HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.5%

Time: 1.0min

Alternatives: 18

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

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

FPCore

(FPCore (u v)
  :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))))))))

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)
use fmin_fmax_functions
    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

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

Math

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

FPCore

(FPCore (u v)
  :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))))))))

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

Julia

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

MATLAB

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

Alternative 1: 99.5% accurate, 1.0× speedup

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

Math

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

FPCore

(FPCore (u v)
  :precision binary32
  :pre (and (and (<= 1e-5 u) (<= u 1.0))
     (and (<= 0.0 v) (<= v 109.746574)))
  (fma v (log (+ u (* (- 1.0 u) (exp (/ -2.0 v))))) 1.0))

C

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

Julia

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

Derivation

1. Initial program 99.5%

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

3. Applied rewrites 99.5%

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

5. Applied rewrites 99.5%

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

Alternative 2: 99.5% accurate, 1.0× speedup

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

Math

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

FPCore

(FPCore (u v)
  :precision binary32
  :pre (and (and (<= 1e-5 u) (<= u 1.0))
     (and (<= 0.0 v) (<= v 109.746574)))
  (fma v (log (fma (exp (/ -2.0 v)) (- 1.0 u) u)) 1.0))

C

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

Julia

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

Derivation

1. Initial program 99.5%

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

3. Applied rewrites 99.5%

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

Alternative 3: 97.6% accurate, 1.1× speedup

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

Math

\[\begin{array}{l} \mathbf{if}\;v \leq 0.38953089714050293:\\ \;\;\;\;\mathsf{fma}\left(v, \log \left(-1 \cdot \left(u \cdot \mathsf{expm1}\left(\frac{-2}{v}\right)\right)\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;u \cdot \left(2 + \left(-\frac{\mathsf{fma}\left(2, u, -\frac{1.3333333333333333}{v}\right) - 2}{v}\right)\right) - 1\\ \end{array} \]

FPCore

(FPCore (u v)
  :precision binary32
  :pre (and (and (<= 1e-5 u) (<= u 1.0))
     (and (<= 0.0 v) (<= v 109.746574)))
  (if (<= v 0.38953089714050293)
  (fma v (log (* -1.0 (* u (expm1 (/ -2.0 v))))) 1.0)
  (-
   (*
    u
    (+
     2.0
     (- (/ (- (fma 2.0 u (- (/ 1.3333333333333333 v))) 2.0) v))))
   1.0)))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.38953089714050293f) {
		tmp = fmaf(v, logf((-1.0f * (u * expm1f((-2.0f / v))))), 1.0f);
	} else {
		tmp = (u * (2.0f + -((fmaf(2.0f, u, -(1.3333333333333333f / v)) - 2.0f) / v))) - 1.0f;
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.38953089714050293))
		tmp = fma(v, log(Float32(Float32(-1.0) * Float32(u * expm1(Float32(Float32(-2.0) / v))))), Float32(1.0));
	else
		tmp = Float32(Float32(u * Float32(Float32(2.0) + Float32(-Float32(Float32(fma(Float32(2.0), u, Float32(-Float32(Float32(1.3333333333333333) / v))) - Float32(2.0)) / v)))) - Float32(1.0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.389530897

   1. Initial program 99.5%

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

   3. Applied rewrites 99.5%

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in v around inf

      \[\leadsto \mathsf{fma}\left(v, \log \left(u + \left(1 - u\right)\right), 1\right) \]

   8. Applied rewrites 86.5%

      \[\leadsto \mathsf{fma}\left(v, \log \left(u + \left(1 - u\right)\right), 1\right) \]

   9. Taylor expanded in u around -inf

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

   11. Applied rewrites 94.3%

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

   if 0.389530897 < v

   1. Initial program 99.5%

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

   2. Taylor expanded in u around 0

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

   4. Applied rewrites 5.4%

      \[\leadsto u \cdot \mathsf{fma}\left(-0.5, \frac{u \cdot \left(v \cdot {\left(1 + -1 \cdot e^{\frac{-2}{v}}\right)}^{2}\right)}{e^{\frac{-4}{v}}}, \frac{v \cdot \left(1 + -1 \cdot e^{\frac{-2}{v}}\right)}{e^{\frac{-2}{v}}}\right) - 1 \]

   5. Taylor expanded in v around -inf

      \[\leadsto u \cdot \left(2 + -1 \cdot \frac{\left(-1 \cdot \frac{\frac{4}{3} + \frac{1}{2} \cdot \left(8 \cdot u - 16 \cdot u\right)}{v} + 2 \cdot u\right) - 2}{v}\right) - 1 \]

