HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.5%

Time: 14.8s

Alternatives: 20

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: t_0
    t_0 = log((u + ((1.0e0 - u) * exp(((-2.0e0) / v)))))
    code = 1.0e0 + (v * (1.0e0 / (t_0 / (t_0 ** 2.0e0))))
end function

Julia

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

MATLAB

function tmp = code(u, v)
	t_0 = log((u + ((single(1.0) - u) * exp((single(-2.0) / v)))));
	tmp = single(1.0) + (v * (single(1.0) / (t_0 / (t_0 ^ single(2.0)))));
end

Derivation

1. Initial program 99.5%

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

   1. flip-+ N/A

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \log \left(\frac{u \cdot u - \left(\left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \cdot \left(\left(1 - u\right) \cdot e^{\frac{-2}{v}}\right)}{u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}}\right)\right)\right) \]

   2. div-inv N/A

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \log \left(\left(u \cdot u - \left(\left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \cdot \left(\left(1 - u\right) \cdot e^{\frac{-2}{v}}\right)\right) \cdot \frac{1}{u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}}\right)\right)\right) \]

   3. difference-of-squares N/A

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \log \left(\left(\left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \cdot \left(u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right)\right) \cdot \frac{1}{u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}}\right)\right)\right) \]

   4. associate-*l* N/A

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \log \left(\left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \cdot \left(\left(u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \cdot \frac{1}{u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}}\right)\right)\right)\right) \]

   5. log-prod N/A

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \left(\log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) + \color{blue}{\log \left(\left(u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \cdot \frac{1}{u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}}\right)}\right)\right)\right) \]

   6. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{+.f32}\left(\log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right), \color{blue}{\log \left(\left(u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \cdot \frac{1}{u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}}\right)}\right)\right)\right) \]

4. Applied egg-rr 99.5%

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

   1. flip-+ N/A

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \left(\frac{\log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) - \log \left(\left(u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \cdot \frac{1}{u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}}\right) \cdot \log \left(\left(u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \cdot \frac{1}{u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}}\right)}{\color{blue}{\log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) - \log \left(\left(u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \cdot \frac{1}{u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}}\right)}}\right)\right)\right) \]

   2. clear-num N/A

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \left(\frac{1}{\color{blue}{\frac{\log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) - \log \left(\left(u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \cdot \frac{1}{u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}}\right)}{\log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) - \log \left(\left(u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \cdot \frac{1}{u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}}\right) \cdot \log \left(\left(u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \cdot \frac{1}{u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}}\right)}}}\right)\right)\right) \]

   3. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{/.f32}\left(1, \color{blue}{\left(\frac{\log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) - \log \left(\left(u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \cdot \frac{1}{u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}}\right)}{\log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) - \log \left(\left(u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \cdot \frac{1}{u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}}\right) \cdot \log \left(\left(u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \cdot \frac{1}{u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}}\right)}\right)}\right)\right)\right) \]

6. Applied egg-rr 99.5%

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

Alternative 2: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.5%

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

   1. flip-+ N/A

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \log \left(\frac{u \cdot u - \left(\left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \cdot \left(\left(1 - u\right) \cdot e^{\frac{-2}{v}}\right)}{u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}}\right)\right)\right) \]

   2. div-inv N/A

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \log \left(\left(u \cdot u - \left(\left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \cdot \left(\left(1 - u\right) \cdot e^{\frac{-2}{v}}\right)\right) \cdot \frac{1}{u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}}\right)\right)\right) \]

   3. difference-of-squares N/A

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \log \left(\left(\left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \cdot \left(u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right)\right) \cdot \frac{1}{u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}}\right)\right)\right) \]

   4. associate-*l* N/A

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \log \left(\left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \cdot \left(\left(u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \cdot \frac{1}{u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}}\right)\right)\right)\right) \]

   5. log-prod N/A

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \left(\log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) + \color{blue}{\log \left(\left(u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \cdot \frac{1}{u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}}\right)}\right)\right)\right) \]

   6. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{+.f32}\left(\log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right), \color{blue}{\log \left(\left(u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \cdot \frac{1}{u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}}\right)}\right)\right)\right) \]

4. Applied egg-rr 99.5%

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

   1. flip-+ N/A

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \left(\frac{\log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) - \log \left(\left(u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \cdot \frac{1}{u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}}\right) \cdot \log \left(\left(u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \cdot \frac{1}{u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}}\right)}{\color{blue}{\log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) - \log \left(\left(u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \cdot \frac{1}{u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}}\right)}}\right)\right)\right) \]

   2. clear-num N/A

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \left(\frac{1}{\color{blue}{\frac{\log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) - \log \left(\left(u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \cdot \frac{1}{u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}}\right)}{\log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) - \log \left(\left(u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \cdot \frac{1}{u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}}\right) \cdot \log \left(\left(u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \cdot \frac{1}{u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}}\right)}}}\right)\right)\right) \]

   3. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{/.f32}\left(1, \color{blue}{\left(\frac{\log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) - \log \left(\left(u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \cdot \frac{1}{u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}}\right)}{\log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) - \log \left(\left(u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \cdot \frac{1}{u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}}\right) \cdot \log \left(\left(u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \cdot \frac{1}{u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}}\right)}\right)}\right)\right)\right) \]

6. Applied egg-rr 99.5%

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

7. Taylor expanded in v around 0

   \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{/.f32}\left(1, \color{blue}{\left(\frac{1}{\log \left(u + e^{\frac{-2}{v}} \cdot \left(1 - u\right)\right)}\right)}\right)\right)\right) \]
8. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(1, \color{blue}{\log \left(u + e^{\frac{-2}{v}} \cdot \left(1 - u\right)\right)}\right)\right)\right)\right) \]

   2. log-lowering-log.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(1, \mathsf{log.f32}\left(\left(u + e^{\frac{-2}{v}} \cdot \left(1 - u\right)\right)\right)\right)\right)\right)\right) \]

   3. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(1, \mathsf{log.f32}\left(\mathsf{+.f32}\left(u, \left(e^{\frac{-2}{v}} \cdot \left(1 - u\right)\right)\right)\right)\right)\right)\right)\right) \]

   4. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(1, \mathsf{log.f32}\left(\mathsf{+.f32}\left(u, \mathsf{*.f32}\left(\left(e^{\frac{-2}{v}}\right), \left(1 - u\right)\right)\right)\right)\right)\right)\right)\right) \]

   5. exp-lowering-exp.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(1, \mathsf{log.f32}\left(\mathsf{+.f32}\left(u, \mathsf{*.f32}\left(\mathsf{exp.f32}\left(\left(\frac{-2}{v}\right)\right), \left(1 - u\right)\right)\right)\right)\right)\right)\right)\right) \]

   6. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(1, \mathsf{log.f32}\left(\mathsf{+.f32}\left(u, \mathsf{*.f32}\left(\mathsf{exp.f32}\left(\mathsf{/.f32}\left(-2, v\right)\right), \left(1 - u\right)\right)\right)\right)\right)\right)\right)\right) \]

   7. --lowering--.f32 99.5%

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(1, \mathsf{log.f32}\left(\mathsf{+.f32}\left(u, \mathsf{*.f32}\left(\mathsf{exp.f32}\left(\mathsf{/.f32}\left(-2, v\right)\right), \mathsf{\_.f32}\left(1, u\right)\right)\right)\right)\right)\right)\right)\right) \]

9. Simplified 99.5%

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

10. Final simplification 99.5%

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

Alternative 3: 98.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - u\right) \cdot \left(1 - u\right)\\ t_1 := \left(1 - u\right) \cdot 16\\ \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1 + v \cdot \log \left(\mathsf{expm1}\left(\frac{-2}{v}\right) \cdot \left(-u\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(1 - u\right) \cdot -2 + \frac{\frac{\left(\left(1 - u\right) \cdot 8 + t\_0 \cdot \left(t\_1 + -24\right)\right) \cdot -0.16666666666666666 + \frac{0.041666666666666664 \cdot \left(\left(t\_1 + t\_0 \cdot \left(t\_0 \cdot -96\right)\right) + \left(1 - u\right) \cdot \left(\left(1 - u\right) \cdot \left(-112 + \left(1 - u\right) \cdot 192\right)\right)\right)}{v}}{v} + -0.5 \cdot \left(\left(1 - u\right) \cdot \left(-4 \cdot \left(u + -1\right) - 4\right)\right)}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (let* ((t_0 (* (- 1.0 u) (- 1.0 u))) (t_1 (* (- 1.0 u) 16.0)))
   (if (<= v 0.20000000298023224)
     (+ 1.0 (* v (log (* (expm1 (/ -2.0 v)) (- u)))))
     (+
      1.0
      (+
       (* (- 1.0 u) -2.0)
       (/
        (+
         (/
          (+
           (* (+ (* (- 1.0 u) 8.0) (* t_0 (+ t_1 -24.0))) -0.16666666666666666)
           (/
            (*
             0.041666666666666664
             (+
              (+ t_1 (* t_0 (* t_0 -96.0)))
              (* (- 1.0 u) (* (- 1.0 u) (+ -112.0 (* (- 1.0 u) 192.0))))))
            v))
          v)
         (* -0.5 (* (- 1.0 u) (- (* -4.0 (+ u -1.0)) 4.0))))
        v))))))

C

float code(float u, float v) {
	float t_0 = (1.0f - u) * (1.0f - u);
	float t_1 = (1.0f - u) * 16.0f;
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f + (v * logf((expm1f((-2.0f / v)) * -u)));
	} else {
		tmp = 1.0f + (((1.0f - u) * -2.0f) + ((((((((1.0f - u) * 8.0f) + (t_0 * (t_1 + -24.0f))) * -0.16666666666666666f) + ((0.041666666666666664f * ((t_1 + (t_0 * (t_0 * -96.0f))) + ((1.0f - u) * ((1.0f - u) * (-112.0f + ((1.0f - u) * 192.0f)))))) / v)) / v) + (-0.5f * ((1.0f - u) * ((-4.0f * (u + -1.0f)) - 4.0f)))) / v));
	}
	return tmp;
}

Julia

function code(u, v)
	t_0 = Float32(Float32(Float32(1.0) - u) * Float32(Float32(1.0) - u))
	t_1 = Float32(Float32(Float32(1.0) - u) * Float32(16.0))
	tmp = Float32(0.0)
	if (v <= Float32(0.20000000298023224))
		tmp = Float32(Float32(1.0) + Float32(v * log(Float32(expm1(Float32(Float32(-2.0) / v)) * Float32(-u)))));
	else
		tmp = Float32(Float32(1.0) + Float32(Float32(Float32(Float32(1.0) - u) * Float32(-2.0)) + Float32(Float32(Float32(Float32(Float32(Float32(Float32(Float32(Float32(1.0) - u) * Float32(8.0)) + Float32(t_0 * Float32(t_1 + Float32(-24.0)))) * Float32(-0.16666666666666666)) + Float32(Float32(Float32(0.041666666666666664) * Float32(Float32(t_1 + Float32(t_0 * Float32(t_0 * Float32(-96.0)))) + Float32(Float32(Float32(1.0) - u) * Float32(Float32(Float32(1.0) - u) * Float32(Float32(-112.0) + Float32(Float32(Float32(1.0) - u) * Float32(192.0))))))) / v)) / v) + Float32(Float32(-0.5) * Float32(Float32(Float32(1.0) - u) * Float32(Float32(Float32(-4.0) * Float32(u + Float32(-1.0))) - Float32(4.0))))) / v)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   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 inf

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{log.f32}\left(\color{blue}{\left(u \cdot \left(1 + -1 \cdot e^{\frac{-2}{v}}\right)\right)}\right)\right)\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{log.f32}\left(\left(\left(1 + -1 \cdot e^{\frac{-2}{v}}\right) \cdot u\right)\right)\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{log.f32}\left(\left(\left(-1 \cdot e^{\frac{-2}{v}} + 1\right) \cdot u\right)\right)\right)\right) \]

      3. mul-1-neg N/A

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{log.f32}\left(\left(\left(\left(\mathsf{neg}\left(e^{\frac{-2}{v}}\right)\right) + 1\right) \cdot u\right)\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{log.f32}\left(\left(\left(\left(\mathsf{neg}\left(e^{\frac{-2}{v}}\right)\right) + \left(\mathsf{neg}\left(-1\right)\right)\right) \cdot u\right)\right)\right)\right) \]

      5. distribute-neg-in N/A

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{log.f32}\left(\left(\left(\mathsf{neg}\left(\left(e^{\frac{-2}{v}} + -1\right)\right)\right) \cdot u\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{log.f32}\left(\left(\left(\mathsf{neg}\left(\left(e^{\frac{-2}{v}} + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right) \cdot u\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{log.f32}\left(\left(\left(\mathsf{neg}\left(\left(e^{\frac{-2}{v}} - 1\right)\right)\right) \cdot u\right)\right)\right)\right) \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{log.f32}\left(\left(\mathsf{neg}\left(\left(e^{\frac{-2}{v}} - 1\right) \cdot u\right)\right)\right)\right)\right) \]

      9. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{log.f32}\left(\left(\left(e^{\frac{-2}{v}} - 1\right) \cdot \left(\mathsf{neg}\left(u\right)\right)\right)\right)\right)\right) \]

      10. neg-mul-1 N/A

          \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{log.f32}\left(\left(\left(e^{\frac{-2}{v}} - 1\right) \cdot \left(-1 \cdot u\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{log.f32}\left(\mathsf{*.f32}\left(\left(e^{\frac{-2}{v}} - 1\right), \left(-1 \cdot u\right)\right)\right)\right)\right) \]

      12. expm1-define N/A

          \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{log.f32}\left(\mathsf{*.f32}\left(\left(\mathsf{expm1}\left(\frac{-2}{v}\right)\right), \left(-1 \cdot u\right)\right)\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{log.f32}\left(\mathsf{*.f32}\left(\left(\mathsf{expm1}\left(\frac{\mathsf{neg}\left(2\right)}{v}\right)\right), \left(-1 \cdot u\right)\right)\right)\right)\right) \]

      14. distribute-neg-frac N/A

          \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{log.f32}\left(\mathsf{*.f32}\left(\left(\mathsf{expm1}\left(\mathsf{neg}\left(\frac{2}{v}\right)\right)\right), \left(-1 \cdot u\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{log.f32}\left(\mathsf{*.f32}\left(\left(\mathsf{expm1}\left(\mathsf{neg}\left(\frac{2 \cdot 1}{v}\right)\right)\right), \left(-1 \cdot u\right)\right)\right)\right)\right) \]

      16. associate-*r/ N/A

          \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{log.f32}\left(\mathsf{*.f32}\left(\left(\mathsf{expm1}\left(\mathsf{neg}\left(2 \cdot \frac{1}{v}\right)\right)\right), \left(-1 \cdot u\right)\right)\right)\right)\right) \]

