HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.5%

Time: 16.2s

Alternatives: 17

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.5%

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

   1. +-commutative 99.5%

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

   2. fma-define 99.5%

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

   3. +-commutative 99.5%

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

   4. fma-define 99.5%

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

3. Simplified 99.5%

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

5. Taylor expanded in u around 0 99.6%

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

   1. mul-1-neg 99.6%

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

7. Simplified 99.6%

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

8. Final simplification 99.6%

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

Alternative 2: 99.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ 1 + v \cdot \log \left(e^{\frac{-2}{v}} - u \cdot \mathsf{expm1}\left(\frac{-2}{v}\right)\right) \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (+ 1.0 (* v (log (- (exp (/ -2.0 v)) (* u (expm1 (/ -2.0 v))))))))

C

float code(float u, float v) {
	return 1.0f + (v * logf((expf((-2.0f / v)) - (u * expm1f((-2.0f / v))))));
}

Julia

function code(u, v)
	return Float32(Float32(1.0) + Float32(v * log(Float32(exp(Float32(Float32(-2.0) / v)) - Float32(u * expm1(Float32(Float32(-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. Step-by-step derivation

   1. +-commutative 99.5%

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

   2. fma-define 99.5%

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

   3. +-commutative 99.5%

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

   4. fma-define 99.5%

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

3. Simplified 99.5%

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

5. Taylor expanded in u around 0 99.6%

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

   1. mul-1-neg 99.6%

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

7. Simplified 99.6%

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

8. Taylor expanded in v around 0 99.6%

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

   1. sub-neg 99.6%

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

   2. mul-1-neg 99.6%

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

   3. +-commutative 99.6%

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

   4. mul-1-neg 99.6%

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

   5. metadata-eval 99.6%

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

   6. distribute-neg-in 99.6%

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

   7. metadata-eval 99.6%

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

   8. sub-neg 99.6%

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

   9. expm1-define 99.6%

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

10. Simplified 99.6%

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

11. Final simplification 99.6%

    \[\leadsto 1 + v \cdot \log \left(e^{\frac{-2}{v}} - u \cdot \mathsf{expm1}\left(\frac{-2}{v}\right)\right) \]
12. Add Preprocessing

Alternative 3: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, v)
	return Float32(Float32(1.0) + Float32(v * log(Float32(u + Float32(Float32(Float32(1.0) - u) * (Float32(exp(1)) ^ Float32(Float32(-2.0) / v)))))))
end

MATLAB

function tmp = code(u, v)
	tmp = single(1.0) + (v * log((u + ((single(1.0) - u) * (single(2.71828182845904523536) ^ (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. *-un-lft-identity 99.5%

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

   2. exp-prod 99.6%

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

4. Applied egg-rr 99.6%

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

   1. exp-1-e 99.6%

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

6. Simplified 99.6%

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

7. Final simplification 99.6%

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

Alternative 4: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Final simplification 99.5%

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

Alternative 5: 96.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(u, v)
	tmp = single(1.0) + (v * log((exp((single(-2.0) / v)) + u)));
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. *-un-lft-identity 99.5%