   7. Applied rewrites 12.1%

      \[\leadsto u \cdot \left(2 + -1 \cdot \frac{\mathsf{fma}\left(-1, \frac{1.3333333333333333 + 0.5 \cdot \left(8 \cdot u - 16 \cdot u\right)}{v}, 2 \cdot u\right) - 2}{v}\right) - 1 \]

   9. Applied rewrites 12.1%

      \[\leadsto u \cdot \left(2 + \left(-\frac{\mathsf{fma}\left(2, u, -\frac{\mathsf{fma}\left(u \cdot -8, 0.5, 1.3333333333333333\right)}{v}\right) - 2}{v}\right)\right) - 1 \]

   10. Taylor expanded in u around 0

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

   12. Applied rewrites 12.6%

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

Alternative 4: 97.6% accurate, 1.1× speedup

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

Math

\[\begin{array}{l} \mathbf{if}\;v \leq 0.38953089714050293:\\ \;\;\;\;1 + v \cdot \log \left(-u \cdot \mathsf{expm1}\left(\frac{-2}{v}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;u \cdot \left(2 + \left(-\frac{\mathsf{fma}\left(2, u, -\frac{1.3333333333333333}{v}\right) - 2}{v}\right)\right) - 1\\ \end{array} \]

FPCore

(FPCore (u v)
  :precision binary32
  :pre (and (and (<= 1e-5 u) (<= u 1.0))
     (and (<= 0.0 v) (<= v 109.746574)))
  (if (<= v 0.38953089714050293)
  (+ 1.0 (* v (log (- (* u (expm1 (/ -2.0 v)))))))
  (-
   (*
    u
    (+
     2.0
     (- (/ (- (fma 2.0 u (- (/ 1.3333333333333333 v))) 2.0) v))))
   1.0)))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.38953089714050293f) {
		tmp = 1.0f + (v * logf(-(u * expm1f((-2.0f / v)))));
	} else {
		tmp = (u * (2.0f + -((fmaf(2.0f, u, -(1.3333333333333333f / v)) - 2.0f) / v))) - 1.0f;
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.38953089714050293))
		tmp = Float32(Float32(1.0) + Float32(v * log(Float32(-Float32(u * expm1(Float32(Float32(-2.0) / v)))))));
	else
		tmp = Float32(Float32(u * Float32(Float32(2.0) + Float32(-Float32(Float32(fma(Float32(2.0), u, Float32(-Float32(Float32(1.3333333333333333) / v))) - Float32(2.0)) / v)))) - Float32(1.0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.389530897

   1. Initial program 99.5%

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

   2. Taylor expanded in u around -inf

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

   4. Applied rewrites 94.3%

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

   6. Applied rewrites 94.3%

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

   if 0.389530897 < v

   1. Initial program 99.5%

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

   2. Taylor expanded in u around 0

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

   4. Applied rewrites 5.4%

      \[\leadsto u \cdot \mathsf{fma}\left(-0.5, \frac{u \cdot \left(v \cdot {\left(1 + -1 \cdot e^{\frac{-2}{v}}\right)}^{2}\right)}{e^{\frac{-4}{v}}}, \frac{v \cdot \left(1 + -1 \cdot e^{\frac{-2}{v}}\right)}{e^{\frac{-2}{v}}}\right) - 1 \]

   5. Taylor expanded in v around -inf

      \[\leadsto u \cdot \left(2 + -1 \cdot \frac{\left(-1 \cdot \frac{\frac{4}{3} + \frac{1}{2} \cdot \left(8 \cdot u - 16 \cdot u\right)}{v} + 2 \cdot u\right) - 2}{v}\right) - 1 \]

   7. Applied rewrites 12.1%

      \[\leadsto u \cdot \left(2 + -1 \cdot \frac{\mathsf{fma}\left(-1, \frac{1.3333333333333333 + 0.5 \cdot \left(8 \cdot u - 16 \cdot u\right)}{v}, 2 \cdot u\right) - 2}{v}\right) - 1 \]

   9. Applied rewrites 12.1%

      \[\leadsto u \cdot \left(2 + \left(-\frac{\mathsf{fma}\left(2, u, -\frac{\mathsf{fma}\left(u \cdot -8, 0.5, 1.3333333333333333\right)}{v}\right) - 2}{v}\right)\right) - 1 \]

   10. Taylor expanded in u around 0

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

   12. Applied rewrites 12.6%

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

Alternative 5: 95.9% accurate, 1.2× speedup

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

Math

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

FPCore

(FPCore (u v)
  :precision binary32
  :pre (and (and (<= 1e-5 u) (<= u 1.0))
     (and (<= 0.0 v) (<= v 109.746574)))
  (fma v (log (+ u (exp (/ -2.0 v)))) 1.0))