      17. expm1-lowering-expm1.f32 N/A

          \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{log.f32}\left(\mathsf{*.f32}\left(\mathsf{expm1.f32}\left(\left(\mathsf{neg}\left(2 \cdot \frac{1}{v}\right)\right)\right), \left(-1 \cdot u\right)\right)\right)\right)\right) \]

      18. associate-*r/ N/A

          \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{log.f32}\left(\mathsf{*.f32}\left(\mathsf{expm1.f32}\left(\left(\mathsf{neg}\left(\frac{2 \cdot 1}{v}\right)\right)\right), \left(-1 \cdot u\right)\right)\right)\right)\right) \]

      19. metadata-eval N/A

          \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{log.f32}\left(\mathsf{*.f32}\left(\mathsf{expm1.f32}\left(\left(\mathsf{neg}\left(\frac{2}{v}\right)\right)\right), \left(-1 \cdot u\right)\right)\right)\right)\right) \]

      20. distribute-neg-frac N/A

          \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{log.f32}\left(\mathsf{*.f32}\left(\mathsf{expm1.f32}\left(\left(\frac{\mathsf{neg}\left(2\right)}{v}\right)\right), \left(-1 \cdot u\right)\right)\right)\right)\right) \]

      21. metadata-eval N/A

          \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{log.f32}\left(\mathsf{*.f32}\left(\mathsf{expm1.f32}\left(\left(\frac{-2}{v}\right)\right), \left(-1 \cdot u\right)\right)\right)\right)\right) \]

      22. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{log.f32}\left(\mathsf{*.f32}\left(\mathsf{expm1.f32}\left(\mathsf{/.f32}\left(-2, v\right)\right), \left(-1 \cdot u\right)\right)\right)\right)\right) \]

      23. neg-mul-1 N/A

          \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{log.f32}\left(\mathsf{*.f32}\left(\mathsf{expm1.f32}\left(\mathsf{/.f32}\left(-2, v\right)\right), \left(\mathsf{neg}\left(u\right)\right)\right)\right)\right)\right) \]

      24. neg-lowering-neg.f32 99.7%

          \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{log.f32}\left(\mathsf{*.f32}\left(\mathsf{expm1.f32}\left(\mathsf{/.f32}\left(-2, v\right)\right), \mathsf{neg.f32}\left(u\right)\right)\right)\right)\right) \]

   5. Simplified 99.7%

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

   if 0.200000003 < v

   1. Initial program 91.1%

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

   3. Taylor expanded in v around -inf

      \[\leadsto \mathsf{+.f32}\left(1, \color{blue}{\left(-2 \cdot \left(1 - u\right) + -1 \cdot \frac{-1 \cdot \frac{\frac{-1}{6} \cdot \left(-24 \cdot {\left(1 - u\right)}^{2} + \left(8 \cdot \left(1 - u\right) + 16 \cdot {\left(1 - u\right)}^{3}\right)\right) + \frac{1}{24} \cdot \frac{-96 \cdot {\left(1 - u\right)}^{4} + \left(-64 \cdot {\left(1 - u\right)}^{2} + \left(-48 \cdot {\left(1 - u\right)}^{2} + \left(16 \cdot \left(1 - u\right) + 192 \cdot {\left(1 - u\right)}^{3}\right)\right)\right)}{v}}{v} + \frac{-1}{2} \cdot \left(-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)\right)}{v}\right)}\right) \]

   5. Simplified 84.9%

      \[\leadsto 1 + \color{blue}{\left(-2 \cdot \left(1 - u\right) - \frac{\left(\left(1 - u\right) \cdot \left(\left(1 - u\right) \cdot -4 + 4\right)\right) \cdot -0.5 - \frac{\left(\left(1 - u\right) \cdot 8 + \left(\left(1 - u\right) \cdot \left(1 - u\right)\right) \cdot \left(\left(1 - u\right) \cdot 16 + -24\right)\right) \cdot -0.16666666666666666 + \frac{0.041666666666666664 \cdot \left(-96 \cdot {\left(1 - u\right)}^{4} + \left(\left(\left(1 - u\right) \cdot \left(1 - u\right)\right) \cdot \left(-112 + \left(1 - u\right) \cdot 192\right) + \left(1 - u\right) \cdot 16\right)\right)}{v}}{v}}{v}\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{\_.f32}\left(1, u\right)\right), \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), -4\right), 4\right)\right), \frac{-1}{2}\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), 8\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{\_.f32}\left(1, u\right)\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), 16\right), -24\right)\right)\right), \frac{-1}{6}\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{24}, \left(-96 \cdot {\left(1 - u\right)}^{4} + \left(\left(1 - u\right) \cdot 16 + \left(\left(1 - u\right) \cdot \left(1 - u\right)\right) \cdot \left(-112 + \left(1 - u\right) \cdot 192\right)\right)\right)\right), v\right)\right), v\right)\right), v\right)\right)\right) \]

      2. associate-+r+ N/A

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{\_.f32}\left(1, u\right)\right), \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), -4\right), 4\right)\right), \frac{-1}{2}\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), 8\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{\_.f32}\left(1, u\right)\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), 16\right), -24\right)\right)\right), \frac{-1}{6}\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{24}, \left(\left(-96 \cdot {\left(1 - u\right)}^{4} + \left(1 - u\right) \cdot 16\right) + \left(\left(1 - u\right) \cdot \left(1 - u\right)\right) \cdot \left(-112 + \left(1 - u\right) \cdot 192\right)\right)\right), v\right)\right), v\right)\right), v\right)\right)\right) \]

      3. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{\_.f32}\left(1, u\right)\right), \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), -4\right), 4\right)\right), \frac{-1}{2}\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), 8\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{\_.f32}\left(1, u\right)\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), 16\right), -24\right)\right)\right), \frac{-1}{6}\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{24}, \mathsf{+.f32}\left(\left(-96 \cdot {\left(1 - u\right)}^{4} + \left(1 - u\right) \cdot 16\right), \left(\left(\left(1 - u\right) \cdot \left(1 - u\right)\right) \cdot \left(-112 + \left(1 - u\right) \cdot 192\right)\right)\right)\right), v\right)\right), v\right)\right), v\right)\right)\right) \]

   7. Applied egg-rr 84.9%

      \[\leadsto 1 + \left(-2 \cdot \left(1 - u\right) - \frac{\left(\left(1 - u\right) \cdot \left(\left(1 - u\right) \cdot -4 + 4\right)\right) \cdot -0.5 - \frac{\left(\left(1 - u\right) \cdot 8 + \left(\left(1 - u\right) \cdot \left(1 - u\right)\right) \cdot \left(\left(1 - u\right) \cdot 16 + -24\right)\right) \cdot -0.16666666666666666 + \frac{0.041666666666666664 \cdot \color{blue}{\left(\left(\left(\left(1 - u\right) \cdot \left(1 - u\right)\right) \cdot \left(\left(\left(1 - u\right) \cdot \left(1 - u\right)\right) \cdot -96\right) + \left(1 - u\right) \cdot 16\right) + \left(1 - u\right) \cdot \left(\left(1 - u\right) \cdot \left(-112 + \left(1 - u\right) \cdot 192\right)\right)\right)}}{v}}{v}}{v}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1 + v \cdot \log \left(\mathsf{expm1}\left(\frac{-2}{v}\right) \cdot \left(-u\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(1 - u\right) \cdot -2 + \frac{\frac{\left(\left(1 - u\right) \cdot 8 + \left(\left(1 - u\right) \cdot \left(1 - u\right)\right) \cdot \left(\left(1 - u\right) \cdot 16 + -24\right)\right) \cdot -0.16666666666666666 + \frac{0.041666666666666664 \cdot \left(\left(\left(1 - u\right) \cdot 16 + \left(\left(1 - u\right) \cdot \left(1 - u\right)\right) \cdot \left(\left(\left(1 - u\right) \cdot \left(1 - u\right)\right) \cdot -96\right)\right) + \left(1 - u\right) \cdot \left(\left(1 - u\right) \cdot \left(-112 + \left(1 - u\right) \cdot 192\right)\right)\right)}{v}}{v} + -0.5 \cdot \left(\left(1 - u\right) \cdot \left(-4 \cdot \left(u + -1\right) - 4\right)\right)}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.5%

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

   1. flip-+ N/A

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \log \left(\frac{u \cdot u - \left(\left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \cdot \left(\left(1 - u\right) \cdot e^{\frac{-2}{v}}\right)}{u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}}\right)\right)\right) \]

   2. div-inv N/A

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \log \left(\left(u \cdot u - \left(\left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \cdot \left(\left(1 - u\right) \cdot e^{\frac{-2}{v}}\right)\right) \cdot \frac{1}{u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}}\right)\right)\right) \]

   3. difference-of-squares N/A

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \log \left(\left(\left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \cdot \left(u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right)\right) \cdot \frac{1}{u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}}\right)\right)\right) \]

   4. associate-*l* N/A

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \log \left(\left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \cdot \left(\left(u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \cdot \frac{1}{u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}}\right)\right)\right)\right) \]

   5. log-prod N/A

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \left(\log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) + \color{blue}{\log \left(\left(u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \cdot \frac{1}{u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}}\right)}\right)\right)\right) \]

   6. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{+.f32}\left(\log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right), \color{blue}{\log \left(\left(u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \cdot \frac{1}{u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}}\right)}\right)\right)\right) \]

4. Applied egg-rr 99.5%

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

   1. flip-+ N/A

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \left(\frac{\log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) - \log \left(\left(u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \cdot \frac{1}{u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}}\right) \cdot \log \left(\left(u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \cdot \frac{1}{u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}}\right)}{\color{blue}{\log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) - \log \left(\left(u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \cdot \frac{1}{u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}}\right)}}\right)\right)\right) \]

   2. clear-num N/A

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \left(\frac{1}{\color{blue}{\frac{\log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) - \log \left(\left(u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \cdot \frac{1}{u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}}\right)}{\log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) - \log \left(\left(u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \cdot \frac{1}{u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}}\right) \cdot \log \left(\left(u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \cdot \frac{1}{u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}}\right)}}}\right)\right)\right) \]

   3. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{/.f32}\left(1, \color{blue}{\left(\frac{\log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) - \log \left(\left(u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \cdot \frac{1}{u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}}\right)}{\log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) - \log \left(\left(u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \cdot \frac{1}{u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}}\right) \cdot \log \left(\left(u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \cdot \frac{1}{u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}}\right)}\right)}\right)\right)\right) \]

6. Applied egg-rr 99.5%

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

7. Taylor expanded in v around 0

   \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{/.f32}\left(1, \color{blue}{\left(\frac{1}{\log \left(u + e^{\frac{-2}{v}} \cdot \left(1 - u\right)\right)}\right)}\right)\right)\right) \]
8. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(1, \color{blue}{\log \left(u + e^{\frac{-2}{v}} \cdot \left(1 - u\right)\right)}\right)\right)\right)\right) \]

   2. log-lowering-log.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(1, \mathsf{log.f32}\left(\left(u + e^{\frac{-2}{v}} \cdot \left(1 - u\right)\right)\right)\right)\right)\right)\right) \]

   3. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(1, \mathsf{log.f32}\left(\mathsf{+.f32}\left(u, \left(e^{\frac{-2}{v}} \cdot \left(1 - u\right)\right)\right)\right)\right)\right)\right)\right) \]

   4. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(1, \mathsf{log.f32}\left(\mathsf{+.f32}\left(u, \mathsf{*.f32}\left(\left(e^{\frac{-2}{v}}\right), \left(1 - u\right)\right)\right)\right)\right)\right)\right)\right) \]

   5. exp-lowering-exp.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(1, \mathsf{log.f32}\left(\mathsf{+.f32}\left(u, \mathsf{*.f32}\left(\mathsf{exp.f32}\left(\left(\frac{-2}{v}\right)\right), \left(1 - u\right)\right)\right)\right)\right)\right)\right)\right) \]

   6. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(1, \mathsf{log.f32}\left(\mathsf{+.f32}\left(u, \mathsf{*.f32}\left(\mathsf{exp.f32}\left(\mathsf{/.f32}\left(-2, v\right)\right), \left(1 - u\right)\right)\right)\right)\right)\right)\right)\right) \]

   7. --lowering--.f32 99.5%

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(1, \mathsf{log.f32}\left(\mathsf{+.f32}\left(u, \mathsf{*.f32}\left(\mathsf{exp.f32}\left(\mathsf{/.f32}\left(-2, v\right)\right), \mathsf{\_.f32}\left(1, u\right)\right)\right)\right)\right)\right)\right)\right) \]

9. Simplified 99.5%

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

    1. un-div-inv N/A

       \[\leadsto \mathsf{+.f32}\left(1, \left(\frac{v}{\color{blue}{\frac{1}{\log \left(u + e^{\frac{-2}{v}} \cdot \left(1 - u\right)\right)}}}\right)\right) \]

    2. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(v, \color{blue}{\left(\frac{1}{\log \left(u + e^{\frac{-2}{v}} \cdot \left(1 - u\right)\right)}\right)}\right)\right) \]

    3. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(v, \mathsf{/.f32}\left(1, \color{blue}{\log \left(u + e^{\frac{-2}{v}} \cdot \left(1 - u\right)\right)}\right)\right)\right) \]

    4. log-lowering-log.f32 N/A

       \[\leadsto \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(v, \mathsf{/.f32}\left(1, \mathsf{log.f32}\left(\left(u + e^{\frac{-2}{v}} \cdot \left(1 - u\right)\right)\right)\right)\right)\right) \]

    5. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(v, \mathsf{/.f32}\left(1, \mathsf{log.f32}\left(\mathsf{+.f32}\left(u, \left(e^{\frac{-2}{v}} \cdot \left(1 - u\right)\right)\right)\right)\right)\right)\right) \]

    6. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(v, \mathsf{/.f32}\left(1, \mathsf{log.f32}\left(\mathsf{+.f32}\left(u, \mathsf{*.f32}\left(\left(e^{\frac{-2}{v}}\right), \left(1 - u\right)\right)\right)\right)\right)\right)\right) \]

    7. exp-lowering-exp.f32 N/A

       \[\leadsto \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(v, \mathsf{/.f32}\left(1, \mathsf{log.f32}\left(\mathsf{+.f32}\left(u, \mathsf{*.f32}\left(\mathsf{exp.f32}\left(\left(\frac{-2}{v}\right)\right), \left(1 - u\right)\right)\right)\right)\right)\right)\right) \]

    8. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(v, \mathsf{/.f32}\left(1, \mathsf{log.f32}\left(\mathsf{+.f32}\left(u, \mathsf{*.f32}\left(\mathsf{exp.f32}\left(\mathsf{/.f32}\left(-2, v\right)\right), \left(1 - u\right)\right)\right)\right)\right)\right)\right) \]

