HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.5%

Time: 4.1s

Alternatives: 15

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

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

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%

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

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 2: 96.1% 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

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

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 (+ (exp (/ -2.0 v)) u)) 1.0))

C

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

Julia

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

Derivation

1. Initial program 99.5%

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

2. Taylor expanded in u around 0

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

4. Applied rewrites 96.1%

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

6. Applied rewrites 96.1%

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

Alternative 3: 94.3% accurate, 1.3× speedup

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

Math

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

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 (- (* (expm1 (/ -2.0 v)) u))) 1.0))

C

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

Julia

function code(u, v)
	return fma(v, log(Float32(-Float32(expm1(Float32(Float32(-2.0) / v)) * u))), Float32(1.0))
end

Derivation

1. Initial program 99.5%

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

2. Taylor expanded in u around -inf

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

4. Applied rewrites 94.3%

   \[\leadsto 1 + v \cdot \log \left(-1 \cdot \left(u \cdot \mathsf{expm1}\left(\frac{-2}{v}\right)\right)\right) \]
5. Step-by-step derivation

6. Applied rewrites 94.3%

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

Alternative 4: 87.5% accurate, 1.4× speedup

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

Math

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

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 (* -1.0 (/ 1.0 (/ v (* -2.0 u))))))))

C

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

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(((-1.0e0) * (1.0e0 / (v / ((-2.0e0) * u))))))
end function

Julia

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

MATLAB

function tmp = code(u, v)
	tmp = single(1.0) + (v * log((single(-1.0) * (single(1.0) / (v / (single(-2.0) * 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. Taylor expanded in u around -inf

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

4. Applied rewrites 94.3%

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

5. Taylor expanded in v around inf

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

7. Applied rewrites 87.5%

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

9. Applied rewrites 87.5%

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

Alternative 5: 87.5% 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

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

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 (- (/ -1.0 (/ v (+ u u))))) 1.0))

C

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

Julia

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

Derivation

1. Initial program 99.5%

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

2. Taylor expanded in u around -inf

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

4. Applied rewrites 94.3%

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

5. Taylor expanded in v around inf

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

7. Applied rewrites 87.5%

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

9. Applied rewrites 87.5%

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

11. Applied rewrites 87.5%

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

Alternative 6: 87.5% 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

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

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 (* -1.0 (* -2.0 (/ u v)))))))

C

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

Julia

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

MATLAB

function tmp = code(u, v)
	tmp = single(1.0) + (v * log((single(-1.0) * (single(-2.0) * (u / 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. Taylor expanded in u around -inf

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

4. Applied rewrites 94.3%

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

5. Taylor expanded in v around inf

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

7. Applied rewrites 87.5%

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

Alternative 7: 87.5% 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

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

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 v) -2.0))) 1.0))

C

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

Julia

function code(u, v)
	return fma(v, log(Float32(-Float32(Float32(u / v) * 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 -inf

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

4. Applied rewrites 94.3%

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

5. Taylor expanded in v around inf

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

7. Applied rewrites 87.5%

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

9. Applied rewrites 87.5%

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

Alternative 8: 86.8% accurate, 2.1× speedup

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

Math

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

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 (+ (- 1.0 u) u)) 1.0))

C

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

Julia

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

Derivation

1. Initial program 99.5%

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

2. Taylor expanded in v around inf

   \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right)\right) \]
3. Step-by-step derivation

4. Applied rewrites 86.8%

   \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right)\right) \]
5. Step-by-step derivation

6. Applied rewrites 86.8%

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

Alternative 9: 86.8% accurate, 2.4× speedup

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

Math

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

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 (+ 1.0 u)) 1.0))

C

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

Julia

function code(u, v)
	return fma(v, log(Float32(Float32(1.0) + u)), Float32(1.0))
end

Derivation

1. Initial program 99.5%

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

2. Taylor expanded in v around inf

   \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right)\right) \]
3. Step-by-step derivation

4. Applied rewrites 86.8%

   \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right)\right) \]
5. Step-by-step derivation

6. Applied rewrites 86.8%

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

7. Taylor expanded in u around 0

   \[\leadsto \mathsf{fma}\left(v, \log \left(1 + u\right), 1\right) \]
8. Step-by-step derivation

9. Applied rewrites 86.8%

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

Alternative 10: 46.6% 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

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

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 (* 2.0 (/ u v)))))

C

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

Julia

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

MATLAB

function tmp = code(u, v)
	tmp = single(1.0) + (v * (single(2.0) * (u / 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. Taylor expanded in v around inf

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

4. Applied rewrites 8.1%

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

5. Taylor expanded in u around inf

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

7. Applied rewrites 46.6%

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

Alternative 11: 46.6% accurate, 2.9× speedup

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

Math

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

FPCore

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

C

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

Julia

function code(u, v)
	return fma(v, Float32(Float32(u + u) / v), Float32(1.0))
end

Derivation

1. Initial program 99.5%

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

2. Taylor expanded in v around inf

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

4. Applied rewrites 8.1%

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

5. Taylor expanded in u around inf

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

7. Applied rewrites 46.6%

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

9. Applied rewrites 46.6%

   \[\leadsto \mathsf{fma}\left(v, \frac{u + u}{v}, 1\right) \]
10. Add Preprocessing

Alternative 12: 46.6% accurate, 5.7× speedup

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

Math

\[1 + \left(u + u\right) \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(u, v)
	tmp = single(1.0) + (u + 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. Taylor expanded in v around inf

   \[\leadsto 1 + -2 \cdot \left(1 - u\right) \]
3. Step-by-step derivation

4. Applied rewrites 8.1%

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

5. Taylor expanded in u around inf

   \[\leadsto 1 + 2 \cdot u \]
6. Step-by-step derivation

7. Applied rewrites 46.6%

   \[\leadsto 1 + 2 \cdot u \]
8. Step-by-step derivation

9. Applied rewrites 46.6%

   \[\leadsto 1 + \left(u + u\right) \]
10. Add Preprocessing

Alternative 13: 19.8% accurate, 8.9× speedup

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

Math

\[u \cdot 2 \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

2. Taylor expanded in u around 0

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

4. Applied rewrites 10.7%

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

5. Taylor expanded in v around inf

   \[\leadsto 2 \cdot u - 1 \]
6. Step-by-step derivation

7. Applied rewrites 8.1%

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

8. Taylor expanded in u around inf

   \[\leadsto u \cdot \left(2 - \frac{1}{u}\right) \]
9. Step-by-step derivation

10. Applied rewrites 8.1%

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

11. Taylor expanded in u around inf

    \[\leadsto u \cdot 2 \]
12. Step-by-step derivation

13. Applied rewrites 19.8%

    \[\leadsto u \cdot 2 \]
14. Add Preprocessing

Alternative 14: 6.2% accurate, 9.9× speedup

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

Math

\[u + -1 \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

2. Taylor expanded in u around 0

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

4. Applied rewrites 10.7%

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

5. Taylor expanded in v around inf

   \[\leadsto 2 \cdot u - 1 \]
6. Step-by-step derivation

7. Applied rewrites 8.1%

   \[\leadsto 2 \cdot u - 1 \]
8. Step-by-step derivation

9. Applied rewrites 8.1%

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

10. Taylor expanded in u around 0

    \[\leadsto u + -1 \]
11. Step-by-step derivation

12. Applied rewrites 6.2%

    \[\leadsto u + -1 \]
13. Add Preprocessing

Alternative 15: 6.0% 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

   \[\leadsto -1 \]
3. Step-by-step derivation

4. Applied rewrites 6.0%

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

Reproduce

herbie shell --seed 2026084 
(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))))))))