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

   2. exp-prod 99.6%

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

4. Applied egg-rr 99.6%

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

   1. exp-1-e 99.6%

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

6. Simplified 99.6%

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

7. Taylor expanded in u around 0 96.1%

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

   1. log-E 96.1%

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

   2. associate-*r/ 96.1%

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

   3. metadata-eval 96.1%

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

9. Simplified 96.1%

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

10. Final simplification 96.1%

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

Alternative 6: 91.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.30000001192092896:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;u \cdot \left(v \cdot \left(\frac{1}{e^{\frac{-2}{v}}} + -1\right)\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.30000001192092896)
   1.0
   (+ (* u (* v (+ (/ 1.0 (exp (/ -2.0 v))) -1.0))) -1.0)))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.30000001192092896f) {
		tmp = 1.0f;
	} else {
		tmp = (u * (v * ((1.0f / expf((-2.0f / v))) + -1.0f))) + -1.0f;
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.30000001192092896e0) then
        tmp = 1.0e0
    else
        tmp = (u * (v * ((1.0e0 / exp(((-2.0e0) / v))) + (-1.0e0)))) + (-1.0e0)
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.30000001192092896))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(u * Float32(v * Float32(Float32(Float32(1.0) / exp(Float32(Float32(-2.0) / v))) + Float32(-1.0)))) + Float32(-1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.30000001192092896))
		tmp = single(1.0);
	else
		tmp = (u * (v * ((single(1.0) / exp((single(-2.0) / v))) + single(-1.0)))) + single(-1.0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.300000012

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-define 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in u around 0 100.0%

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

      1. mul-1-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in v around 0 92.7%

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

   if 0.300000012 < v

   1. Initial program 93.1%

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

   3. Taylor expanded in u around 0 69.3%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.30000001192092896:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;u \cdot \left(v \cdot \left(\frac{1}{e^{\frac{-2}{v}}} + -1\right)\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.30000001192092896:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;u \cdot \left(v \cdot \mathsf{expm1}\left(\frac{2}{v}\right)\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.30000001192092896) 1.0 (+ (* u (* v (expm1 (/ 2.0 v)))) -1.0)))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.30000001192092896f) {
		tmp = 1.0f;
	} else {
		tmp = (u * (v * expm1f((2.0f / v)))) + -1.0f;
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.30000001192092896))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(u * Float32(v * expm1(Float32(Float32(2.0) / v)))) + Float32(-1.0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.300000012

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-define 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in u around 0 100.0%

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

      1. mul-1-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in v around 0 92.7%

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

   if 0.300000012 < v

   1. Initial program 93.1%

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

   3. Taylor expanded in u around 0 69.1%

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

      1. +-commutative 69.1%

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

      2. associate-+l- 69.3%

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

      3. rec-exp 69.3%

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

      4. expm1-define 69.3%

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

      5. distribute-neg-frac 69.3%

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

      6. metadata-eval 69.3%

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

      7. metadata-eval 69.3%

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

   5. Applied egg-rr 69.3%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.30000001192092896:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;u \cdot \left(v \cdot \mathsf{expm1}\left(\frac{2}{v}\right)\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 91.1% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(\frac{\frac{0.6666666666666666 \cdot \frac{u}{v} + u \cdot 1.3333333333333333}{v} - -2 \cdot u}{v} + u \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224)
   1.0
   (+
    -1.0
    (+
     (/
      (-
       (/ (+ (* 0.6666666666666666 (/ u v)) (* u 1.3333333333333333)) v)
       (* -2.0 u))
      v)
     (* u 2.0)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + ((((((0.6666666666666666f * (u / v)) + (u * 1.3333333333333333f)) / v) - (-2.0f * u)) / v) + (u * 2.0f));
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.20000000298023224e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + ((((((0.6666666666666666e0 * (u / v)) + (u * 1.3333333333333333e0)) / v) - ((-2.0e0) * u)) / v) + (u * 2.0e0))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.20000000298023224))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(Float32(Float32(Float32(Float32(Float32(Float32(0.6666666666666666) * Float32(u / v)) + Float32(u * Float32(1.3333333333333333))) / v) - Float32(Float32(-2.0) * u)) / v) + Float32(u * Float32(2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.20000000298023224))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + ((((((single(0.6666666666666666) * (u / v)) + (u * single(1.3333333333333333))) / v) - (single(-2.0) * u)) / v) + (u * single(2.0)));
	end
	tmp_2 = 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. Step-by-step derivation

      1. +-commutative 100.0%

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

      2. fma-define 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in u around 0 100.0%

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

      1. mul-1-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in v around 0 93.1%