C

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

Julia

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

Derivation

1. Initial program 99.5%

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

3. Applied rewrites 99.5%

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

5. Applied rewrites 99.5%

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

6. Taylor expanded in u around 0

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

8. Applied rewrites 95.9%

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

Alternative 6: 90.6% accurate, 1.2× speedup

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

Math

\[\begin{array}{l} \mathbf{if}\;v \leq 0.38953089714050293:\\ \;\;\;\;\mathsf{fma}\left(v, \log \left(u + \left(1 - u\right)\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;u \cdot \left(2 + \left(-\frac{\mathsf{fma}\left(2, u, -\frac{1.3333333333333333}{v}\right) - 2}{v}\right)\right) - 1\\ \end{array} \]

FPCore

(FPCore (u v)
  :precision binary32
  :pre (and (and (<= 1e-5 u) (<= u 1.0))
     (and (<= 0.0 v) (<= v 109.746574)))
  (if (<= v 0.38953089714050293)
  (fma v (log (+ u (- 1.0 u))) 1.0)
  (-
   (*
    u
    (+
     2.0
     (- (/ (- (fma 2.0 u (- (/ 1.3333333333333333 v))) 2.0) v))))
   1.0)))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.38953089714050293f) {
		tmp = fmaf(v, logf((u + (1.0f - u))), 1.0f);
	} else {
		tmp = (u * (2.0f + -((fmaf(2.0f, u, -(1.3333333333333333f / v)) - 2.0f) / v))) - 1.0f;
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.38953089714050293))
		tmp = fma(v, log(Float32(u + Float32(Float32(1.0) - u))), Float32(1.0));
	else
		tmp = Float32(Float32(u * Float32(Float32(2.0) + Float32(-Float32(Float32(fma(Float32(2.0), u, Float32(-Float32(Float32(1.3333333333333333) / v))) - Float32(2.0)) / v)))) - Float32(1.0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.389530897

   1. Initial program 99.5%

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

   3. Applied rewrites 99.5%

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in v around inf

      \[\leadsto \mathsf{fma}\left(v, \log \left(u + \left(1 - u\right)\right), 1\right) \]

   8. Applied rewrites 86.5%

      \[\leadsto \mathsf{fma}\left(v, \log \left(u + \left(1 - u\right)\right), 1\right) \]

   if 0.389530897 < v

   1. Initial program 99.5%

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

   2. Taylor expanded in u around 0

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

   4. Applied rewrites 5.4%

      \[\leadsto u \cdot \mathsf{fma}\left(-0.5, \frac{u \cdot \left(v \cdot {\left(1 + -1 \cdot e^{\frac{-2}{v}}\right)}^{2}\right)}{e^{\frac{-4}{v}}}, \frac{v \cdot \left(1 + -1 \cdot e^{\frac{-2}{v}}\right)}{e^{\frac{-2}{v}}}\right) - 1 \]

   5. Taylor expanded in v around -inf

      \[\leadsto u \cdot \left(2 + -1 \cdot \frac{\left(-1 \cdot \frac{\frac{4}{3} + \frac{1}{2} \cdot \left(8 \cdot u - 16 \cdot u\right)}{v} + 2 \cdot u\right) - 2}{v}\right) - 1 \]

   7. Applied rewrites 12.1%

      \[\leadsto u \cdot \left(2 + -1 \cdot \frac{\mathsf{fma}\left(-1, \frac{1.3333333333333333 + 0.5 \cdot \left(8 \cdot u - 16 \cdot u\right)}{v}, 2 \cdot u\right) - 2}{v}\right) - 1 \]

   9. Applied rewrites 12.1%

      \[\leadsto u \cdot \left(2 + \left(-\frac{\mathsf{fma}\left(2, u, -\frac{\mathsf{fma}\left(u \cdot -8, 0.5, 1.3333333333333333\right)}{v}\right) - 2}{v}\right)\right) - 1 \]

   10. Taylor expanded in u around 0

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

   12. Applied rewrites 12.6%

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

Alternative 7: 90.6% accurate, 1.5× speedup

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

Math

\[\begin{array}{l} \mathbf{if}\;v \leq 0.38953089714050293:\\ \;\;\;\;\mathsf{fma}\left(v, \log \left(u + \left(1 - u\right)\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;u \cdot \left(v \cdot \mathsf{expm1}\left(\frac{2}{v}\right)\right) - 1\\ \end{array} \]

FPCore

(FPCore (u v)
  :precision binary32
  :pre (and (and (<= 1e-5 u) (<= u 1.0))
     (and (<= 0.0 v) (<= v 109.746574)))
  (if (<= v 0.38953089714050293)
  (fma v (log (+ u (- 1.0 u))) 1.0)
  (- (* u (* v (expm1 (/ 2.0 v)))) 1.0)))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.38953089714050293f) {
		tmp = fmaf(v, logf((u + (1.0f - u))), 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(0.38953089714050293))
		tmp = fma(v, log(Float32(u + Float32(Float32(1.0) - u))), 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 < 0.389530897