    9. --lowering--.f32 99.5%

       \[\leadsto \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(v, \mathsf{/.f32}\left(1, \mathsf{log.f32}\left(\mathsf{+.f32}\left(u, \mathsf{*.f32}\left(\mathsf{exp.f32}\left(\mathsf{/.f32}\left(-2, v\right)\right), \mathsf{\_.f32}\left(1, u\right)\right)\right)\right)\right)\right)\right) \]

11. Applied egg-rr 99.5%

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

12. Final simplification 99.5%

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

Alternative 5: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

Alternative 6: 91.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - u\right) \cdot \left(1 - u\right)\\ t_1 := \left(1 - u\right) \cdot t\_0\\ t_2 := -4 \cdot t\_0 + \left(1 - u\right) \cdot 4\\ 1 + v \cdot \frac{1}{v \cdot \left(\frac{\frac{-0.03125 \cdot \left(t\_2 \cdot t\_2\right)}{v \cdot t\_1} + \left(\frac{0.041666666666666664 \cdot \left(t\_0 \cdot -24 + \left(\left(1 - u\right) \cdot 8 + 16 \cdot t\_1\right)\right)}{v \cdot t\_0} + 0.125 \cdot \frac{t\_2}{\left(1 - u\right) \cdot \left(u + -1\right)}\right)}{v} + \frac{0.5}{u + -1}\right)} \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (let* ((t_0 (* (- 1.0 u) (- 1.0 u)))
        (t_1 (* (- 1.0 u) t_0))
        (t_2 (+ (* -4.0 t_0) (* (- 1.0 u) 4.0))))
   (+
    1.0
    (*
     v
     (/
      1.0
      (*
       v
       (+
        (/
         (+
          (/ (* -0.03125 (* t_2 t_2)) (* v t_1))
          (+
           (/
            (*
             0.041666666666666664
             (+ (* t_0 -24.0) (+ (* (- 1.0 u) 8.0) (* 16.0 t_1))))
            (* v t_0))
           (* 0.125 (/ t_2 (* (- 1.0 u) (+ u -1.0))))))
         v)
        (/ 0.5 (+ u -1.0)))))))))

C

float code(float u, float v) {
	float t_0 = (1.0f - u) * (1.0f - u);
	float t_1 = (1.0f - u) * t_0;
	float t_2 = (-4.0f * t_0) + ((1.0f - u) * 4.0f);
	return 1.0f + (v * (1.0f / (v * (((((-0.03125f * (t_2 * t_2)) / (v * t_1)) + (((0.041666666666666664f * ((t_0 * -24.0f) + (((1.0f - u) * 8.0f) + (16.0f * t_1)))) / (v * t_0)) + (0.125f * (t_2 / ((1.0f - u) * (u + -1.0f)))))) / v) + (0.5f / (u + -1.0f))))));
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: t_0
    real(4) :: t_1
    real(4) :: t_2
    t_0 = (1.0e0 - u) * (1.0e0 - u)
    t_1 = (1.0e0 - u) * t_0
    t_2 = ((-4.0e0) * t_0) + ((1.0e0 - u) * 4.0e0)
    code = 1.0e0 + (v * (1.0e0 / (v * ((((((-0.03125e0) * (t_2 * t_2)) / (v * t_1)) + (((0.041666666666666664e0 * ((t_0 * (-24.0e0)) + (((1.0e0 - u) * 8.0e0) + (16.0e0 * t_1)))) / (v * t_0)) + (0.125e0 * (t_2 / ((1.0e0 - u) * (u + (-1.0e0))))))) / v) + (0.5e0 / (u + (-1.0e0)))))))
end function

Julia

function code(u, v)
	t_0 = Float32(Float32(Float32(1.0) - u) * Float32(Float32(1.0) - u))
	t_1 = Float32(Float32(Float32(1.0) - u) * t_0)
	t_2 = Float32(Float32(Float32(-4.0) * t_0) + Float32(Float32(Float32(1.0) - u) * Float32(4.0)))
	return Float32(Float32(1.0) + Float32(v * Float32(Float32(1.0) / Float32(v * Float32(Float32(Float32(Float32(Float32(Float32(-0.03125) * Float32(t_2 * t_2)) / Float32(v * t_1)) + Float32(Float32(Float32(Float32(0.041666666666666664) * Float32(Float32(t_0 * Float32(-24.0)) + Float32(Float32(Float32(Float32(1.0) - u) * Float32(8.0)) + Float32(Float32(16.0) * t_1)))) / Float32(v * t_0)) + Float32(Float32(0.125) * Float32(t_2 / Float32(Float32(Float32(1.0) - u) * Float32(u + Float32(-1.0))))))) / v) + Float32(Float32(0.5) / Float32(u + Float32(-1.0))))))))
end

MATLAB

function tmp = code(u, v)
	t_0 = (single(1.0) - u) * (single(1.0) - u);
	t_1 = (single(1.0) - u) * t_0;
	t_2 = (single(-4.0) * t_0) + ((single(1.0) - u) * single(4.0));
	tmp = single(1.0) + (v * (single(1.0) / (v * (((((single(-0.03125) * (t_2 * t_2)) / (v * t_1)) + (((single(0.041666666666666664) * ((t_0 * single(-24.0)) + (((single(1.0) - u) * single(8.0)) + (single(16.0) * t_1)))) / (v * t_0)) + (single(0.125) * (t_2 / ((single(1.0) - u) * (u + single(-1.0))))))) / v) + (single(0.5) / (u + single(-1.0)))))));
end

Derivation

1. Initial program 99.5%

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

   1. flip-+ N/A

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \log \left(\frac{u \cdot u - \left(\left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \cdot \left(\left(1 - u\right) \cdot e^{\frac{-2}{v}}\right)}{u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}}\right)\right)\right) \]

   2. div-inv N/A

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \log \left(\left(u \cdot u - \left(\left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \cdot \left(\left(1 - u\right) \cdot e^{\frac{-2}{v}}\right)\right) \cdot \frac{1}{u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}}\right)\right)\right) \]

   3. difference-of-squares N/A

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \log \left(\left(\left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \cdot \left(u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right)\right) \cdot \frac{1}{u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}}\right)\right)\right) \]

   4. associate-*l* N/A

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \log \left(\left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \cdot \left(\left(u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \cdot \frac{1}{u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}}\right)\right)\right)\right) \]

   5. log-prod N/A

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \left(\log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) + \color{blue}{\log \left(\left(u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \cdot \frac{1}{u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}}\right)}\right)\right)\right) \]

   6. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{+.f32}\left(\log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right), \color{blue}{\log \left(\left(u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \cdot \frac{1}{u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}}\right)}\right)\right)\right) \]

4. Applied egg-rr 99.5%

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

   1. flip-+ N/A

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \left(\frac{\log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) - \log \left(\left(u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \cdot \frac{1}{u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}}\right) \cdot \log \left(\left(u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \cdot \frac{1}{u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}}\right)}{\color{blue}{\log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) - \log \left(\left(u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \cdot \frac{1}{u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}}\right)}}\right)\right)\right) \]

   2. clear-num N/A

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \left(\frac{1}{\color{blue}{\frac{\log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) - \log \left(\left(u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \cdot \frac{1}{u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}}\right)}{\log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) - \log \left(\left(u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \cdot \frac{1}{u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}}\right) \cdot \log \left(\left(u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \cdot \frac{1}{u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}}\right)}}}\right)\right)\right) \]

   3. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{/.f32}\left(1, \color{blue}{\left(\frac{\log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) - \log \left(\left(u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \cdot \frac{1}{u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}}\right)}{\log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) - \log \left(\left(u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \cdot \frac{1}{u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}}\right) \cdot \log \left(\left(u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \cdot \frac{1}{u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}}\right)}\right)}\right)\right)\right) \]

6. Applied egg-rr 99.5%

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

7. Taylor expanded in v around -inf

   \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{/.f32}\left(1, \color{blue}{\left(-1 \cdot \left(v \cdot \left(-1 \cdot \frac{\left(\frac{-1}{32} \cdot \frac{{\left(-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)\right)}^{2}}{v \cdot {\left(1 - u\right)}^{3}} + \frac{1}{24} \cdot \frac{-24 \cdot {\left(1 - u\right)}^{2} + \left(8 \cdot \left(1 - u\right) + 16 \cdot {\left(1 - u\right)}^{3}\right)}{v \cdot {\left(1 - u\right)}^{2}}\right) - \frac{1}{8} \cdot \frac{-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)}{{\left(1 - u\right)}^{2}}}{v} + \frac{1}{2} \cdot \frac{1}{1 - u}\right)\right)\right)}\right)\right)\right) \]

9. Simplified 92.8%

   \[\leadsto 1 + v \cdot \frac{1}{\color{blue}{\left(-v\right) \cdot \left(\frac{0.5}{1 - u} - \frac{\frac{-0.03125 \cdot \left(\left(-4 \cdot \left(\left(1 - u\right) \cdot \left(1 - u\right)\right) + \left(1 - u\right) \cdot 4\right) \cdot \left(-4 \cdot \left(\left(1 - u\right) \cdot \left(1 - u\right)\right) + \left(1 - u\right) \cdot 4\right)\right)}{v \cdot \left(\left(1 - u\right) \cdot \left(\left(1 - u\right) \cdot \left(1 - u\right)\right)\right)} + \left(\frac{0.041666666666666664 \cdot \left(\left(\left(1 - u\right) \cdot \left(1 - u\right)\right) \cdot -24 + \left(\left(1 - u\right) \cdot 8 + \left(\left(1 - u\right) \cdot \left(\left(1 - u\right) \cdot \left(1 - u\right)\right)\right) \cdot 16\right)\right)}{v \cdot \left(\left(1 - u\right) \cdot \left(1 - u\right)\right)} - 0.125 \cdot \frac{-4 \cdot \left(\left(1 - u\right) \cdot \left(1 - u\right)\right) + \left(1 - u\right) \cdot 4}{\left(1 - u\right) \cdot \left(1 - u\right)}\right)}{v}\right)}} \]

10. Final simplification 92.8%

    \[\leadsto 1 + v \cdot \frac{1}{v \cdot \left(\frac{\frac{-0.03125 \cdot \left(\left(-4 \cdot \left(\left(1 - u\right) \cdot \left(1 - u\right)\right) + \left(1 - u\right) \cdot 4\right) \cdot \left(-4 \cdot \left(\left(1 - u\right) \cdot \left(1 - u\right)\right) + \left(1 - u\right) \cdot 4\right)\right)}{v \cdot \left(\left(1 - u\right) \cdot \left(\left(1 - u\right) \cdot \left(1 - u\right)\right)\right)} + \left(\frac{0.041666666666666664 \cdot \left(\left(\left(1 - u\right) \cdot \left(1 - u\right)\right) \cdot -24 + \left(\left(1 - u\right) \cdot 8 + 16 \cdot \left(\left(1 - u\right) \cdot \left(\left(1 - u\right) \cdot \left(1 - u\right)\right)\right)\right)\right)}{v \cdot \left(\left(1 - u\right) \cdot \left(1 - u\right)\right)} + 0.125 \cdot \frac{-4 \cdot \left(\left(1 - u\right) \cdot \left(1 - u\right)\right) + \left(1 - u\right) \cdot 4}{\left(1 - u\right) \cdot \left(u + -1\right)}\right)}{v} + \frac{0.5}{u + -1}\right)} \]
11. Add Preprocessing

Alternative 7: 91.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - u\right) \cdot \left(1 - u\right)\\ t_1 := \left(1 - u\right) \cdot 16\\ \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(1 - u\right) \cdot -2 + \frac{\frac{\left(\left(1 - u\right) \cdot 8 + t\_0 \cdot \left(t\_1 + -24\right)\right) \cdot -0.16666666666666666 + \frac{0.041666666666666664 \cdot \left(\left(t\_1 + t\_0 \cdot \left(t\_0 \cdot -96\right)\right) + \left(1 - u\right) \cdot \left(\left(1 - u\right) \cdot \left(-112 + \left(1 - u\right) \cdot 192\right)\right)\right)}{v}}{v} + -0.5 \cdot \left(\left(1 - u\right) \cdot \left(-4 \cdot \left(u + -1\right) - 4\right)\right)}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (let* ((t_0 (* (- 1.0 u) (- 1.0 u))) (t_1 (* (- 1.0 u) 16.0)))
   (if (<= v 0.10000000149011612)
     1.0
     (+
      1.0
      (+
       (* (- 1.0 u) -2.0)
       (/
        (+
         (/
          (+
           (* (+ (* (- 1.0 u) 8.0) (* t_0 (+ t_1 -24.0))) -0.16666666666666666)
           (/
            (*
             0.041666666666666664
             (+
              (+ t_1 (* t_0 (* t_0 -96.0)))
              (* (- 1.0 u) (* (- 1.0 u) (+ -112.0 (* (- 1.0 u) 192.0))))))
            v))
          v)
         (* -0.5 (* (- 1.0 u) (- (* -4.0 (+ u -1.0)) 4.0))))
        v))))))

C

float code(float u, float v) {
	float t_0 = (1.0f - u) * (1.0f - u);
	float t_1 = (1.0f - u) * 16.0f;
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + (((1.0f - u) * -2.0f) + ((((((((1.0f - u) * 8.0f) + (t_0 * (t_1 + -24.0f))) * -0.16666666666666666f) + ((0.041666666666666664f * ((t_1 + (t_0 * (t_0 * -96.0f))) + ((1.0f - u) * ((1.0f - u) * (-112.0f + ((1.0f - u) * 192.0f)))))) / v)) / v) + (-0.5f * ((1.0f - u) * ((-4.0f * (u + -1.0f)) - 4.0f)))) / v));
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: t_0
    real(4) :: t_1
    real(4) :: tmp
    t_0 = (1.0e0 - u) * (1.0e0 - u)
    t_1 = (1.0e0 - u) * 16.0e0
    if (v <= 0.10000000149011612e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 + (((1.0e0 - u) * (-2.0e0)) + ((((((((1.0e0 - u) * 8.0e0) + (t_0 * (t_1 + (-24.0e0)))) * (-0.16666666666666666e0)) + ((0.041666666666666664e0 * ((t_1 + (t_0 * (t_0 * (-96.0e0)))) + ((1.0e0 - u) * ((1.0e0 - u) * ((-112.0e0) + ((1.0e0 - u) * 192.0e0)))))) / v)) / v) + ((-0.5e0) * ((1.0e0 - u) * (((-4.0e0) * (u + (-1.0e0))) - 4.0e0)))) / v))
    end if
    code = tmp
end function