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

   if 0.200000003 < v

   1. Initial program 93.5%

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

   3. Taylor expanded in u around 0 65.7%

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

   4. Taylor expanded in v around -inf 64.2%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(\frac{\frac{0.6666666666666666 \cdot \frac{u}{v} + u \cdot 1.3333333333333333}{v} - -2 \cdot u}{v} + u \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 90.9% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\left(2 + -2 \cdot u\right) + \frac{-2 \cdot u + \frac{u}{v} \cdot -1.3333333333333333}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224)
   1.0
   (-
    1.0
    (+
     (+ 2.0 (* -2.0 u))
     (/ (+ (* -2.0 u) (* (/ u v) -1.3333333333333333)) v)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f - ((2.0f + (-2.0f * u)) + (((-2.0f * u) + ((u / v) * -1.3333333333333333f)) / v));
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.20000000298023224e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 - ((2.0e0 + ((-2.0e0) * u)) + ((((-2.0e0) * u) + ((u / v) * (-1.3333333333333333e0))) / v))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.20000000298023224))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) - Float32(Float32(Float32(2.0) + Float32(Float32(-2.0) * u)) + Float32(Float32(Float32(Float32(-2.0) * u) + Float32(Float32(u / v) * Float32(-1.3333333333333333))) / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.20000000298023224))
		tmp = single(1.0);
	else
		tmp = single(1.0) - ((single(2.0) + (single(-2.0) * u)) + (((single(-2.0) * u) + ((u / v) * single(-1.3333333333333333))) / v));
	end
	tmp_2 = 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. Step-by-step derivation

      1. +-commutative 100.0%

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

      2. fma-define 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in u around 0 100.0%

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

      1. mul-1-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in v around 0 93.1%

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

   if 0.200000003 < v

   1. Initial program 93.5%

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

      1. +-commutative 93.5%

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

      2. fma-define 93.6%

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

      3. +-commutative 93.6%

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

      4. fma-define 93.2%

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

   3. Simplified 93.2%

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

   5. Taylor expanded in u around 0 94.3%

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

      1. mul-1-neg 94.3%

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

   7. Simplified 94.3%

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

   8. Taylor expanded in u around 0 65.6%

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

   9. Taylor expanded in v around -inf 63.0%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\left(2 + -2 \cdot u\right) + \frac{-2 \cdot u + \frac{u}{v} \cdot -1.3333333333333333}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 91.1% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 + \frac{2 + \frac{1.3333333333333333 + 0.6666666666666666 \cdot \frac{1}{v}}{v}}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224)
   1.0
   (+
    -1.0
    (*
     u
     (+
      2.0
      (/
       (+ 2.0 (/ (+ 1.3333333333333333 (* 0.6666666666666666 (/ 1.0 v))) v))
       v))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + (u * (2.0f + ((2.0f + ((1.3333333333333333f + (0.6666666666666666f * (1.0f / 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 <= 0.20000000298023224e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + (u * (2.0e0 + ((2.0e0 + ((1.3333333333333333e0 + (0.6666666666666666e0 * (1.0e0 / v))) / v)) / v)))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.20000000298023224))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(u * Float32(Float32(2.0) + Float32(Float32(Float32(2.0) + Float32(Float32(Float32(1.3333333333333333) + Float32(Float32(0.6666666666666666) * Float32(Float32(1.0) / v))) / v)) / v))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.20000000298023224))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + (u * (single(2.0) + ((single(2.0) + ((single(1.3333333333333333) + (single(0.6666666666666666) * (single(1.0) / v))) / v)) / v)));
	end
	tmp_2 = 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. Step-by-step derivation

      1. +-commutative 100.0%

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

      2. fma-define 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in u around 0 100.0%

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

      1. mul-1-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in v around 0 93.1%

