Average Error: 0.1 → 0.1
Time: 13.5s
Precision: binary32
Cost: 10016
\[\left(\left(\left(\left(-1 \leq cosTheta_i \land cosTheta_i \leq 1\right) \land \left(-1 \leq cosTheta_O \land cosTheta_O \leq 1\right)\right) \land \left(-1 \leq sinTheta_i \land sinTheta_i \leq 1\right)\right) \land \left(-1 \leq sinTheta_O \land sinTheta_O \leq 1\right)\right) \land \left(-1.5707964 \leq v \land v \leq 0.1\right)\]
\[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
↓
\[0.5 \cdot {e}^{\left(\left(0.6931 + \frac{-1 - sinTheta_i \cdot sinTheta_O}{v}\right) - \log v\right)} \]
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (exp
  (+
   (+
    (-
     (- (/ (* cosTheta_i cosTheta_O) v) (/ (* sinTheta_i sinTheta_O) v))
     (/ 1.0 v))
    0.6931)
   (log (/ 1.0 (* 2.0 v))))))
↓
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  0.5
  (pow E (- (+ 0.6931 (/ (- -1.0 (* sinTheta_i sinTheta_O)) v)) (log v)))))
float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return expf(((((((cosTheta_i * cosTheta_O) / v) - ((sinTheta_i * sinTheta_O) / v)) - (1.0f / v)) + 0.6931f) + logf((1.0f / (2.0f * v)))));
}
↓
float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return 0.5f * powf(((float) M_E), ((0.6931f + ((-1.0f - (sinTheta_i * sinTheta_O)) / v)) - logf(v)));
}

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return exp(Float32(Float32(Float32(Float32(Float32(Float32(cosTheta_i * cosTheta_O) / v) - Float32(Float32(sinTheta_i * sinTheta_O) / v)) - Float32(Float32(1.0) / v)) + Float32(0.6931)) + log(Float32(Float32(1.0) / Float32(Float32(2.0) * v)))))
end
↓
function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(0.5) * (Float32(exp(1)) ^ Float32(Float32(Float32(0.6931) + Float32(Float32(Float32(-1.0) - Float32(sinTheta_i * sinTheta_O)) / v)) - log(v))))
end

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = exp(((((((cosTheta_i * cosTheta_O) / v) - ((sinTheta_i * sinTheta_O) / v)) - (single(1.0) / v)) + single(0.6931)) + log((single(1.0) / (single(2.0) * v)))));
end
↓
function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.5) * (single(2.71828182845904523536) ^ ((single(0.6931) + ((single(-1.0) - (sinTheta_i * sinTheta_O)) / v)) - log(v)));
end

e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)}
↓
0.5 \cdot {e}^{\left(\left(0.6931 + \frac{-1 - sinTheta_i \cdot sinTheta_O}{v}\right) - \log v\right)}

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.1

    \[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]