Julia

function code(u, v)
	t_0 = Float32(Float32(Float32(1.0) - u) * Float32(Float32(1.0) - u))
	t_1 = Float32(Float32(Float32(1.0) - u) * Float32(16.0))
	tmp = Float32(0.0)
	if (v <= Float32(0.10000000149011612))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + Float32(Float32(Float32(Float32(1.0) - u) * Float32(-2.0)) + Float32(Float32(Float32(Float32(Float32(Float32(Float32(Float32(Float32(1.0) - u) * Float32(8.0)) + Float32(t_0 * Float32(t_1 + Float32(-24.0)))) * Float32(-0.16666666666666666)) + Float32(Float32(Float32(0.041666666666666664) * Float32(Float32(t_1 + Float32(t_0 * Float32(t_0 * Float32(-96.0)))) + Float32(Float32(Float32(1.0) - u) * Float32(Float32(Float32(1.0) - u) * Float32(Float32(-112.0) + Float32(Float32(Float32(1.0) - u) * Float32(192.0))))))) / v)) / v) + Float32(Float32(-0.5) * Float32(Float32(Float32(1.0) - u) * Float32(Float32(Float32(-4.0) * Float32(u + Float32(-1.0))) - Float32(4.0))))) / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	t_0 = (single(1.0) - u) * (single(1.0) - u);
	t_1 = (single(1.0) - u) * single(16.0);
	tmp = single(0.0);
	if (v <= single(0.10000000149011612))
		tmp = single(1.0);
	else
		tmp = single(1.0) + (((single(1.0) - u) * single(-2.0)) + ((((((((single(1.0) - u) * single(8.0)) + (t_0 * (t_1 + single(-24.0)))) * single(-0.16666666666666666)) + ((single(0.041666666666666664) * ((t_1 + (t_0 * (t_0 * single(-96.0)))) + ((single(1.0) - u) * ((single(1.0) - u) * (single(-112.0) + ((single(1.0) - u) * single(192.0))))))) / v)) / v) + (single(-0.5) * ((single(1.0) - u) * ((single(-4.0) * (u + single(-1.0))) - single(4.0))))) / 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. Taylor expanded in v around 0

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 94.6%

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

   if 0.100000001 < v

   1. Initial program 91.7%

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

   3. Taylor expanded in v around -inf

      \[\leadsto \mathsf{+.f32}\left(1, \color{blue}{\left(-2 \cdot \left(1 - u\right) + -1 \cdot \frac{-1 \cdot \frac{\frac{-1}{6} \cdot \left(-24 \cdot {\left(1 - u\right)}^{2} + \left(8 \cdot \left(1 - u\right) + 16 \cdot {\left(1 - u\right)}^{3}\right)\right) + \frac{1}{24} \cdot \frac{-96 \cdot {\left(1 - u\right)}^{4} + \left(-64 \cdot {\left(1 - u\right)}^{2} + \left(-48 \cdot {\left(1 - u\right)}^{2} + \left(16 \cdot \left(1 - u\right) + 192 \cdot {\left(1 - u\right)}^{3}\right)\right)\right)}{v}}{v} + \frac{-1}{2} \cdot \left(-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)\right)}{v}\right)}\right) \]

   5. Simplified 81.3%

      \[\leadsto 1 + \color{blue}{\left(-2 \cdot \left(1 - u\right) - \frac{\left(\left(1 - u\right) \cdot \left(\left(1 - u\right) \cdot -4 + 4\right)\right) \cdot -0.5 - \frac{\left(\left(1 - u\right) \cdot 8 + \left(\left(1 - u\right) \cdot \left(1 - u\right)\right) \cdot \left(\left(1 - u\right) \cdot 16 + -24\right)\right) \cdot -0.16666666666666666 + \frac{0.041666666666666664 \cdot \left(-96 \cdot {\left(1 - u\right)}^{4} + \left(\left(\left(1 - u\right) \cdot \left(1 - u\right)\right) \cdot \left(-112 + \left(1 - u\right) \cdot 192\right) + \left(1 - u\right) \cdot 16\right)\right)}{v}}{v}}{v}\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{\_.f32}\left(1, u\right)\right), \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), -4\right), 4\right)\right), \frac{-1}{2}\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), 8\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{\_.f32}\left(1, u\right)\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), 16\right), -24\right)\right)\right), \frac{-1}{6}\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{24}, \left(-96 \cdot {\left(1 - u\right)}^{4} + \left(\left(1 - u\right) \cdot 16 + \left(\left(1 - u\right) \cdot \left(1 - u\right)\right) \cdot \left(-112 + \left(1 - u\right) \cdot 192\right)\right)\right)\right), v\right)\right), v\right)\right), v\right)\right)\right) \]

      2. associate-+r+ N/A

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{\_.f32}\left(1, u\right)\right), \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), -4\right), 4\right)\right), \frac{-1}{2}\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), 8\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{\_.f32}\left(1, u\right)\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), 16\right), -24\right)\right)\right), \frac{-1}{6}\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{24}, \left(\left(-96 \cdot {\left(1 - u\right)}^{4} + \left(1 - u\right) \cdot 16\right) + \left(\left(1 - u\right) \cdot \left(1 - u\right)\right) \cdot \left(-112 + \left(1 - u\right) \cdot 192\right)\right)\right), v\right)\right), v\right)\right), v\right)\right)\right) \]

      3. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{\_.f32}\left(1, u\right)\right), \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), -4\right), 4\right)\right), \frac{-1}{2}\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), 8\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{\_.f32}\left(1, u\right)\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), 16\right), -24\right)\right)\right), \frac{-1}{6}\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{24}, \mathsf{+.f32}\left(\left(-96 \cdot {\left(1 - u\right)}^{4} + \left(1 - u\right) \cdot 16\right), \left(\left(\left(1 - u\right) \cdot \left(1 - u\right)\right) \cdot \left(-112 + \left(1 - u\right) \cdot 192\right)\right)\right)\right), v\right)\right), v\right)\right), v\right)\right)\right) \]

   7. Applied egg-rr 81.3%

      \[\leadsto 1 + \left(-2 \cdot \left(1 - u\right) - \frac{\left(\left(1 - u\right) \cdot \left(\left(1 - u\right) \cdot -4 + 4\right)\right) \cdot -0.5 - \frac{\left(\left(1 - u\right) \cdot 8 + \left(\left(1 - u\right) \cdot \left(1 - u\right)\right) \cdot \left(\left(1 - u\right) \cdot 16 + -24\right)\right) \cdot -0.16666666666666666 + \frac{0.041666666666666664 \cdot \color{blue}{\left(\left(\left(\left(1 - u\right) \cdot \left(1 - u\right)\right) \cdot \left(\left(\left(1 - u\right) \cdot \left(1 - u\right)\right) \cdot -96\right) + \left(1 - u\right) \cdot 16\right) + \left(1 - u\right) \cdot \left(\left(1 - u\right) \cdot \left(-112 + \left(1 - u\right) \cdot 192\right)\right)\right)}}{v}}{v}}{v}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(1 - u\right) \cdot -2 + \frac{\frac{\left(\left(1 - u\right) \cdot 8 + \left(\left(1 - u\right) \cdot \left(1 - u\right)\right) \cdot \left(\left(1 - u\right) \cdot 16 + -24\right)\right) \cdot -0.16666666666666666 + \frac{0.041666666666666664 \cdot \left(\left(\left(1 - u\right) \cdot 16 + \left(\left(1 - u\right) \cdot \left(1 - u\right)\right) \cdot \left(\left(\left(1 - u\right) \cdot \left(1 - u\right)\right) \cdot -96\right)\right) + \left(1 - u\right) \cdot \left(\left(1 - u\right) \cdot \left(-112 + \left(1 - u\right) \cdot 192\right)\right)\right)}{v}}{v} + -0.5 \cdot \left(\left(1 - u\right) \cdot \left(-4 \cdot \left(u + -1\right) - 4\right)\right)}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 91.7% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(1 - u\right) \cdot -2 + \frac{u \cdot \left(\left(\frac{0.6666666666666666}{v \cdot v} + \left(2 + \frac{1.3333333333333333}{v}\right)\right) + u \cdot \left(\left(\left(u \cdot \left(\left(\frac{2.6666666666666665}{v} + \frac{8}{v \cdot v}\right) - 4 \cdot \frac{u}{v \cdot v}\right) - \frac{4.666666666666667}{v \cdot v}\right) - \frac{4}{v}\right) - 2\right)\right)}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.10000000149011612)
   1.0
   (+
    1.0
    (+
     (* (- 1.0 u) -2.0)
     (/
      (*
       u
       (+
        (+ (/ 0.6666666666666666 (* v v)) (+ 2.0 (/ 1.3333333333333333 v)))
        (*
         u
         (-
          (-
           (-
            (*
             u
             (-
              (+ (/ 2.6666666666666665 v) (/ 8.0 (* v v)))
              (* 4.0 (/ u (* v v)))))
            (/ 4.666666666666667 (* v v)))
           (/ 4.0 v))
          2.0))))
      v)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + (((1.0f - u) * -2.0f) + ((u * (((0.6666666666666666f / (v * v)) + (2.0f + (1.3333333333333333f / v))) + (u * ((((u * (((2.6666666666666665f / v) + (8.0f / (v * v))) - (4.0f * (u / (v * v))))) - (4.666666666666667f / (v * v))) - (4.0f / 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 + (((1.0e0 - u) * (-2.0e0)) + ((u * (((0.6666666666666666e0 / (v * v)) + (2.0e0 + (1.3333333333333333e0 / v))) + (u * ((((u * (((2.6666666666666665e0 / v) + (8.0e0 / (v * v))) - (4.0e0 * (u / (v * v))))) - (4.666666666666667e0 / (v * v))) - (4.0e0 / 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(Float32(Float32(Float32(1.0) - u) * Float32(-2.0)) + Float32(Float32(u * Float32(Float32(Float32(Float32(0.6666666666666666) / Float32(v * v)) + Float32(Float32(2.0) + Float32(Float32(1.3333333333333333) / v))) + Float32(u * Float32(Float32(Float32(Float32(u * Float32(Float32(Float32(Float32(2.6666666666666665) / v) + Float32(Float32(8.0) / Float32(v * v))) - Float32(Float32(4.0) * Float32(u / Float32(v * v))))) - Float32(Float32(4.666666666666667) / Float32(v * v))) - Float32(Float32(4.0) / 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) + (((single(1.0) - u) * single(-2.0)) + ((u * (((single(0.6666666666666666) / (v * v)) + (single(2.0) + (single(1.3333333333333333) / v))) + (u * ((((u * (((single(2.6666666666666665) / v) + (single(8.0) / (v * v))) - (single(4.0) * (u / (v * v))))) - (single(4.666666666666667) / (v * v))) - (single(4.0) / 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. Taylor expanded in v around 0

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 94.6%

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

   if 0.100000001 < v

   1. Initial program 91.7%

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

   3. Taylor expanded in v around -inf

      \[\leadsto \mathsf{+.f32}\left(1, \color{blue}{\left(-2 \cdot \left(1 - u\right) + -1 \cdot \frac{-1 \cdot \frac{\frac{-1}{6} \cdot \left(-24 \cdot {\left(1 - u\right)}^{2} + \left(8 \cdot \left(1 - u\right) + 16 \cdot {\left(1 - u\right)}^{3}\right)\right) + \frac{1}{24} \cdot \frac{-96 \cdot {\left(1 - u\right)}^{4} + \left(-64 \cdot {\left(1 - u\right)}^{2} + \left(-48 \cdot {\left(1 - u\right)}^{2} + \left(16 \cdot \left(1 - u\right) + 192 \cdot {\left(1 - u\right)}^{3}\right)\right)\right)}{v}}{v} + \frac{-1}{2} \cdot \left(-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)\right)}{v}\right)}\right) \]

   5. Simplified 81.3%

      \[\leadsto 1 + \color{blue}{\left(-2 \cdot \left(1 - u\right) - \frac{\left(\left(1 - u\right) \cdot \left(\left(1 - u\right) \cdot -4 + 4\right)\right) \cdot -0.5 - \frac{\left(\left(1 - u\right) \cdot 8 + \left(\left(1 - u\right) \cdot \left(1 - u\right)\right) \cdot \left(\left(1 - u\right) \cdot 16 + -24\right)\right) \cdot -0.16666666666666666 + \frac{0.041666666666666664 \cdot \left(-96 \cdot {\left(1 - u\right)}^{4} + \left(\left(\left(1 - u\right) \cdot \left(1 - u\right)\right) \cdot \left(-112 + \left(1 - u\right) \cdot 192\right) + \left(1 - u\right) \cdot 16\right)\right)}{v}}{v}}{v}\right)} \]

   6. Taylor expanded in u around 0

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{\_.f32}\left(1, u\right)\right), \mathsf{/.f32}\left(\color{blue}{\left(u \cdot \left(u \cdot \left(2 + \left(4 \cdot \frac{1}{v} + \left(\frac{14}{3} \cdot \frac{1}{{v}^{2}} + u \cdot \left(4 \cdot \frac{u}{{v}^{2}} - \left(\frac{8}{3} \cdot \frac{1}{v} + 8 \cdot \frac{1}{{v}^{2}}\right)\right)\right)\right)\right) - \left(2 + \left(\frac{2}{3} \cdot \frac{1}{{v}^{2}} + \frac{4}{3} \cdot \frac{1}{v}\right)\right)\right)\right)}, v\right)\right)\right) \]

   8. Simplified 81.3%

      \[\leadsto 1 + \left(-2 \cdot \left(1 - u\right) - \frac{\color{blue}{u \cdot \left(u \cdot \left(2 + \left(\frac{4}{v} + \left(\frac{4.666666666666667}{v \cdot v} + u \cdot \left(4 \cdot \frac{u}{v \cdot v} - \left(\frac{2.6666666666666665}{v} + \frac{8}{v \cdot v}\right)\right)\right)\right)\right) - \left(\frac{0.6666666666666666}{v \cdot v} + \left(\frac{1.3333333333333333}{v} + 2\right)\right)\right)}}{v}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(1 - u\right) \cdot -2 + \frac{u \cdot \left(\left(\frac{0.6666666666666666}{v \cdot v} + \left(2 + \frac{1.3333333333333333}{v}\right)\right) + u \cdot \left(\left(\left(u \cdot \left(\left(\frac{2.6666666666666665}{v} + \frac{8}{v \cdot v}\right) - 4 \cdot \frac{u}{v \cdot v}\right) - \frac{4.666666666666667}{v \cdot v}\right) - \frac{4}{v}\right) - 2\right)\right)}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 91.6% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(1 - u\right) \cdot -2 + \frac{u \cdot \left(\left(\frac{0.6666666666666666}{v \cdot v} + \left(2 + \frac{1.3333333333333333}{v}\right)\right) + u \cdot \left(\left(u \cdot \left(\frac{2.6666666666666665}{v} + \frac{8}{v \cdot v}\right) - \left(\frac{4}{v} + \frac{4.666666666666667}{v \cdot v}\right)\right) - 2\right)\right)}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.10000000149011612)
   1.0
   (+
    1.0
    (+
     (* (- 1.0 u) -2.0)
     (/
      (*
       u
       (+
        (+ (/ 0.6666666666666666 (* v v)) (+ 2.0 (/ 1.3333333333333333 v)))
        (*
         u
         (-
          (-
           (* u (+ (/ 2.6666666666666665 v) (/ 8.0 (* v v))))
           (+ (/ 4.0 v) (/ 4.666666666666667 (* v v))))
          2.0))))
      v)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + (((1.0f - u) * -2.0f) + ((u * (((0.6666666666666666f / (v * v)) + (2.0f + (1.3333333333333333f / v))) + (u * (((u * ((2.6666666666666665f / v) + (8.0f / (v * v)))) - ((4.0f / v) + (4.666666666666667f / (v * 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 + (((1.0e0 - u) * (-2.0e0)) + ((u * (((0.6666666666666666e0 / (v * v)) + (2.0e0 + (1.3333333333333333e0 / v))) + (u * (((u * ((2.6666666666666665e0 / v) + (8.0e0 / (v * v)))) - ((4.0e0 / v) + (4.666666666666667e0 / (v * 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(Float32(Float32(Float32(1.0) - u) * Float32(-2.0)) + Float32(Float32(u * Float32(Float32(Float32(Float32(0.6666666666666666) / Float32(v * v)) + Float32(Float32(2.0) + Float32(Float32(1.3333333333333333) / v))) + Float32(u * Float32(Float32(Float32(u * Float32(Float32(Float32(2.6666666666666665) / v) + Float32(Float32(8.0) / Float32(v * v)))) - Float32(Float32(Float32(4.0) / v) + Float32(Float32(4.666666666666667) / Float32(v * 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) + (((single(1.0) - u) * single(-2.0)) + ((u * (((single(0.6666666666666666) / (v * v)) + (single(2.0) + (single(1.3333333333333333) / v))) + (u * (((u * ((single(2.6666666666666665) / v) + (single(8.0) / (v * v)))) - ((single(4.0) / v) + (single(4.666666666666667) / (v * 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. Taylor expanded in v around 0