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

   if 0.200000003 < v

   1. Initial program 93.5%

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

      1. +-commutative 93.5%

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

      2. fma-define 93.6%

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

      3. +-commutative 93.6%

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

      4. fma-define 93.2%

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

   3. Simplified 93.2%

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

   5. Taylor expanded in u around 0 94.3%

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

      1. mul-1-neg 94.3%

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

   7. Simplified 94.3%

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

   8. Taylor expanded in u around 0 65.6%

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

   9. Taylor expanded in v around -inf 64.1%

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

   10. Taylor expanded in u around 0 64.2%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 + \frac{2 + \frac{1.3333333333333333 + 0.6666666666666666 \cdot \frac{1}{v}}{v}}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 90.9% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(u \cdot 2 - \frac{-2 \cdot u + \frac{u}{v} \cdot -1.3333333333333333}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224)
   1.0
   (+
    -1.0
    (- (* u 2.0) (/ (+ (* -2.0 u) (* (/ u v) -1.3333333333333333)) v)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + ((u * 2.0f) - (((-2.0f * u) + ((u / v) * -1.3333333333333333f)) / v));
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.20000000298023224e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + ((u * 2.0e0) - ((((-2.0e0) * u) + ((u / v) * (-1.3333333333333333e0))) / v))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.20000000298023224))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(Float32(u * Float32(2.0)) - Float32(Float32(Float32(Float32(-2.0) * u) + Float32(Float32(u / v) * Float32(-1.3333333333333333))) / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.20000000298023224))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + ((u * single(2.0)) - (((single(-2.0) * u) + ((u / v) * single(-1.3333333333333333))) / v));
	end
	tmp_2 = 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. Step-by-step derivation

      1. +-commutative 100.0%

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

      2. fma-define 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in u around 0 100.0%

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

      1. mul-1-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in v around 0 93.1%

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

   if 0.200000003 < v

   1. Initial program 93.5%

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

   3. Taylor expanded in u around 0 65.7%

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

   4. Taylor expanded in v around -inf 63.0%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(u \cdot 2 - \frac{-2 \cdot u + \frac{u}{v} \cdot -1.3333333333333333}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 90.7% accurate, 10.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(u \cdot \left(v \cdot \frac{2 + \frac{2}{v}}{v}\right) - 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224)
   1.0
   (+ 1.0 (- (* u (* v (/ (+ 2.0 (/ 2.0 v)) v))) 2.0))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + ((u * (v * ((2.0f + (2.0f / v)) / v))) - 2.0f);
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.20000000298023224e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 + ((u * (v * ((2.0e0 + (2.0e0 / v)) / v))) - 2.0e0)
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.20000000298023224))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + Float32(Float32(u * Float32(v * Float32(Float32(Float32(2.0) + Float32(Float32(2.0) / v)) / v))) - Float32(2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.20000000298023224))
		tmp = single(1.0);
	else
		tmp = single(1.0) + ((u * (v * ((single(2.0) + (single(2.0) / v)) / v))) - single(2.0));
	end
	tmp_2 = 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. Step-by-step derivation

      1. +-commutative 100.0%

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

      2. fma-define 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in u around 0 100.0%

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

      1. mul-1-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in v around 0 93.1%

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

   if 0.200000003 < v

   1. Initial program 93.5%

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

   3. Taylor expanded in u around 0 65.7%

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

   4. Taylor expanded in v around inf 57.7%

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

      1. associate-*r/ 57.7%

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

      2. metadata-eval 57.7%

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

   6. Simplified 57.7%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(u \cdot \left(v \cdot \frac{2 + \frac{2}{v}}{v}\right) - 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 90.7% accurate, 13.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{u}{v} + \left(-1 - -2 \cdot u\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224) 1.0 (+ (* 2.0 (/ u v)) (- -1.0 (* -2.0 u)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f;
	} else {
		tmp = (2.0f * (u / v)) + (-1.0f - (-2.0f * u));
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.20000000298023224e0) then
        tmp = 1.0e0
    else
        tmp = (2.0e0 * (u / v)) + ((-1.0e0) - ((-2.0e0) * u))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.20000000298023224))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(Float32(2.0) * Float32(u / v)) + Float32(Float32(-1.0) - Float32(Float32(-2.0) * u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.20000000298023224))
		tmp = single(1.0);
	else
		tmp = (single(2.0) * (u / v)) + (single(-1.0) - (single(-2.0) * u));
	end
	tmp_2 = 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. Step-by-step derivation