   1. Initial program 99.5%

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

   3. Applied rewrites 99.5%

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in v around inf

      \[\leadsto \mathsf{fma}\left(v, \log \left(u + \left(1 - u\right)\right), 1\right) \]

   8. Applied rewrites 86.5%

      \[\leadsto \mathsf{fma}\left(v, \log \left(u + \left(1 - u\right)\right), 1\right) \]

   if 0.389530897 < v

   1. Initial program 99.5%

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

   2. Taylor expanded in u around 0

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

   4. Applied rewrites 5.4%

      \[\leadsto u \cdot \mathsf{fma}\left(-0.5, \frac{u \cdot \left(v \cdot {\left(1 + -1 \cdot e^{\frac{-2}{v}}\right)}^{2}\right)}{e^{\frac{-4}{v}}}, \frac{v \cdot \left(1 + -1 \cdot e^{\frac{-2}{v}}\right)}{e^{\frac{-2}{v}}}\right) - 1 \]

   5. Taylor expanded in v around 0

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

   7. Applied rewrites 5.4%

      \[\leadsto u \cdot \left(v \cdot \left(\mathsf{fma}\left(-0.5, \frac{u \cdot {\left(1 + -1 \cdot e^{\frac{-2}{v}}\right)}^{2}}{e^{\frac{-4}{v}}}, \frac{1}{e^{\frac{-2}{v}}}\right) - 1\right)\right) - 1 \]

   9. Applied rewrites 5.4%

      \[\leadsto u \cdot \left(v \cdot \mathsf{fma}\left(u \cdot \frac{{\left(1 + \left(-e^{\frac{-2}{v}}\right)\right)}^{2}}{e^{\frac{-4}{v}}}, -0.5, \mathsf{expm1}\left(-\frac{-2}{v}\right)\right)\right) - 1 \]

   10. Taylor expanded in u around 0

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

   12. Applied rewrites 10.6%

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

Alternative 8: 90.4% accurate, 1.6× speedup

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

Math

\[\begin{array}{l} \mathbf{if}\;v \leq 0.2006438821554184:\\ \;\;\;\;\mathsf{fma}\left(v, \log \left(u + \left(1 - u\right)\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;u \cdot \left(2 - \frac{\mathsf{fma}\left(u, 2, -2\right)}{v}\right) - 1\\ \end{array} \]

FPCore

(FPCore (u v)
  :precision binary32
  :pre (and (and (<= 1e-5 u) (<= u 1.0))
     (and (<= 0.0 v) (<= v 109.746574)))
  (if (<= v 0.2006438821554184)
  (fma v (log (+ u (- 1.0 u))) 1.0)
  (- (* u (- 2.0 (/ (fma u 2.0 -2.0) v))) 1.0)))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.2006438821554184f) {
		tmp = fmaf(v, logf((u + (1.0f - u))), 1.0f);
	} else {
		tmp = (u * (2.0f - (fmaf(u, 2.0f, -2.0f) / v))) - 1.0f;
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.2006438821554184))
		tmp = fma(v, log(Float32(u + Float32(Float32(1.0) - u))), Float32(1.0));
	else
		tmp = Float32(Float32(u * Float32(Float32(2.0) - Float32(fma(u, Float32(2.0), Float32(-2.0)) / v))) - Float32(1.0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.200643882

   1. Initial program 99.5%

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

   3. Applied rewrites 99.5%

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in v around inf

      \[\leadsto \mathsf{fma}\left(v, \log \left(u + \left(1 - u\right)\right), 1\right) \]

   8. Applied rewrites 86.5%

      \[\leadsto \mathsf{fma}\left(v, \log \left(u + \left(1 - u\right)\right), 1\right) \]

   if 0.200643882 < v

   1. Initial program 99.5%

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

   2. Taylor expanded in u around 0

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

   4. Applied rewrites 5.4%

      \[\leadsto u \cdot \mathsf{fma}\left(-0.5, \frac{u \cdot \left(v \cdot {\left(1 + -1 \cdot e^{\frac{-2}{v}}\right)}^{2}\right)}{e^{\frac{-4}{v}}}, \frac{v \cdot \left(1 + -1 \cdot e^{\frac{-2}{v}}\right)}{e^{\frac{-2}{v}}}\right) - 1 \]

   5. Taylor expanded in v around -inf

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

   7. Applied rewrites 14.4%

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

   9. Applied rewrites 14.4%

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

   11. Applied rewrites 14.4%

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

Alternative 9: 90.3% accurate, 0.7× speedup

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

Math

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

FPCore

(FPCore (u v)
  :precision binary32
  :pre (and (and (<= 1e-5 u) (<= u 1.0))
     (and (<= 0.0 v) (<= v 109.746574)))
  (if (<= (* v (log (+ u (* (- 1.0 u) (exp (/ -2.0 v)))))) -1.0)
  (+ 1.0 (fma -2.0 (- 1.0 u) (/ (+ u u) v)))
  (fma v (log (+ u (- 1.0 u))) 1.0)))