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 94.6%

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

   if 0.100000001 < v

   1. Initial program 91.7%

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

   3. Taylor expanded in v around -inf

      \[\leadsto \mathsf{+.f32}\left(1, \color{blue}{\left(-2 \cdot \left(1 - u\right) + -1 \cdot \frac{-1 \cdot \frac{\frac{-1}{6} \cdot \left(-24 \cdot {\left(1 - u\right)}^{2} + \left(8 \cdot \left(1 - u\right) + 16 \cdot {\left(1 - u\right)}^{3}\right)\right) + \frac{1}{24} \cdot \frac{-96 \cdot {\left(1 - u\right)}^{4} + \left(-64 \cdot {\left(1 - u\right)}^{2} + \left(-48 \cdot {\left(1 - u\right)}^{2} + \left(16 \cdot \left(1 - u\right) + 192 \cdot {\left(1 - u\right)}^{3}\right)\right)\right)}{v}}{v} + \frac{-1}{2} \cdot \left(-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)\right)}{v}\right)}\right) \]

   5. Simplified 81.3%

      \[\leadsto 1 + \color{blue}{\left(-2 \cdot \left(1 - u\right) - \frac{\left(\left(1 - u\right) \cdot \left(\left(1 - u\right) \cdot -4 + 4\right)\right) \cdot -0.5 - \frac{\left(\left(1 - u\right) \cdot 8 + \left(\left(1 - u\right) \cdot \left(1 - u\right)\right) \cdot \left(\left(1 - u\right) \cdot 16 + -24\right)\right) \cdot -0.16666666666666666 + \frac{0.041666666666666664 \cdot \left(-96 \cdot {\left(1 - u\right)}^{4} + \left(\left(\left(1 - u\right) \cdot \left(1 - u\right)\right) \cdot \left(-112 + \left(1 - u\right) \cdot 192\right) + \left(1 - u\right) \cdot 16\right)\right)}{v}}{v}}{v}\right)} \]

   6. Taylor expanded in u around 0

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{\_.f32}\left(1, u\right)\right), \mathsf{/.f32}\left(\color{blue}{\left(u \cdot \left(u \cdot \left(2 + \left(-1 \cdot \left(u \cdot \left(\frac{8}{3} \cdot \frac{1}{v} + 8 \cdot \frac{1}{{v}^{2}}\right)\right) + \left(4 \cdot \frac{1}{v} + \frac{14}{3} \cdot \frac{1}{{v}^{2}}\right)\right)\right) - \left(2 + \left(\frac{2}{3} \cdot \frac{1}{{v}^{2}} + \frac{4}{3} \cdot \frac{1}{v}\right)\right)\right)\right)}, v\right)\right)\right) \]

   8. Simplified 80.2%

      \[\leadsto 1 + \left(-2 \cdot \left(1 - u\right) - \frac{\color{blue}{u \cdot \left(u \cdot \left(2 + \left(\left(\frac{4}{v} + \frac{4.666666666666667}{v \cdot v}\right) - u \cdot \left(\frac{2.6666666666666665}{v} + \frac{8}{v \cdot v}\right)\right)\right) - \left(\frac{0.6666666666666666}{v \cdot v} + \left(\frac{1.3333333333333333}{v} + 2\right)\right)\right)}}{v}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(1 - u\right) \cdot -2 + \frac{u \cdot \left(\left(\frac{0.6666666666666666}{v \cdot v} + \left(2 + \frac{1.3333333333333333}{v}\right)\right) + u \cdot \left(\left(u \cdot \left(\frac{2.6666666666666665}{v} + \frac{8}{v \cdot v}\right) - \left(\frac{4}{v} + \frac{4.666666666666667}{v \cdot v}\right)\right) - 2\right)\right)}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 91.3% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := v \cdot \left(v \cdot v\right)\\ \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;u \cdot \left(\left(2 - u \cdot \left(\frac{2}{v} + \left(\frac{4}{v \cdot v} + \frac{4.666666666666667}{t\_0}\right)\right)\right) + \left(\frac{2}{v} + \left(\frac{1.3333333333333333}{v \cdot v} + \frac{0.6666666666666666}{t\_0}\right)\right)\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (let* ((t_0 (* v (* v v))))
   (if (<= v 0.10000000149011612)
     1.0
     (+
      (*
       u
       (+
        (-
         2.0
         (* u (+ (/ 2.0 v) (+ (/ 4.0 (* v v)) (/ 4.666666666666667 t_0)))))
        (+
         (/ 2.0 v)
         (+ (/ 1.3333333333333333 (* v v)) (/ 0.6666666666666666 t_0)))))
      -1.0))))

C

float code(float u, float v) {
	float t_0 = v * (v * v);
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f;
	} else {
		tmp = (u * ((2.0f - (u * ((2.0f / v) + ((4.0f / (v * v)) + (4.666666666666667f / t_0))))) + ((2.0f / v) + ((1.3333333333333333f / (v * v)) + (0.6666666666666666f / t_0))))) + -1.0f;
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: t_0
    real(4) :: tmp
    t_0 = v * (v * v)
    if (v <= 0.10000000149011612e0) then
        tmp = 1.0e0
    else
        tmp = (u * ((2.0e0 - (u * ((2.0e0 / v) + ((4.0e0 / (v * v)) + (4.666666666666667e0 / t_0))))) + ((2.0e0 / v) + ((1.3333333333333333e0 / (v * v)) + (0.6666666666666666e0 / t_0))))) + (-1.0e0)
    end if
    code = tmp
end function

Julia

function code(u, v)
	t_0 = Float32(v * Float32(v * v))
	tmp = Float32(0.0)
	if (v <= Float32(0.10000000149011612))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(u * Float32(Float32(Float32(2.0) - Float32(u * Float32(Float32(Float32(2.0) / v) + Float32(Float32(Float32(4.0) / Float32(v * v)) + Float32(Float32(4.666666666666667) / t_0))))) + Float32(Float32(Float32(2.0) / v) + Float32(Float32(Float32(1.3333333333333333) / Float32(v * v)) + Float32(Float32(0.6666666666666666) / t_0))))) + Float32(-1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	t_0 = v * (v * v);
	tmp = single(0.0);
	if (v <= single(0.10000000149011612))
		tmp = single(1.0);
	else
		tmp = (u * ((single(2.0) - (u * ((single(2.0) / v) + ((single(4.0) / (v * v)) + (single(4.666666666666667) / t_0))))) + ((single(2.0) / v) + ((single(1.3333333333333333) / (v * v)) + (single(0.6666666666666666) / t_0))))) + 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. Taylor expanded in v around 0

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 94.6%

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

   if 0.100000001 < v

   1. Initial program 91.7%

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

   3. Taylor expanded in v around -inf

      \[\leadsto \mathsf{+.f32}\left(1, \color{blue}{\left(-2 \cdot \left(1 - u\right) + -1 \cdot \frac{-1 \cdot \frac{\frac{-1}{6} \cdot \left(-24 \cdot {\left(1 - u\right)}^{2} + \left(8 \cdot \left(1 - u\right) + 16 \cdot {\left(1 - u\right)}^{3}\right)\right) + \frac{1}{24} \cdot \frac{-96 \cdot {\left(1 - u\right)}^{4} + \left(-64 \cdot {\left(1 - u\right)}^{2} + \left(-48 \cdot {\left(1 - u\right)}^{2} + \left(16 \cdot \left(1 - u\right) + 192 \cdot {\left(1 - u\right)}^{3}\right)\right)\right)}{v}}{v} + \frac{-1}{2} \cdot \left(-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)\right)}{v}\right)}\right) \]

   5. Simplified 81.3%

      \[\leadsto 1 + \color{blue}{\left(-2 \cdot \left(1 - u\right) - \frac{\left(\left(1 - u\right) \cdot \left(\left(1 - u\right) \cdot -4 + 4\right)\right) \cdot -0.5 - \frac{\left(\left(1 - u\right) \cdot 8 + \left(\left(1 - u\right) \cdot \left(1 - u\right)\right) \cdot \left(\left(1 - u\right) \cdot 16 + -24\right)\right) \cdot -0.16666666666666666 + \frac{0.041666666666666664 \cdot \left(-96 \cdot {\left(1 - u\right)}^{4} + \left(\left(\left(1 - u\right) \cdot \left(1 - u\right)\right) \cdot \left(-112 + \left(1 - u\right) \cdot 192\right) + \left(1 - u\right) \cdot 16\right)\right)}{v}}{v}}{v}\right)} \]

   6. Taylor expanded in u around 0

      \[\leadsto \color{blue}{u \cdot \left(2 + \left(-1 \cdot \left(u \cdot \left(2 \cdot \frac{1}{v} + \left(4 \cdot \frac{1}{{v}^{2}} + \frac{14}{3} \cdot \frac{1}{{v}^{3}}\right)\right)\right) + \left(\frac{2}{3} \cdot \frac{1}{{v}^{3}} + \left(\frac{4}{3} \cdot \frac{1}{{v}^{2}} + 2 \cdot \frac{1}{v}\right)\right)\right)\right) - 1} \]

   8. Simplified 77.5%

      \[\leadsto \color{blue}{u \cdot \left(\left(2 - u \cdot \left(\frac{2}{v} + \left(\frac{4}{v \cdot v} + \frac{4.666666666666667}{v \cdot \left(v \cdot v\right)}\right)\right)\right) + \left(\frac{2}{v} + \left(\frac{1.3333333333333333}{v \cdot v} + \frac{0.6666666666666666}{v \cdot \left(v \cdot v\right)}\right)\right)\right) + -1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 91.3% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(1 - u\right) \cdot -2 + \frac{u \cdot \left(\left(\frac{0.6666666666666666}{v \cdot v} + \left(2 + \frac{1.3333333333333333}{v}\right)\right) - u \cdot \left(2 + \left(\frac{4}{v} + \frac{4.666666666666667}{v \cdot v}\right)\right)\right)}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.10000000149011612)
   1.0
   (+
    1.0
    (+
     (* (- 1.0 u) -2.0)
     (/
      (*
       u
       (-
        (+ (/ 0.6666666666666666 (* v v)) (+ 2.0 (/ 1.3333333333333333 v)))
        (* u (+ 2.0 (+ (/ 4.0 v) (/ 4.666666666666667 (* v v)))))))
      v)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + (((1.0f - u) * -2.0f) + ((u * (((0.6666666666666666f / (v * v)) + (2.0f + (1.3333333333333333f / v))) - (u * (2.0f + ((4.0f / v) + (4.666666666666667f / (v * v))))))) / v));
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.10000000149011612e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 + (((1.0e0 - u) * (-2.0e0)) + ((u * (((0.6666666666666666e0 / (v * v)) + (2.0e0 + (1.3333333333333333e0 / v))) - (u * (2.0e0 + ((4.0e0 / v) + (4.666666666666667e0 / (v * v))))))) / 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(Float32(Float32(1.0) - u) * Float32(-2.0)) + Float32(Float32(u * Float32(Float32(Float32(Float32(0.6666666666666666) / Float32(v * v)) + Float32(Float32(2.0) + Float32(Float32(1.3333333333333333) / v))) - Float32(u * Float32(Float32(2.0) + Float32(Float32(Float32(4.0) / v) + Float32(Float32(4.666666666666667) / Float32(v * v))))))) / 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(1.0) - u) * single(-2.0)) + ((u * (((single(0.6666666666666666) / (v * v)) + (single(2.0) + (single(1.3333333333333333) / v))) - (u * (single(2.0) + ((single(4.0) / v) + (single(4.666666666666667) / (v * v))))))) / 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. Taylor expanded in v around 0

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 94.6%

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

   if 0.100000001 < v

   1. Initial program 91.7%

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

   3. Taylor expanded in v around -inf

      \[\leadsto \mathsf{+.f32}\left(1, \color{blue}{\left(-2 \cdot \left(1 - u\right) + -1 \cdot \frac{-1 \cdot \frac{\frac{-1}{6} \cdot \left(-24 \cdot {\left(1 - u\right)}^{2} + \left(8 \cdot \left(1 - u\right) + 16 \cdot {\left(1 - u\right)}^{3}\right)\right) + \frac{1}{24} \cdot \frac{-96 \cdot {\left(1 - u\right)}^{4} + \left(-64 \cdot {\left(1 - u\right)}^{2} + \left(-48 \cdot {\left(1 - u\right)}^{2} + \left(16 \cdot \left(1 - u\right) + 192 \cdot {\left(1 - u\right)}^{3}\right)\right)\right)}{v}}{v} + \frac{-1}{2} \cdot \left(-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)\right)}{v}\right)}\right) \]

   5. Simplified 81.3%

      \[\leadsto 1 + \color{blue}{\left(-2 \cdot \left(1 - u\right) - \frac{\left(\left(1 - u\right) \cdot \left(\left(1 - u\right) \cdot -4 + 4\right)\right) \cdot -0.5 - \frac{\left(\left(1 - u\right) \cdot 8 + \left(\left(1 - u\right) \cdot \left(1 - u\right)\right) \cdot \left(\left(1 - u\right) \cdot 16 + -24\right)\right) \cdot -0.16666666666666666 + \frac{0.041666666666666664 \cdot \left(-96 \cdot {\left(1 - u\right)}^{4} + \left(\left(\left(1 - u\right) \cdot \left(1 - u\right)\right) \cdot \left(-112 + \left(1 - u\right) \cdot 192\right) + \left(1 - u\right) \cdot 16\right)\right)}{v}}{v}}{v}\right)} \]