      1. +-commutative 100.0%

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

      2. fma-define 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in u around 0 100.0%

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

      1. mul-1-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in v around 0 93.1%

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

   if 0.200000003 < v

   1. Initial program 93.5%

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

      1. +-commutative 93.5%

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

      2. fma-define 93.6%

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

      3. +-commutative 93.6%

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

      4. fma-define 93.2%

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

   3. Simplified 93.2%

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

   5. Taylor expanded in u around 0 94.3%

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

      1. mul-1-neg 94.3%

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

   7. Simplified 94.3%

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

   8. Taylor expanded in u around 0 65.6%

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

   9. Taylor expanded in v around -inf 57.7%

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

       1. +-commutative 57.7%

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

       2. +-commutative 57.7%

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

       3. associate-+l+ 57.7%

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

       4. +-commutative 57.7%

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

       5. distribute-lft-in 57.7%

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

       6. metadata-eval 57.7%

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

       7. associate-+r+ 57.7%

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

       8. metadata-eval 57.7%

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

       9. mul-1-neg 57.7%

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

   11. Simplified 57.7%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{u}{v} + \left(-1 - -2 \cdot u\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 90.7% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + 2 \cdot \left(u + \frac{u}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224) 1.0 (+ -1.0 (* 2.0 (+ u (/ u v))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + (2.0f * (u + (u / v)));
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.20000000298023224e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + (2.0e0 * (u + (u / v)))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.20000000298023224))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(Float32(2.0) * Float32(u + Float32(u / v))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.20000000298023224))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + (single(2.0) * (u + (u / v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-define 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in u around 0 100.0%

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

      1. mul-1-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in v around 0 93.1%

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

   if 0.200000003 < v

   1. Initial program 93.5%

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

   3. Taylor expanded in u around 0 65.7%

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

   4. Taylor expanded in v around inf 57.7%

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

      1. sub-neg 57.7%

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

      2. distribute-lft-out 57.7%

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

      3. metadata-eval 57.7%

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

   6. Simplified 57.7%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + 2 \cdot \left(u + \frac{u}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 90.1% accurate, 21.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224) 1.0 (+ -1.0 (* u 2.0))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + (u * 2.0f);
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.20000000298023224e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + (u * 2.0e0)
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.20000000298023224))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(u * Float32(2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.20000000298023224))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + (u * single(2.0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-define 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in u around 0 100.0%

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

      1. mul-1-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in v around 0 93.1%

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

   if 0.200000003 < v

   1. Initial program 93.5%

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

      1. +-commutative 93.5%

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

      2. fma-define 93.6%

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

      3. +-commutative 93.6%

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

      4. fma-define 93.2%

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

   3. Simplified 93.2%

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

   5. Taylor expanded in u around 0 94.3%

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

      1. mul-1-neg 94.3%

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

   7. Simplified 94.3%

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

   8. Taylor expanded in v around inf 51.1%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 5.9% accurate, 213.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in u around 0 6.2%

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

4. Final simplification 6.2%

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

Alternative 17: 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. Step-by-step derivation

   1. +-commutative 99.5%

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

   2. fma-define 99.5%

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

   3. +-commutative 99.5%

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

   4. fma-define 99.5%

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

3. Simplified 99.5%

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

5. Taylor expanded in u around 0 99.6%

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

   1. mul-1-neg 99.6%

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

7. Simplified 99.6%

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

8. Taylor expanded in v around 0 86.5%

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

9. Final simplification 86.5%

   \[\leadsto 1 \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024074 
(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))))))))