C

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

Julia

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

Derivation

1. Split input into 2 regimes

2. if (*.f32 v (log.f32 (+.f32 u (*.f32 (-.f32 #s(literal 1 binary32) u) (exp.f32 (/.f32 #s(literal -2 binary32) v)))))) < -1

   1. Initial program 99.5%

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

   2. Taylor expanded in v around inf

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

   4. Applied rewrites 14.4%

      \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, 0.5 \cdot \frac{\mathsf{fma}\left(-4, {\left(1 - u\right)}^{2}, 4 \cdot \left(1 - u\right)\right)}{v}\right) \]

   5. Taylor expanded in u around 0

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

   7. Applied rewrites 14.2%

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

   9. Applied rewrites 14.2%

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

   if -1 < (*.f32 v (log.f32 (+.f32 u (*.f32 (-.f32 #s(literal 1 binary32) u) (exp.f32 (/.f32 #s(literal -2 binary32) v))))))

   1. Initial program 99.5%

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

   3. Applied rewrites 99.5%

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in v around inf

      \[\leadsto \mathsf{fma}\left(v, \log \left(u + \left(1 - u\right)\right), 1\right) \]

   8. Applied rewrites 86.5%

      \[\leadsto \mathsf{fma}\left(v, \log \left(u + \left(1 - u\right)\right), 1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 90.3% accurate, 0.7× speedup

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

Math

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

FPCore

(FPCore (u v)
  :precision binary32
  :pre (and (and (<= 1e-5 u) (<= u 1.0))
     (and (<= 0.0 v) (<= v 109.746574)))
  (if (<= (* v (log (+ u (* (- 1.0 u) (exp (/ -2.0 v)))))) -1.0)
  (- (* u (+ 2.0 (* -1.0 (/ -2.0 v)))) 1.0)
  (fma v (log (+ u (- 1.0 u))) 1.0)))

C

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

Julia

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

Derivation

1. Split input into 2 regimes

2. if (*.f32 v (log.f32 (+.f32 u (*.f32 (-.f32 #s(literal 1 binary32) u) (exp.f32 (/.f32 #s(literal -2 binary32) v)))))) < -1

   1. Initial program 99.5%

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

   2. Taylor expanded in u around 0

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

   4. Applied rewrites 5.4%

      \[\leadsto u \cdot \mathsf{fma}\left(-0.5, \frac{u \cdot \left(v \cdot {\left(1 + -1 \cdot e^{\frac{-2}{v}}\right)}^{2}\right)}{e^{\frac{-4}{v}}}, \frac{v \cdot \left(1 + -1 \cdot e^{\frac{-2}{v}}\right)}{e^{\frac{-2}{v}}}\right) - 1 \]

   5. Taylor expanded in v around -inf

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

   7. Applied rewrites 14.4%

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

   8. Taylor expanded in u around 0

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

   10. Applied rewrites 14.2%

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

   if -1 < (*.f32 v (log.f32 (+.f32 u (*.f32 (-.f32 #s(literal 1 binary32) u) (exp.f32 (/.f32 #s(literal -2 binary32) v))))))

   1. Initial program 99.5%

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

   3. Applied rewrites 99.5%

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in v around inf

      \[\leadsto \mathsf{fma}\left(v, \log \left(u + \left(1 - u\right)\right), 1\right) \]

   8. Applied rewrites 86.5%

      \[\leadsto \mathsf{fma}\left(v, \log \left(u + \left(1 - u\right)\right), 1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 90.2% accurate, 1.8× speedup

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

Math

\[\begin{array}{l} \mathbf{if}\;v \leq 0.2006438821554184:\\ \;\;\;\;\mathsf{fma}\left(v, \log \left(u + 1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;u \cdot \left(2 + -1 \cdot \frac{-2}{v}\right) - 1\\ \end{array} \]

FPCore

(FPCore (u v)
  :precision binary32
  :pre (and (and (<= 1e-5 u) (<= u 1.0))
     (and (<= 0.0 v) (<= v 109.746574)))
  (if (<= v 0.2006438821554184)
  (fma v (log (+ u 1.0)) 1.0)
  (- (* u (+ 2.0 (* -1.0 (/ -2.0 v)))) 1.0)))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.2006438821554184f) {
		tmp = fmaf(v, logf((u + 1.0f)), 1.0f);
	} else {
		tmp = (u * (2.0f + (-1.0f * (-2.0f / v)))) - 1.0f;
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.2006438821554184))
		tmp = fma(v, log(Float32(u + Float32(1.0))), Float32(1.0));
	else
		tmp = Float32(Float32(u * Float32(Float32(2.0) + Float32(Float32(-1.0) * Float32(Float32(-2.0) / v)))) - Float32(1.0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.200643882