   6. Taylor expanded in u around 0

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{\_.f32}\left(1, u\right)\right), \mathsf{/.f32}\left(\color{blue}{\left(u \cdot \left(u \cdot \left(2 + \left(4 \cdot \frac{1}{v} + \frac{14}{3} \cdot \frac{1}{{v}^{2}}\right)\right) - \left(2 + \left(\frac{2}{3} \cdot \frac{1}{{v}^{2}} + \frac{4}{3} \cdot \frac{1}{v}\right)\right)\right)\right)}, v\right)\right)\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{\_.f32}\left(1, u\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(u, \left(u \cdot \left(2 + \left(4 \cdot \frac{1}{v} + \frac{14}{3} \cdot \frac{1}{{v}^{2}}\right)\right) - \left(2 + \left(\frac{2}{3} \cdot \frac{1}{{v}^{2}} + \frac{4}{3} \cdot \frac{1}{v}\right)\right)\right)\right), v\right)\right)\right) \]

      2. --lowering--.f32 N/A

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{\_.f32}\left(1, u\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(u, \mathsf{\_.f32}\left(\left(u \cdot \left(2 + \left(4 \cdot \frac{1}{v} + \frac{14}{3} \cdot \frac{1}{{v}^{2}}\right)\right)\right), \left(2 + \left(\frac{2}{3} \cdot \frac{1}{{v}^{2}} + \frac{4}{3} \cdot \frac{1}{v}\right)\right)\right)\right), v\right)\right)\right) \]

   8. Simplified 77.3%

      \[\leadsto 1 + \left(-2 \cdot \left(1 - u\right) - \frac{\color{blue}{u \cdot \left(u \cdot \left(2 + \left(\frac{4}{v} + \frac{4.666666666666667}{v \cdot v}\right)\right) - \left(\frac{0.6666666666666666}{v \cdot v} + \left(\frac{1.3333333333333333}{v} + 2\right)\right)\right)}}{v}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(1 - u\right) \cdot -2 + \frac{u \cdot \left(\left(\frac{0.6666666666666666}{v \cdot v} + \left(2 + \frac{1.3333333333333333}{v}\right)\right) - u \cdot \left(2 + \left(\frac{4}{v} + \frac{4.666666666666667}{v \cdot v}\right)\right)\right)}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 91.1% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(1 - u\right) \cdot -2\right) + \frac{\frac{\left(1 - u\right) \cdot \left(u \cdot 1.3333333333333333\right)}{v} + \left(\left(1 - u\right) \cdot -4 + 4\right) \cdot \left(-0.5 \cdot \left(u + -1\right)\right)}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.10000000149011612)
   1.0
   (+
    (+ 1.0 (* (- 1.0 u) -2.0))
    (/
     (+
      (/ (* (- 1.0 u) (* u 1.3333333333333333)) v)
      (* (+ (* (- 1.0 u) -4.0) 4.0) (* -0.5 (+ u -1.0))))
     v))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f;
	} else {
		tmp = (1.0f + ((1.0f - u) * -2.0f)) + (((((1.0f - u) * (u * 1.3333333333333333f)) / v) + ((((1.0f - u) * -4.0f) + 4.0f) * (-0.5f * (u + -1.0f)))) / 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 + ((1.0e0 - u) * (-2.0e0))) + (((((1.0e0 - u) * (u * 1.3333333333333333e0)) / v) + ((((1.0e0 - u) * (-4.0e0)) + 4.0e0) * ((-0.5e0) * (u + (-1.0e0))))) / 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(Float32(1.0) + Float32(Float32(Float32(1.0) - u) * Float32(-2.0))) + Float32(Float32(Float32(Float32(Float32(Float32(1.0) - u) * Float32(u * Float32(1.3333333333333333))) / v) + Float32(Float32(Float32(Float32(Float32(1.0) - u) * Float32(-4.0)) + Float32(4.0)) * Float32(Float32(-0.5) * Float32(u + Float32(-1.0))))) / 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) + ((single(1.0) - u) * single(-2.0))) + (((((single(1.0) - u) * (u * single(1.3333333333333333))) / v) + ((((single(1.0) - u) * single(-4.0)) + single(4.0)) * (single(-0.5) * (u + single(-1.0))))) / 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. Taylor expanded in v around 0

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 94.6%

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

   if 0.100000001 < v

   1. Initial program 91.7%

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

   3. Taylor expanded in v around -inf

      \[\leadsto \mathsf{+.f32}\left(1, \color{blue}{\left(-2 \cdot \left(1 - u\right) + -1 \cdot \frac{-1 \cdot \frac{\frac{-1}{6} \cdot \left(-24 \cdot {\left(1 - u\right)}^{2} + \left(8 \cdot \left(1 - u\right) + 16 \cdot {\left(1 - u\right)}^{3}\right)\right) + \frac{1}{24} \cdot \frac{-96 \cdot {\left(1 - u\right)}^{4} + \left(-64 \cdot {\left(1 - u\right)}^{2} + \left(-48 \cdot {\left(1 - u\right)}^{2} + \left(16 \cdot \left(1 - u\right) + 192 \cdot {\left(1 - u\right)}^{3}\right)\right)\right)}{v}}{v} + \frac{-1}{2} \cdot \left(-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)\right)}{v}\right)}\right) \]

   5. Simplified 81.3%

      \[\leadsto 1 + \color{blue}{\left(-2 \cdot \left(1 - u\right) - \frac{\left(\left(1 - u\right) \cdot \left(\left(1 - u\right) \cdot -4 + 4\right)\right) \cdot -0.5 - \frac{\left(\left(1 - u\right) \cdot 8 + \left(\left(1 - u\right) \cdot \left(1 - u\right)\right) \cdot \left(\left(1 - u\right) \cdot 16 + -24\right)\right) \cdot -0.16666666666666666 + \frac{0.041666666666666664 \cdot \left(-96 \cdot {\left(1 - u\right)}^{4} + \left(\left(\left(1 - u\right) \cdot \left(1 - u\right)\right) \cdot \left(-112 + \left(1 - u\right) \cdot 192\right) + \left(1 - u\right) \cdot 16\right)\right)}{v}}{v}}{v}\right)} \]

   7. Applied egg-rr 80.6%

      \[\leadsto \color{blue}{\left(1 + \left(1 - u\right) \cdot -2\right) - \frac{\left(\left(1 - u\right) \cdot -4 + 4\right) \cdot \left(\left(1 - u\right) \cdot -0.5\right) - \frac{\left(\left(1 - u\right) \cdot \left(8 + \left(1 - u\right) \cdot \left(\left(1 - u\right) \cdot 16 + -24\right)\right)\right) \cdot -0.16666666666666666 + \left(\left(\left(1 - u\right) \cdot \left(1 - u\right)\right) \cdot \left(\left(\left(1 - u\right) \cdot \left(1 - u\right)\right) \cdot -96\right) + \left(1 - u\right) \cdot \left(16 + \left(1 - u\right) \cdot \left(-112 + \left(1 - u\right) \cdot 192\right)\right)\right) \cdot \frac{0.041666666666666664}{v}}{v}}{v}} \]

   8. Taylor expanded in u around 0

      \[\leadsto \mathsf{\_.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), -2\right)\right), \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), -4\right), 4\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \frac{-1}{2}\right)\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \color{blue}{\left(-8 \cdot u\right)}\right), \frac{-1}{6}\right), \mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{\_.f32}\left(1, u\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{\_.f32}\left(1, u\right)\right), -96\right)\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{+.f32}\left(16, \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{+.f32}\left(-112, \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), 192\right)\right)\right)\right)\right)\right), \mathsf{/.f32}\left(\frac{1}{24}, v\right)\right)\right), v\right)\right), v\right)\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{\_.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), -2\right)\right), \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), -4\right), 4\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \frac{-1}{2}\right)\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \left(u \cdot -8\right)\right), \frac{-1}{6}\right), \mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{\_.f32}\left(1, u\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{\_.f32}\left(1, u\right)\right), -96\right)\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{+.f32}\left(16, \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{+.f32}\left(-112, \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), 192\right)\right)\right)\right)\right)\right), \mathsf{/.f32}\left(\frac{1}{24}, v\right)\right)\right), v\right)\right), v\right)\right) \]

      2. *-lowering-*.f32 76.1%

         \[\leadsto \mathsf{\_.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), -2\right)\right), \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), -4\right), 4\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \frac{-1}{2}\right)\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{*.f32}\left(u, -8\right)\right), \frac{-1}{6}\right), \mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{\_.f32}\left(1, u\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{\_.f32}\left(1, u\right)\right), -96\right)\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{+.f32}\left(16, \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{+.f32}\left(-112, \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), 192\right)\right)\right)\right)\right)\right), \mathsf{/.f32}\left(\frac{1}{24}, v\right)\right)\right), v\right)\right), v\right)\right) \]

   10. Simplified 76.1%

       \[\leadsto \left(1 + \left(1 - u\right) \cdot -2\right) - \frac{\left(\left(1 - u\right) \cdot -4 + 4\right) \cdot \left(\left(1 - u\right) \cdot -0.5\right) - \frac{\left(\left(1 - u\right) \cdot \color{blue}{\left(u \cdot -8\right)}\right) \cdot -0.16666666666666666 + \left(\left(\left(1 - u\right) \cdot \left(1 - u\right)\right) \cdot \left(\left(\left(1 - u\right) \cdot \left(1 - u\right)\right) \cdot -96\right) + \left(1 - u\right) \cdot \left(16 + \left(1 - u\right) \cdot \left(-112 + \left(1 - u\right) \cdot 192\right)\right)\right) \cdot \frac{0.041666666666666664}{v}}{v}}{v} \]

   11. Taylor expanded in v around inf

       \[\leadsto \mathsf{\_.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), -2\right)\right), \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), -4\right), 4\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \frac{-1}{2}\right)\right), \color{blue}{\left(\frac{4}{3} \cdot \frac{u \cdot \left(1 - u\right)}{v}\right)}\right), v\right)\right) \]
   12. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \mathsf{\_.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), -2\right)\right), \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), -4\right), 4\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \frac{-1}{2}\right)\right), \left(\frac{\frac{4}{3} \cdot \left(u \cdot \left(1 - u\right)\right)}{v}\right)\right), v\right)\right) \]

       2. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{\_.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), -2\right)\right), \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), -4\right), 4\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \frac{-1}{2}\right)\right), \mathsf{/.f32}\left(\left(\frac{4}{3} \cdot \left(u \cdot \left(1 - u\right)\right)\right), v\right)\right), v\right)\right) \]

       3. associate-*r* N/A

          \[\leadsto \mathsf{\_.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), -2\right)\right), \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), -4\right), 4\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \frac{-1}{2}\right)\right), \mathsf{/.f32}\left(\left(\left(\frac{4}{3} \cdot u\right) \cdot \left(1 - u\right)\right), v\right)\right), v\right)\right) \]

       4. metadata-eval N/A

          \[\leadsto \mathsf{\_.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), -2\right)\right), \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), -4\right), 4\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \frac{-1}{2}\right)\right), \mathsf{/.f32}\left(\left(\left(\left(\mathsf{neg}\left(\frac{-4}{3}\right)\right) \cdot u\right) \cdot \left(1 - u\right)\right), v\right)\right), v\right)\right) \]

       5. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{\_.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), -2\right)\right), \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), -4\right), 4\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \frac{-1}{2}\right)\right), \mathsf{/.f32}\left(\left(\left(\mathsf{neg}\left(\frac{-4}{3} \cdot u\right)\right) \cdot \left(1 - u\right)\right), v\right)\right), v\right)\right) \]

       6. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{\_.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), -2\right)\right), \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), -4\right), 4\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \frac{-1}{2}\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(\left(\mathsf{neg}\left(\frac{-4}{3} \cdot u\right)\right), \left(1 - u\right)\right), v\right)\right), v\right)\right) \]

       7. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{\_.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), -2\right)\right), \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), -4\right), 4\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \frac{-1}{2}\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(\left(\left(\mathsf{neg}\left(\frac{-4}{3}\right)\right) \cdot u\right), \left(1 - u\right)\right), v\right)\right), v\right)\right) \]

       8. metadata-eval N/A

          \[\leadsto \mathsf{\_.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), -2\right)\right), \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), -4\right), 4\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \frac{-1}{2}\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(\left(\frac{4}{3} \cdot u\right), \left(1 - u\right)\right), v\right)\right), v\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{\_.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), -2\right)\right), \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), -4\right), 4\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \frac{-1}{2}\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(\left(u \cdot \frac{4}{3}\right), \left(1 - u\right)\right), v\right)\right), v\right)\right) \]

       10. *-lowering-*.f32 N/A

           \[\leadsto \mathsf{\_.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), -2\right)\right), \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), -4\right), 4\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \frac{-1}{2}\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(u, \frac{4}{3}\right), \left(1 - u\right)\right), v\right)\right), v\right)\right) \]

       11. --lowering--.f32 75.9%

           \[\leadsto \mathsf{\_.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), -2\right)\right), \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), -4\right), 4\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \frac{-1}{2}\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(u, \frac{4}{3}\right), \mathsf{\_.f32}\left(1, u\right)\right), v\right)\right), v\right)\right) \]

   13. Simplified 75.9%

       \[\leadsto \left(1 + \left(1 - u\right) \cdot -2\right) - \frac{\left(\left(1 - u\right) \cdot -4 + 4\right) \cdot \left(\left(1 - u\right) \cdot -0.5\right) - \color{blue}{\frac{\left(u \cdot 1.3333333333333333\right) \cdot \left(1 - u\right)}{v}}}{v} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(1 - u\right) \cdot -2\right) + \frac{\frac{\left(1 - u\right) \cdot \left(u \cdot 1.3333333333333333\right)}{v} + \left(\left(1 - u\right) \cdot -4 + 4\right) \cdot \left(-0.5 \cdot \left(u + -1\right)\right)}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 91.2% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(1 - u\right) \cdot -2\right) + \frac{\frac{u \cdot \left(1.3333333333333333 + \frac{0.6666666666666666}{v}\right)}{v} + \left(\left(1 - u\right) \cdot -4 + 4\right) \cdot \left(-0.5 \cdot \left(u + -1\right)\right)}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.10000000149011612)
   1.0
   (+
    (+ 1.0 (* (- 1.0 u) -2.0))
    (/
     (+
      (/ (* u (+ 1.3333333333333333 (/ 0.6666666666666666 v))) v)
      (* (+ (* (- 1.0 u) -4.0) 4.0) (* -0.5 (+ u -1.0))))
     v))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f;
	} else {
		tmp = (1.0f + ((1.0f - u) * -2.0f)) + ((((u * (1.3333333333333333f + (0.6666666666666666f / v))) / v) + ((((1.0f - u) * -4.0f) + 4.0f) * (-0.5f * (u + -1.0f)))) / 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 + ((1.0e0 - u) * (-2.0e0))) + ((((u * (1.3333333333333333e0 + (0.6666666666666666e0 / v))) / v) + ((((1.0e0 - u) * (-4.0e0)) + 4.0e0) * ((-0.5e0) * (u + (-1.0e0))))) / 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(Float32(1.0) + Float32(Float32(Float32(1.0) - u) * Float32(-2.0))) + Float32(Float32(Float32(Float32(u * Float32(Float32(1.3333333333333333) + Float32(Float32(0.6666666666666666) / v))) / v) + Float32(Float32(Float32(Float32(Float32(1.0) - u) * Float32(-4.0)) + Float32(4.0)) * Float32(Float32(-0.5) * Float32(u + Float32(-1.0))))) / 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) + ((single(1.0) - u) * single(-2.0))) + ((((u * (single(1.3333333333333333) + (single(0.6666666666666666) / v))) / v) + ((((single(1.0) - u) * single(-4.0)) + single(4.0)) * (single(-0.5) * (u + single(-1.0))))) / 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. Taylor expanded in v around 0

