 91.8%

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

   if 0.100000001 < v

   1. Initial program 91.5%

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

   3. Taylor expanded in u around 0 68.8%

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

   4. Taylor expanded in v around inf 62.3%

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

      1. sub-neg 62.3%

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

      2. distribute-lft-out 62.3%

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

      3. metadata-eval 62.3%

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

   6. Simplified 62.3%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(-2 + 2 \cdot \left(u + \frac{u}{v}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 90.4% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + 2 \cdot \left(u + \frac{u}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.10000000149011612) 1.0 (+ -1.0 (* 2.0 (+ u (/ u v))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		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 <= 0.10000000149011612e0) 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(0.10000000149011612))
		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(0.10000000149011612))
		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 < 0.100000001

   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. Step-by-step derivation

      1. add-sqr-sqrt 100.0%

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

      2. log-prod 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-undefine 100.0%

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

      5. +-commutative 100.0%

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

      6. fma-undefine 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. count-2 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in v around 0 91.8%

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

   if 0.100000001 < v

   1. Initial program 91.5%

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

   3. Taylor expanded in u around 0 68.8%

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

   4. Taylor expanded in v around inf 62.2%

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

      1. sub-neg 62.2%

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

      2. distribute-lft-out 62.2%

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

      3. metadata-eval 62.2%

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

   6. Simplified 62.2%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + 2 \cdot \left(u + \frac{u}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 89.8% accurate, 17.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + -2 \cdot \left(1 - u\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.10000000149011612) 1.0 (+ 1.0 (* -2.0 (- 1.0 u)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + (-2.0f * (1.0f - u));
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.10000000149011612e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 + ((-2.0e0) * (1.0e0 - u))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.10000000149011612))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + Float32(Float32(-2.0) * Float32(Float32(1.0) - u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.10000000149011612))
		tmp = single(1.0);
	else
		tmp = single(1.0) + (single(-2.0) * (single(1.0) - u));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.100000001

   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. Step-by-step derivation

      1. add-sqr-sqrt 100.0%

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

      2. log-prod 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-undefine 100.0%

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

      5. +-commutative 100.0%

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

      6. fma-undefine 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. count-2 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in v around 0 91.8%

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

   if 0.100000001 < v

   1. Initial program 91.5%

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

   3. Taylor expanded in v around inf 53.2%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + -2 \cdot \left(1 - u\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 89.8% accurate, 21.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.10000000149011612) 1.0 (+ -1.0 (* u 2.0))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		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 <= 0.10000000149011612e0) 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(0.10000000149011612))
		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(0.10000000149011612))
		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 < 0.100000001

   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. Step-by-step derivation

      1. add-sqr-sqrt 100.0%

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

      2. log-prod 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-undefine 100.0%

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

      5. +-commutative 100.0%

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

      6. fma-undefine 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. count-2 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in v around 0 91.8%

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

   if 0.100000001 < v

   1. Initial program 91.5%

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

   3. Taylor expanded in u around 0 68.8%

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

   4. Taylor expanded in v around inf 53.2%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 89.1% accurate, 35.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v) :precision binary32 (if (<= v 0.10000000149011612) 1.0 -1.0))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		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 <= 0.10000000149011612e0) 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(0.10000000149011612))
		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(0.10000000149011612))
		tmp = single(1.0);
	else
		tmp = single(-1.0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.100000001

   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. Step-by-step derivation

      1. add-sqr-sqrt 100.0%

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

      2. log-prod 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-undefine 100.0%

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

      5. +-commutative 100.0%

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

      6. fma-undefine 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. count-2 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in v around 0 91.8%

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

   if 0.100000001 < v

   1. Initial program 91.5%

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

   3. Taylor expanded in u around 0 43.6%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 6.0% accurate, 213.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in u around 0 6.5%

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

4. Final simplification 6.5%

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

Reproduce

herbie shell --seed 2024059 
(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))))))))