   1. Initial program 99.5%

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

   3. Applied rewrites 99.5%

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in v around inf

      \[\leadsto \mathsf{fma}\left(v, \log \left(u + \left(1 - u\right)\right), 1\right) \]

   8. Applied rewrites 86.5%

      \[\leadsto \mathsf{fma}\left(v, \log \left(u + \left(1 - u\right)\right), 1\right) \]

   9. Taylor expanded in u around 0

      \[\leadsto \mathsf{fma}\left(v, \log \left(u + 1\right), 1\right) \]

   11. Applied rewrites 86.4%

       \[\leadsto \mathsf{fma}\left(v, \log \left(u + 1\right), 1\right) \]

   if 0.200643882 < v

   1. Initial program 99.5%

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

   2. Taylor expanded in u around 0

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

   4. Applied rewrites 5.4%

      \[\leadsto u \cdot \mathsf{fma}\left(-0.5, \frac{u \cdot \left(v \cdot {\left(1 + -1 \cdot e^{\frac{-2}{v}}\right)}^{2}\right)}{e^{\frac{-4}{v}}}, \frac{v \cdot \left(1 + -1 \cdot e^{\frac{-2}{v}}\right)}{e^{\frac{-2}{v}}}\right) - 1 \]

   5. Taylor expanded in v around -inf

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

   7. Applied rewrites 14.4%

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

   8. Taylor expanded in u around 0

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

   10. Applied rewrites 14.2%

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

Alternative 12: 89.5% accurate, 1.9× speedup

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

Math

\[\begin{array}{l} \mathbf{if}\;v \leq 0.38953089714050293:\\ \;\;\;\;\mathsf{fma}\left(v, \log \left(u + 1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(v, -2 \cdot \frac{1 - u}{v}, 1\right)\\ \end{array} \]

FPCore

(FPCore (u v)
  :precision binary32
  :pre (and (and (<= 1e-5 u) (<= u 1.0))
     (and (<= 0.0 v) (<= v 109.746574)))
  (if (<= v 0.38953089714050293)
  (fma v (log (+ u 1.0)) 1.0)
  (fma v (* -2.0 (/ (- 1.0 u) v)) 1.0)))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.38953089714050293f) {
		tmp = fmaf(v, logf((u + 1.0f)), 1.0f);
	} else {
		tmp = fmaf(v, (-2.0f * ((1.0f - u) / v)), 1.0f);
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.38953089714050293))
		tmp = fma(v, log(Float32(u + Float32(1.0))), Float32(1.0));
	else
		tmp = fma(v, Float32(Float32(-2.0) * Float32(Float32(Float32(1.0) - u) / v)), Float32(1.0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.389530897

   1. Initial program 99.5%

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

   3. Applied rewrites 99.5%

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in v around inf

      \[\leadsto \mathsf{fma}\left(v, \log \left(u + \left(1 - u\right)\right), 1\right) \]

   8. Applied rewrites 86.5%

      \[\leadsto \mathsf{fma}\left(v, \log \left(u + \left(1 - u\right)\right), 1\right) \]

   9. Taylor expanded in u around 0

      \[\leadsto \mathsf{fma}\left(v, \log \left(u + 1\right), 1\right) \]

   11. Applied rewrites 86.4%

       \[\leadsto \mathsf{fma}\left(v, \log \left(u + 1\right), 1\right) \]

   if 0.389530897 < v

   1. Initial program 99.5%

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

   3. Applied rewrites 99.5%

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in v around inf

      \[\leadsto \mathsf{fma}\left(v, \log \left(u + \left(1 - u\right)\right), 1\right) \]

   8. Applied rewrites 86.5%

      \[\leadsto \mathsf{fma}\left(v, \log \left(u + \left(1 - u\right)\right), 1\right) \]

   9. Taylor expanded in v around inf

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

   11. Applied rewrites 8.0%

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

Alternative 13: 89.5% accurate, 1.9× speedup

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

Math

\[\begin{array}{l} \mathbf{if}\;v \leq 0.38953089714050293:\\ \;\;\;\;\mathsf{fma}\left(v, \log \left(u + 1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 + -2 \cdot \left(1 - u\right)\\ \end{array} \]