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 94.6%

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

   if 0.100000001 < v

   1. Initial program 91.7%

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

   3. Taylor expanded in v around -inf

      \[\leadsto \mathsf{+.f32}\left(1, \color{blue}{\left(-2 \cdot \left(1 - u\right) + -1 \cdot \frac{-1 \cdot \frac{\frac{-1}{6} \cdot \left(-24 \cdot {\left(1 - u\right)}^{2} + \left(8 \cdot \left(1 - u\right) + 16 \cdot {\left(1 - u\right)}^{3}\right)\right) + \frac{1}{24} \cdot \frac{-96 \cdot {\left(1 - u\right)}^{4} + \left(-64 \cdot {\left(1 - u\right)}^{2} + \left(-48 \cdot {\left(1 - u\right)}^{2} + \left(16 \cdot \left(1 - u\right) + 192 \cdot {\left(1 - u\right)}^{3}\right)\right)\right)}{v}}{v} + \frac{-1}{2} \cdot \left(-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)\right)}{v}\right)}\right) \]

   5. Simplified 81.3%

      \[\leadsto 1 + \color{blue}{\left(-2 \cdot \left(1 - u\right) - \frac{\left(\left(1 - u\right) \cdot \left(\left(1 - u\right) \cdot -4 + 4\right)\right) \cdot -0.5 - \frac{\left(\left(1 - u\right) \cdot 8 + \left(\left(1 - u\right) \cdot \left(1 - u\right)\right) \cdot \left(\left(1 - u\right) \cdot 16 + -24\right)\right) \cdot -0.16666666666666666 + \frac{0.041666666666666664 \cdot \left(-96 \cdot {\left(1 - u\right)}^{4} + \left(\left(\left(1 - u\right) \cdot \left(1 - u\right)\right) \cdot \left(-112 + \left(1 - u\right) \cdot 192\right) + \left(1 - u\right) \cdot 16\right)\right)}{v}}{v}}{v}\right)} \]

   7. Applied egg-rr 80.6%

      \[\leadsto \color{blue}{\left(1 + \left(1 - u\right) \cdot -2\right) - \frac{\left(\left(1 - u\right) \cdot -4 + 4\right) \cdot \left(\left(1 - u\right) \cdot -0.5\right) - \frac{\left(\left(1 - u\right) \cdot \left(8 + \left(1 - u\right) \cdot \left(\left(1 - u\right) \cdot 16 + -24\right)\right)\right) \cdot -0.16666666666666666 + \left(\left(\left(1 - u\right) \cdot \left(1 - u\right)\right) \cdot \left(\left(\left(1 - u\right) \cdot \left(1 - u\right)\right) \cdot -96\right) + \left(1 - u\right) \cdot \left(16 + \left(1 - u\right) \cdot \left(-112 + \left(1 - u\right) \cdot 192\right)\right)\right) \cdot \frac{0.041666666666666664}{v}}{v}}{v}} \]

   8. Taylor expanded in u around 0

      \[\leadsto \mathsf{\_.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), -2\right)\right), \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), -4\right), 4\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \frac{-1}{2}\right)\right), \color{blue}{\left(\frac{u \cdot \left(\frac{4}{3} + \frac{2}{3} \cdot \frac{1}{v}\right)}{v}\right)}\right), v\right)\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{\_.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), -2\right)\right), \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), -4\right), 4\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \frac{-1}{2}\right)\right), \mathsf{/.f32}\left(\left(u \cdot \left(\frac{4}{3} + \frac{2}{3} \cdot \frac{1}{v}\right)\right), v\right)\right), v\right)\right) \]

      2. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{\_.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), -2\right)\right), \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), -4\right), 4\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \frac{-1}{2}\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(u, \left(\frac{4}{3} + \frac{2}{3} \cdot \frac{1}{v}\right)\right), v\right)\right), v\right)\right) \]

      3. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{\_.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), -2\right)\right), \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), -4\right), 4\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \frac{-1}{2}\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\frac{4}{3}, \left(\frac{2}{3} \cdot \frac{1}{v}\right)\right)\right), v\right)\right), v\right)\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), -2\right)\right), \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), -4\right), 4\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \frac{-1}{2}\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\frac{4}{3}, \left(\frac{\frac{2}{3} \cdot 1}{v}\right)\right)\right), v\right)\right), v\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{\_.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), -2\right)\right), \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), -4\right), 4\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \frac{-1}{2}\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\frac{4}{3}, \left(\frac{\frac{2}{3}}{v}\right)\right)\right), v\right)\right), v\right)\right) \]

      6. /-lowering-/.f32 75.4%

         \[\leadsto \mathsf{\_.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), -2\right)\right), \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), -4\right), 4\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \frac{-1}{2}\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\frac{4}{3}, \mathsf{/.f32}\left(\frac{2}{3}, v\right)\right)\right), v\right)\right), v\right)\right) \]

   10. Simplified 75.4%

       \[\leadsto \left(1 + \left(1 - u\right) \cdot -2\right) - \frac{\left(\left(1 - u\right) \cdot -4 + 4\right) \cdot \left(\left(1 - u\right) \cdot -0.5\right) - \color{blue}{\frac{u \cdot \left(1.3333333333333333 + \frac{0.6666666666666666}{v}\right)}{v}}}{v} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(1 - u\right) \cdot -2\right) + \frac{\frac{u \cdot \left(1.3333333333333333 + \frac{0.6666666666666666}{v}\right)}{v} + \left(\left(1 - u\right) \cdot -4 + 4\right) \cdot \left(-0.5 \cdot \left(u + -1\right)\right)}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 90.9% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(1 - u\right) \cdot -2 + \frac{u \cdot \left(\frac{0.6666666666666666}{v \cdot v} + \left(2 + \frac{1.3333333333333333}{v}\right)\right)}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.10000000149011612)
   1.0
   (+
    1.0
    (+
     (* (- 1.0 u) -2.0)
     (/
      (* u (+ (/ 0.6666666666666666 (* v v)) (+ 2.0 (/ 1.3333333333333333 v))))
      v)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + (((1.0f - u) * -2.0f) + ((u * ((0.6666666666666666f / (v * v)) + (2.0f + (1.3333333333333333f / v)))) / v));
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.10000000149011612e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 + (((1.0e0 - u) * (-2.0e0)) + ((u * ((0.6666666666666666e0 / (v * v)) + (2.0e0 + (1.3333333333333333e0 / v)))) / 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(Float32(Float32(1.0) - u) * Float32(-2.0)) + Float32(Float32(u * Float32(Float32(Float32(0.6666666666666666) / Float32(v * v)) + Float32(Float32(2.0) + Float32(Float32(1.3333333333333333) / v)))) / 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(1.0) - u) * single(-2.0)) + ((u * ((single(0.6666666666666666) / (v * v)) + (single(2.0) + (single(1.3333333333333333) / v)))) / 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. Taylor expanded in v around 0

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 94.6%

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

   if 0.100000001 < v

   1. Initial program 91.7%

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

   3. Taylor expanded in v around -inf

      \[\leadsto \mathsf{+.f32}\left(1, \color{blue}{\left(-2 \cdot \left(1 - u\right) + -1 \cdot \frac{-1 \cdot \frac{\frac{-1}{6} \cdot \left(-24 \cdot {\left(1 - u\right)}^{2} + \left(8 \cdot \left(1 - u\right) + 16 \cdot {\left(1 - u\right)}^{3}\right)\right) + \frac{1}{24} \cdot \frac{-96 \cdot {\left(1 - u\right)}^{4} + \left(-64 \cdot {\left(1 - u\right)}^{2} + \left(-48 \cdot {\left(1 - u\right)}^{2} + \left(16 \cdot \left(1 - u\right) + 192 \cdot {\left(1 - u\right)}^{3}\right)\right)\right)}{v}}{v} + \frac{-1}{2} \cdot \left(-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)\right)}{v}\right)}\right) \]

   5. Simplified 81.3%

      \[\leadsto 1 + \color{blue}{\left(-2 \cdot \left(1 - u\right) - \frac{\left(\left(1 - u\right) \cdot \left(\left(1 - u\right) \cdot -4 + 4\right)\right) \cdot -0.5 - \frac{\left(\left(1 - u\right) \cdot 8 + \left(\left(1 - u\right) \cdot \left(1 - u\right)\right) \cdot \left(\left(1 - u\right) \cdot 16 + -24\right)\right) \cdot -0.16666666666666666 + \frac{0.041666666666666664 \cdot \left(-96 \cdot {\left(1 - u\right)}^{4} + \left(\left(\left(1 - u\right) \cdot \left(1 - u\right)\right) \cdot \left(-112 + \left(1 - u\right) \cdot 192\right) + \left(1 - u\right) \cdot 16\right)\right)}{v}}{v}}{v}\right)} \]

   6. Taylor expanded in u around 0

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{\_.f32}\left(1, u\right)\right), \color{blue}{\left(-1 \cdot \frac{u \cdot \left(2 + \left(\frac{2}{3} \cdot \frac{1}{{v}^{2}} + \frac{4}{3} \cdot \frac{1}{v}\right)\right)}{v}\right)}\right)\right) \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{\_.f32}\left(1, u\right)\right), \left(\frac{-1 \cdot \left(u \cdot \left(2 + \left(\frac{2}{3} \cdot \frac{1}{{v}^{2}} + \frac{4}{3} \cdot \frac{1}{v}\right)\right)\right)}{\color{blue}{v}}\right)\right)\right) \]

      2. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{\_.f32}\left(1, u\right)\right), \mathsf{/.f32}\left(\left(-1 \cdot \left(u \cdot \left(2 + \left(\frac{2}{3} \cdot \frac{1}{{v}^{2}} + \frac{4}{3} \cdot \frac{1}{v}\right)\right)\right)\right), \color{blue}{v}\right)\right)\right) \]

   8. Simplified 72.2%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(1 - u\right) \cdot -2 + \frac{u \cdot \left(\frac{0.6666666666666666}{v \cdot v} + \left(2 + \frac{1.3333333333333333}{v}\right)\right)}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 91.0% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 + \frac{2 + \left(\frac{0.6666666666666666}{v \cdot v} + \frac{1.3333333333333333}{v}\right)}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.10000000149011612)
   1.0
   (+
    -1.0
    (*
     u
     (+
      2.0
      (/
       (+ 2.0 (+ (/ 0.6666666666666666 (* v v)) (/ 1.3333333333333333 v)))
       v))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + (u * (2.0f + ((2.0f + ((0.6666666666666666f / (v * v)) + (1.3333333333333333f / v))) / v)));
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.10000000149011612e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + (u * (2.0e0 + ((2.0e0 + ((0.6666666666666666e0 / (v * v)) + (1.3333333333333333e0 / v))) / 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(2.0) + Float32(Float32(Float32(0.6666666666666666) / Float32(v * v)) + Float32(Float32(1.3333333333333333) / v))) / 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(2.0) + ((single(0.6666666666666666) / (v * v)) + (single(1.3333333333333333) / v))) / 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. Taylor expanded in v around 0

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 94.6%

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

   if 0.100000001 < v

   1. Initial program 91.7%

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

   3. Taylor expanded in v around -inf

      \[\leadsto \mathsf{+.f32}\left(1, \color{blue}{\left(-2 \cdot \left(1 - u\right) + -1 \cdot \frac{-1 \cdot \frac{\frac{-1}{6} \cdot \left(-24 \cdot {\left(1 - u\right)}^{2} + \left(8 \cdot \left(1 - u\right) + 16 \cdot {\left(1 - u\right)}^{3}\right)\right) + \frac{1}{24} \cdot \frac{-96 \cdot {\left(1 - u\right)}^{4} + \left(-64 \cdot {\left(1 - u\right)}^{2} + \left(-48 \cdot {\left(1 - u\right)}^{2} + \left(16 \cdot \left(1 - u\right) + 192 \cdot {\left(1 - u\right)}^{3}\right)\right)\right)}{v}}{v} + \frac{-1}{2} \cdot \left(-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)\right)}{v}\right)}\right) \]

   5. Simplified 81.3%

      \[\leadsto 1 + \color{blue}{\left(-2 \cdot \left(1 - u\right) - \frac{\left(\left(1 - u\right) \cdot \left(\left(1 - u\right) \cdot -4 + 4\right)\right) \cdot -0.5 - \frac{\left(\left(1 - u\right) \cdot 8 + \left(\left(1 - u\right) \cdot \left(1 - u\right)\right) \cdot \left(\left(1 - u\right) \cdot 16 + -24\right)\right) \cdot -0.16666666666666666 + \frac{0.041666666666666664 \cdot \left(-96 \cdot {\left(1 - u\right)}^{4} + \left(\left(\left(1 - u\right) \cdot \left(1 - u\right)\right) \cdot \left(-112 + \left(1 - u\right) \cdot 192\right) + \left(1 - u\right) \cdot 16\right)\right)}{v}}{v}}{v}\right)} \]

   6. Taylor expanded in u around 0

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{\_.f32}\left(1, u\right)\right), \color{blue}{\left(-1 \cdot \frac{u \cdot \left(2 + \left(\frac{2}{3} \cdot \frac{1}{{v}^{2}} + \frac{4}{3} \cdot \frac{1}{v}\right)\right)}{v}\right)}\right)\right) \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{\_.f32}\left(1, u\right)\right), \left(\frac{-1 \cdot \left(u \cdot \left(2 + \left(\frac{2}{3} \cdot \frac{1}{{v}^{2}} + \frac{4}{3} \cdot \frac{1}{v}\right)\right)\right)}{\color{blue}{v}}\right)\right)\right) \]

      2. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{\_.f32}\left(1, u\right)\right), \mathsf{/.f32}\left(\left(-1 \cdot \left(u \cdot \left(2 + \left(\frac{2}{3} \cdot \frac{1}{{v}^{2}} + \frac{4}{3} \cdot \frac{1}{v}\right)\right)\right)\right), \color{blue}{v}\right)\right)\right) \]

   8. Simplified 72.2%

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

   9. Taylor expanded in u around 0

      \[\leadsto \color{blue}{u \cdot \left(2 - -1 \cdot \frac{2 + \left(\frac{2}{3} \cdot \frac{1}{{v}^{2}} + \frac{4}{3} \cdot \frac{1}{v}\right)}{v}\right) - 1} \]
   10. Step-by-step derivation

       1. sub-neg N/A

          \[\leadsto u \cdot \left(2 - -1 \cdot \frac{2 + \left(\frac{2}{3} \cdot \frac{1}{{v}^{2}} + \frac{4}{3} \cdot \frac{1}{v}\right)}{v}\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} \]

       2. metadata-eval N/A

          \[\leadsto u \cdot \left(2 - -1 \cdot \frac{2 + \left(\frac{2}{3} \cdot \frac{1}{{v}^{2}} + \frac{4}{3} \cdot \frac{1}{v}\right)}{v}\right) + -1 \]

       3. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{+.f32}\left(\left(u \cdot \left(2 - -1 \cdot \frac{2 + \left(\frac{2}{3} \cdot \frac{1}{{v}^{2}} + \frac{4}{3} \cdot \frac{1}{v}\right)}{v}\right)\right), \color{blue}{-1}\right) \]

   11. Simplified 72.1%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 + \frac{2 + \left(\frac{0.6666666666666666}{v \cdot v} + \frac{1.3333333333333333}{v}\right)}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 90.8% accurate, 8.9× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.10000000149011612e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 + (((1.0e0 - u) * (-2.0e0)) - (((u * (-2.0e0)) + ((u * (-1.3333333333333333e0)) / v)) / 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(Float32(Float32(1.0) - u) * Float32(-2.0)) - Float32(Float32(Float32(u * Float32(-2.0)) + Float32(Float32(u * Float32(-1.3333333333333333)) / v)) / 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(1.0) - u) * single(-2.0)) - (((u * single(-2.0)) + ((u * single(-1.3333333333333333)) / v)) / 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. Taylor expanded in v around 0

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 94.6%

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

   if 0.100000001 < v

   1. Initial program 91.7%

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

   3. Taylor expanded in v around -inf

      \[\leadsto \mathsf{+.f32}\left(1, \color{blue}{\left(-2 \cdot \left(1 - u\right) + -1 \cdot \frac{-1 \cdot \frac{\frac{-1}{6} \cdot \left(-24 \cdot {\left(1 - u\right)}^{2} + \left(8 \cdot \left(1 - u\right) + 16 \cdot {\left(1 - u\right)}^{3}\right)\right) + \frac{1}{24} \cdot \frac{-96 \cdot {\left(1 - u\right)}^{4} + \left(-64 \cdot {\left(1 - u\right)}^{2} + \left(-48 \cdot {\left(1 - u\right)}^{2} + \left(16 \cdot \left(1 - u\right) + 192 \cdot {\left(1 - u\right)}^{3}\right)\right)\right)}{v}}{v} + \frac{-1}{2} \cdot \left(-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)\right)}{v}\right)}\right) \]

   5. Simplified 81.3%

      \[\leadsto 1 + \color{blue}{\left(-2 \cdot \left(1 - u\right) - \frac{\left(\left(1 - u\right) \cdot \left(\left(1 - u\right) \cdot -4 + 4\right)\right) \cdot -0.5 - \frac{\left(\left(1 - u\right) \cdot 8 + \left(\left(1 - u\right) \cdot \left(1 - u\right)\right) \cdot \left(\left(1 - u\right) \cdot 16 + -24\right)\right) \cdot -0.16666666666666666 + \frac{0.041666666666666664 \cdot \left(-96 \cdot {\left(1 - u\right)}^{4} + \left(\left(\left(1 - u\right) \cdot \left(1 - u\right)\right) \cdot \left(-112 + \left(1 - u\right) \cdot 192\right) + \left(1 - u\right) \cdot 16\right)\right)}{v}}{v}}{v}\right)} \]

   6. Taylor expanded in u around 0

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{\_.f32}\left(1, u\right)\right), \color{blue}{\left(-1 \cdot \frac{u \cdot \left(2 + \left(\frac{2}{3} \cdot \frac{1}{{v}^{2}} + \frac{4}{3} \cdot \frac{1}{v}\right)\right)}{v}\right)}\right)\right) \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{\_.f32}\left(1, u\right)\right), \left(\frac{-1 \cdot \left(u \cdot \left(2 + \left(\frac{2}{3} \cdot \frac{1}{{v}^{2}} + \frac{4}{3} \cdot \frac{1}{v}\right)\right)\right)}{\color{blue}{v}}\right)\right)\right) \]

      2. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{\_.f32}\left(1, u\right)\right), \mathsf{/.f32}\left(\left(-1 \cdot \left(u \cdot \left(2 + \left(\frac{2}{3} \cdot \frac{1}{{v}^{2}} + \frac{4}{3} \cdot \frac{1}{v}\right)\right)\right)\right), \color{blue}{v}\right)\right)\right) \]

   8. Simplified 72.2%

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

   9. Taylor expanded in v around -inf

      \[\leadsto \mathsf{+.f32}\left(1, \color{blue}{\left(-2 \cdot \left(1 - u\right) + -1 \cdot \frac{\frac{-4}{3} \cdot \frac{u}{v} - 2 \cdot u}{v}\right)}\right) \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{+.f32}\left(1, \left(-2 \cdot \left(1 - u\right) + \left(\mathsf{neg}\left(\frac{\frac{-4}{3} \cdot \frac{u}{v} - 2 \cdot u}{v}\right)\right)\right)\right) \]

       2. unsub-neg N/A

          \[\leadsto \mathsf{+.f32}\left(1, \left(-2 \cdot \left(1 - u\right) - \color{blue}{\frac{\frac{-4}{3} \cdot \frac{u}{v} - 2 \cdot u}{v}}\right)\right) \]

       3. --lowering--.f32 N/A

          \[\leadsto \mathsf{+.f32}\left(1, \mathsf{\_.f32}\left(\left(-2 \cdot \left(1 - u\right)\right), \color{blue}{\left(\frac{\frac{-4}{3} \cdot \frac{u}{v} - 2 \cdot u}{v}\right)}\right)\right) \]

       4. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{+.f32}\left(1, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(-2, \left(1 - u\right)\right), \left(\frac{\color{blue}{\frac{-4}{3} \cdot \frac{u}{v} - 2 \cdot u}}{v}\right)\right)\right) \]

       5. --lowering--.f32 N/A

          \[\leadsto \mathsf{+.f32}\left(1, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{\_.f32}\left(1, u\right)\right), \left(\frac{\frac{-4}{3} \cdot \frac{u}{v} - \color{blue}{2 \cdot u}}{v}\right)\right)\right) \]

       6. cancel-sign-sub-inv N/A

          \[\leadsto \mathsf{+.f32}\left(1, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{\_.f32}\left(1, u\right)\right), \left(\frac{\frac{-4}{3} \cdot \frac{u}{v} + \left(\mathsf{neg}\left(2\right)\right) \cdot u}{v}\right)\right)\right) \]

       7. metadata-eval N/A

          \[\leadsto \mathsf{+.f32}\left(1, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{\_.f32}\left(1, u\right)\right), \left(\frac{\frac{-4}{3} \cdot \frac{u}{v} + -2 \cdot u}{v}\right)\right)\right) \]

       8. +-commutative N/A

          \[\leadsto \mathsf{+.f32}\left(1, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{\_.f32}\left(1, u\right)\right), \left(\frac{-2 \cdot u + \frac{-4}{3} \cdot \frac{u}{v}}{v}\right)\right)\right) \]

       9. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{+.f32}\left(1, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{\_.f32}\left(1, u\right)\right), \mathsf{/.f32}\left(\left(-2 \cdot u + \frac{-4}{3} \cdot \frac{u}{v}\right), \color{blue}{v}\right)\right)\right) \]

       10. +-lowering-+.f32 N/A

           \[\leadsto \mathsf{+.f32}\left(1, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{\_.f32}\left(1, u\right)\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\left(-2 \cdot u\right), \left(\frac{-4}{3} \cdot \frac{u}{v}\right)\right), v\right)\right)\right) \]

       11. *-commutative N/A

           \[\leadsto \mathsf{+.f32}\left(1, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{\_.f32}\left(1, u\right)\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\left(u \cdot -2\right), \left(\frac{-4}{3} \cdot \frac{u}{v}\right)\right), v\right)\right)\right) \]

       12. *-lowering-*.f32 N/A

           \[\leadsto \mathsf{+.f32}\left(1, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{\_.f32}\left(1, u\right)\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(u, -2\right), \left(\frac{-4}{3} \cdot \frac{u}{v}\right)\right), v\right)\right)\right) \]

       13. associate-*r/ N/A

           \[\leadsto \mathsf{+.f32}\left(1, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{\_.f32}\left(1, u\right)\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(u, -2\right), \left(\frac{\frac{-4}{3} \cdot u}{v}\right)\right), v\right)\right)\right) \]

       14. /-lowering-/.f32 N/A

           \[\leadsto \mathsf{+.f32}\left(1, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{\_.f32}\left(1, u\right)\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(u, -2\right), \mathsf{/.f32}\left(\left(\frac{-4}{3} \cdot u\right), v\right)\right), v\right)\right)\right) \]

       15. *-commutative N/A

           \[\leadsto \mathsf{+.f32}\left(1, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{\_.f32}\left(1, u\right)\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(u, -2\right), \mathsf{/.f32}\left(\left(u \cdot \frac{-4}{3}\right), v\right)\right), v\right)\right)\right) \]

       16. *-lowering-*.f32 71.5%

           \[\leadsto \mathsf{+.f32}\left(1, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{\_.f32}\left(1, u\right)\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(u, -2\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(u, \frac{-4}{3}\right), v\right)\right), v\right)\right)\right) \]

   11. Simplified 71.5%

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

4. Final simplification 93.3%

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

Alternative 17: 90.6% accurate, 13.3× speedup

Math

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

FPCore

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

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + (-2.0f + (u * (2.0f + (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 + ((-2.0e0) + (u * (2.0e0 + (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(Float32(-2.0) + Float32(u * Float32(Float32(2.0) + Float32(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) + (single(-2.0) + (u * (single(2.0) + (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. Taylor expanded in v around 0

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 94.6%

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

   if 0.100000001 < v

   1. Initial program 91.7%

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

   3. Taylor expanded in v around inf

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \color{blue}{\left(\frac{-2 \cdot \left(1 - u\right) + \frac{1}{2} \cdot \frac{-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)}{v}}{v}\right)}\right)\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{/.f32}\left(\left(-2 \cdot \left(1 - u\right) + \frac{1}{2} \cdot \frac{-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)}{v}\right), \color{blue}{v}\right)\right)\right) \]

   5. Simplified 69.4%

      \[\leadsto 1 + v \cdot \color{blue}{\frac{-2 \cdot \left(1 - u\right) + \left(\left(1 - u\right) \cdot \left(\left(1 - u\right) \cdot -4 + 4\right)\right) \cdot \frac{0.5}{v}}{v}} \]

   6. Taylor expanded in u around 0

      \[\leadsto \mathsf{+.f32}\left(1, \color{blue}{\left(u \cdot \left(2 + 2 \cdot \frac{1}{v}\right) - 2\right)}\right) \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \mathsf{+.f32}\left(1, \left(u \cdot \left(2 + 2 \cdot \frac{1}{v}\right) + \color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{+.f32}\left(1, \left(u \cdot \left(2 + 2 \cdot \frac{1}{v}\right) + -2\right)\right) \]

      3. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\left(u \cdot \left(2 + 2 \cdot \frac{1}{v}\right)\right), \color{blue}{-2}\right)\right) \]

      4. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \left(2 + 2 \cdot \frac{1}{v}\right)\right), -2\right)\right) \]

      5. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(2, \left(2 \cdot \frac{1}{v}\right)\right)\right), -2\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(2, \left(\frac{2 \cdot 1}{v}\right)\right)\right), -2\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(2, \left(\frac{2}{v}\right)\right)\right), -2\right)\right) \]

      8. /-lowering-/.f32 70.1%

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(2, \mathsf{/.f32}\left(2, v\right)\right)\right), -2\right)\right) \]

   8. Simplified 70.1%

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

4. Final simplification 93.2%

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

Alternative 18: 90.6% accurate, 15.2× speedup

Math

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

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.10000000149011612) 1.0 (+ -1.0 (* u (+ 2.0 (/ 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 + (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 + (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(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(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. Taylor expanded in v around 0

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 94.6%

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

   if 0.100000001 < v

   1. Initial program 91.7%

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

   3. Taylor expanded in v around inf

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \color{blue}{\left(\frac{-2 \cdot \left(1 - u\right) + \frac{1}{2} \cdot \frac{-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)}{v}}{v}\right)}\right)\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{/.f32}\left(\left(-2 \cdot \left(1 - u\right) + \frac{1}{2} \cdot \frac{-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)}{v}\right), \color{blue}{v}\right)\right)\right) \]

   5. Simplified 69.4%

      \[\leadsto 1 + v \cdot \color{blue}{\frac{-2 \cdot \left(1 - u\right) + \left(\left(1 - u\right) \cdot \left(\left(1 - u\right) \cdot -4 + 4\right)\right) \cdot \frac{0.5}{v}}{v}} \]

   6. Taylor expanded in u around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{+.f32}\left(\left(u \cdot \left(2 + 2 \cdot \frac{1}{v}\right)\right), \color{blue}{-1}\right) \]

      4. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \left(2 + 2 \cdot \frac{1}{v}\right)\right), -1\right) \]

      5. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(2, \left(2 \cdot \frac{1}{v}\right)\right)\right), -1\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(2, \left(\frac{2 \cdot 1}{v}\right)\right)\right), -1\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(2, \left(\frac{2}{v}\right)\right)\right), -1\right) \]

      8. /-lowering-/.f32 70.1%

         \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(2, \mathsf{/.f32}\left(2, v\right)\right)\right), -1\right) \]

   8. Simplified 70.1%

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

4. Final simplification 93.2%

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

Alternative 19: 86.9% accurate, 213.0× speedup

Math

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

FPCore

(FPCore (u v) :precision binary32 1.0)

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in v around 0

   \[\leadsto \color{blue}{1} \]
4. Step-by-step derivation

5. Simplified 89.3%

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

Alternative 20: 5.8% accurate, 213.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in u around 0

   \[\leadsto \color{blue}{-1} \]
4. Step-by-step derivation

5. Simplified 5.7%

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

Reproduce

herbie shell --seed 2024149 
(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))))))))