FPCore

(FPCore (u v)
  :precision binary32
  :pre (and (and (<= 1e-5 u) (<= u 1.0))
     (and (<= 0.0 v) (<= v 109.746574)))
  (if (<= v 0.38953089714050293)
  (fma v (log (+ u 1.0)) 1.0)
  (+ 1.0 (* -2.0 (- 1.0 u)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.38953089714050293f) {
		tmp = fmaf(v, logf((u + 1.0f)), 1.0f);
	} else {
		tmp = 1.0f + (-2.0f * (1.0f - u));
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.38953089714050293))
		tmp = fma(v, log(Float32(u + Float32(1.0))), Float32(1.0));
	else
		tmp = Float32(Float32(1.0) + Float32(Float32(-2.0) * Float32(Float32(1.0) - u)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.389530897

   1. Initial program 99.5%

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

   3. Applied rewrites 99.5%

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in v around inf

      \[\leadsto \mathsf{fma}\left(v, \log \left(u + \left(1 - u\right)\right), 1\right) \]

   8. Applied rewrites 86.5%

      \[\leadsto \mathsf{fma}\left(v, \log \left(u + \left(1 - u\right)\right), 1\right) \]

   9. Taylor expanded in u around 0

      \[\leadsto \mathsf{fma}\left(v, \log \left(u + 1\right), 1\right) \]

   11. Applied rewrites 86.4%

       \[\leadsto \mathsf{fma}\left(v, \log \left(u + 1\right), 1\right) \]

   if 0.389530897 < v

   1. Initial program 99.5%

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

   2. Taylor expanded in v around inf

      \[\leadsto 1 + -2 \cdot \left(1 - u\right) \]

   4. Applied rewrites 8.0%

      \[\leadsto 1 + -2 \cdot \left(1 - u\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 49.7% accurate, 2.7× speedup

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

Math

\[\begin{array}{l} \mathbf{if}\;v \leq 0.38953089714050293:\\ \;\;\;\;1 + -2 \cdot \left(-u\right)\\ \mathbf{else}:\\ \;\;\;\;1 + -2 \cdot \left(1 - u\right)\\ \end{array} \]

FPCore

(FPCore (u v)
  :precision binary32
  :pre (and (and (<= 1e-5 u) (<= u 1.0))
     (and (<= 0.0 v) (<= v 109.746574)))
  (if (<= v 0.38953089714050293)
  (+ 1.0 (* -2.0 (- u)))
  (+ 1.0 (* -2.0 (- 1.0 u)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.38953089714050293f) {
		tmp = 1.0f + (-2.0f * -u);
	} else {
		tmp = 1.0f + (-2.0f * (1.0f - u));
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
use fmin_fmax_functions
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.38953089714050293e0) then
        tmp = 1.0e0 + ((-2.0e0) * -u)
    else
        tmp = 1.0e0 + ((-2.0e0) * (1.0e0 - u))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.38953089714050293))
		tmp = Float32(Float32(1.0) + Float32(Float32(-2.0) * Float32(-u)));
	else
		tmp = Float32(Float32(1.0) + Float32(Float32(-2.0) * Float32(Float32(1.0) - u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.38953089714050293))
		tmp = single(1.0) + (single(-2.0) * -u);
	else
		tmp = single(1.0) + (single(-2.0) * (single(1.0) - u));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.389530897

   1. Initial program 99.5%

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

   2. Taylor expanded in v around inf

      \[\leadsto 1 + -2 \cdot \left(1 - u\right) \]

   4. Applied rewrites 8.0%

      \[\leadsto 1 + -2 \cdot \left(1 - u\right) \]

   5. Taylor expanded in u around inf

      \[\leadsto 1 + -2 \cdot \left(-1 \cdot u\right) \]

   7. Applied rewrites 46.6%

      \[\leadsto 1 + -2 \cdot \left(-1 \cdot u\right) \]

   9. Applied rewrites 46.6%

      \[\leadsto 1 + -2 \cdot \left(-u\right) \]

   if 0.389530897 < v

   1. Initial program 99.5%

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

   2. Taylor expanded in v around inf

      \[\leadsto 1 + -2 \cdot \left(1 - u\right) \]

   4. Applied rewrites 8.0%

      \[\leadsto 1 + -2 \cdot \left(1 - u\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 49.7% accurate, 3.1× speedup

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

Math

\[\begin{array}{l} \mathbf{if}\;v \leq 0.38953089714050293:\\ \;\;\;\;1 + -2 \cdot \left(-u\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(u, 2, -1\right)\\ \end{array} \]

FPCore

(FPCore (u v)
  :precision binary32
  :pre (and (and (<= 1e-5 u) (<= u 1.0))
     (and (<= 0.0 v) (<= v 109.746574)))
  (if (<= v 0.38953089714050293)
  (+ 1.0 (* -2.0 (- u)))
  (fma u 2.0 -1.0)))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.38953089714050293f) {
		tmp = 1.0f + (-2.0f * -u);
	} else {
		tmp = fmaf(u, 2.0f, -1.0f);
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.38953089714050293))
		tmp = Float32(Float32(1.0) + Float32(Float32(-2.0) * Float32(-u)));
	else
		tmp = fma(u, Float32(2.0), Float32(-1.0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.389530897

   1. Initial program 99.5%

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

   2. Taylor expanded in v around inf

      \[\leadsto 1 + -2 \cdot \left(1 - u\right) \]

   4. Applied rewrites 8.0%

      \[\leadsto 1 + -2 \cdot \left(1 - u\right) \]

   5. Taylor expanded in u around inf

      \[\leadsto 1 + -2 \cdot \left(-1 \cdot u\right) \]

   7. Applied rewrites 46.6%

      \[\leadsto 1 + -2 \cdot \left(-1 \cdot u\right) \]

   9. Applied rewrites 46.6%

      \[\leadsto 1 + -2 \cdot \left(-u\right) \]

   if 0.389530897 < v

   1. Initial program 99.5%

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

   2. Taylor expanded in u around 0

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

   4. Applied rewrites 10.6%

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

   5. Taylor expanded in v around inf

      \[\leadsto 2 \cdot u - 1 \]

   7. Applied rewrites 8.0%

      \[\leadsto 2 \cdot u - 1 \]

   9. Applied rewrites 8.0%

      \[\leadsto \mathsf{fma}\left(u, 2, -1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 22.9% accurate, 3.6× speedup

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

Math

\[\begin{array}{l} \mathbf{if}\;v \leq 0.38953089714050293:\\ \;\;\;\;\left(u + u\right) - 0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(u, 2, -1\right)\\ \end{array} \]

FPCore

(FPCore (u v)
  :precision binary32
  :pre (and (and (<= 1e-5 u) (<= u 1.0))
     (and (<= 0.0 v) (<= v 109.746574)))
  (if (<= v 0.38953089714050293) (- (+ u u) 0.0) (fma u 2.0 -1.0)))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.38953089714050293f) {
		tmp = (u + u) - 0.0f;
	} else {
		tmp = fmaf(u, 2.0f, -1.0f);
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.38953089714050293))
		tmp = Float32(Float32(u + u) - Float32(0.0));
	else
		tmp = fma(u, Float32(2.0), Float32(-1.0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.389530897

   1. Initial program 99.5%

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

   2. Taylor expanded in u around 0

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

   4. Applied rewrites 10.6%

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

   5. Taylor expanded in v around inf

      \[\leadsto 2 \cdot u - 1 \]

   7. Applied rewrites 8.0%

      \[\leadsto 2 \cdot u - 1 \]

   8. Taylor expanded in undef-var around zero

      \[\leadsto 2 \cdot u - 0 \]

   10. Applied rewrites 19.9%

       \[\leadsto 2 \cdot u - 0 \]

   12. Applied rewrites 19.9%

       \[\leadsto \left(u + u\right) - 0 \]

   if 0.389530897 < v

   1. Initial program 99.5%

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

   2. Taylor expanded in u around 0

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

   4. Applied rewrites 10.6%

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

   5. Taylor expanded in v around inf

      \[\leadsto 2 \cdot u - 1 \]

   7. Applied rewrites 8.0%

      \[\leadsto 2 \cdot u - 1 \]

   9. Applied rewrites 8.0%

      \[\leadsto \mathsf{fma}\left(u, 2, -1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 8.0% accurate, 5.9× speedup

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

Math

\[\mathsf{fma}\left(u, 2, -1\right) \]

FPCore

(FPCore (u v)
  :precision binary32
  :pre (and (and (<= 1e-5 u) (<= u 1.0))
     (and (<= 0.0 v) (<= v 109.746574)))
  (fma u 2.0 -1.0))

C

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

Julia

function code(u, v)
	return fma(u, Float32(2.0), 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. Taylor expanded in u around 0

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

4. Applied rewrites 10.6%

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

5. Taylor expanded in v around inf

   \[\leadsto 2 \cdot u - 1 \]

7. Applied rewrites 8.0%

   \[\leadsto 2 \cdot u - 1 \]

9. Applied rewrites 8.0%

   \[\leadsto \mathsf{fma}\left(u, 2, -1\right) \]
10. Add Preprocessing

Alternative 18: 5.9% accurate, 35.6× speedup

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

Math

\[-1 \]

FPCore

(FPCore (u v)
  :precision binary32
  :pre (and (and (<= 1e-5 u) (<= u 1.0))
     (and (<= 0.0 v) (<= v 109.746574)))
  -1.0)

C

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

Fortran

real(4) function code(u, v)
use fmin_fmax_functions
    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. Taylor expanded in u around 0

   \[\leadsto -1 \]

4. Applied rewrites 5.9%

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

Reproduce

herbie shell --seed 2026089 +o generate:egglog
(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))))))))