  2. Simplified0.1

    \[\leadsto \color{blue}{e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \frac{sinTheta_i}{\frac{v}{sinTheta_O}}\right) + \left(\frac{-1}{v} + 0.6931\right)} \cdot \frac{0.5}{v}} \]
    Proof
    (*.f32 (exp.f32 (+.f32 (-.f32 (/.f32 cosTheta_i (/.f32 v cosTheta_O)) (/.f32 sinTheta_i (/.f32 v sinTheta_O))) (+.f32 (/.f32 -1 v) 6931/10000))) (/.f32 1/2 v)): 0 points increase in error, 0 points decrease in error
    (*.f32 (exp.f32 (+.f32 (-.f32 (Rewrite<= associate-/l*_binary32 (/.f32 (*.f32 cosTheta_i cosTheta_O) v)) (/.f32 sinTheta_i (/.f32 v sinTheta_O))) (+.f32 (/.f32 -1 v) 6931/10000))) (/.f32 1/2 v)): 0 points increase in error, 0 points decrease in error
    (*.f32 (exp.f32 (+.f32 (-.f32 (/.f32 (*.f32 cosTheta_i cosTheta_O) v) (Rewrite<= associate-/l*_binary32 (/.f32 (*.f32 sinTheta_i sinTheta_O) v))) (+.f32 (/.f32 -1 v) 6931/10000))) (/.f32 1/2 v)): 0 points increase in error, 0 points decrease in error
    (*.f32 (exp.f32 (+.f32 (-.f32 (/.f32 (*.f32 cosTheta_i cosTheta_O) v) (/.f32 (*.f32 sinTheta_i sinTheta_O) v)) (+.f32 (/.f32 (Rewrite<= metadata-eval (neg.f32 1)) v) 6931/10000))) (/.f32 1/2 v)): 0 points increase in error, 0 points decrease in error
    (*.f32 (exp.f32 (+.f32 (-.f32 (/.f32 (*.f32 cosTheta_i cosTheta_O) v) (/.f32 (*.f32 sinTheta_i sinTheta_O) v)) (+.f32 (Rewrite<= distribute-neg-frac_binary32 (neg.f32 (/.f32 1 v))) 6931/10000))) (/.f32 1/2 v)): 0 points increase in error, 0 points decrease in error
    (*.f32 (exp.f32 (Rewrite<= associate-+l+_binary32 (+.f32 (+.f32 (-.f32 (/.f32 (*.f32 cosTheta_i cosTheta_O) v) (/.f32 (*.f32 sinTheta_i sinTheta_O) v)) (neg.f32 (/.f32 1 v))) 6931/10000))) (/.f32 1/2 v)): 0 points increase in error, 0 points decrease in error
    (*.f32 (exp.f32 (+.f32 (Rewrite<= sub-neg_binary32 (-.f32 (-.f32 (/.f32 (*.f32 cosTheta_i cosTheta_O) v) (/.f32 (*.f32 sinTheta_i sinTheta_O) v)) (/.f32 1 v))) 6931/10000)) (/.f32 1/2 v)): 0 points increase in error, 0 points decrease in error
    (*.f32 (exp.f32 (Rewrite=> associate-+l-_binary32 (-.f32 (-.f32 (/.f32 (*.f32 cosTheta_i cosTheta_O) v) (/.f32 (*.f32 sinTheta_i sinTheta_O) v)) (-.f32 (/.f32 1 v) 6931/10000)))) (/.f32 1/2 v)): 0 points increase in error, 0 points decrease in error
    (*.f32 (exp.f32 (-.f32 (-.f32 (/.f32 (*.f32 cosTheta_i cosTheta_O) v) (/.f32 (*.f32 sinTheta_i sinTheta_O) v)) (Rewrite<= --rgt-identity_binary32 (-.f32 (-.f32 (/.f32 1 v) 6931/10000) 0)))) (/.f32 1/2 v)): 0 points increase in error, 0 points decrease in error
    (*.f32 (exp.f32 (-.f32 (-.f32 (/.f32 (*.f32 cosTheta_i cosTheta_O) v) (/.f32 (*.f32 sinTheta_i sinTheta_O) v)) (-.f32 (-.f32 (/.f32 1 v) 6931/10000) (Rewrite<= metadata-eval (log.f32 1))))) (/.f32 1/2 v)): 0 points increase in error, 0 points decrease in error
    (*.f32 (exp.f32 (-.f32 (-.f32 (/.f32 (*.f32 cosTheta_i cosTheta_O) v) (/.f32 (*.f32 sinTheta_i sinTheta_O) v)) (-.f32 (-.f32 (/.f32 1 v) 6931/10000) (Rewrite=> metadata-eval 0)))) (/.f32 1/2 v)): 0 points increase in error, 0 points decrease in error
    (*.f32 (exp.f32 (-.f32 (-.f32 (/.f32 (*.f32 cosTheta_i cosTheta_O) v) (/.f32 (*.f32 sinTheta_i sinTheta_O) v)) (Rewrite=> --rgt-identity_binary32 (-.f32 (/.f32 1 v) 6931/10000)))) (/.f32 1/2 v)): 0 points increase in error, 0 points decrease in error
    (*.f32 (exp.f32 (Rewrite<= associate-+l-_binary32 (+.f32 (-.f32 (-.f32 (/.f32 (*.f32 cosTheta_i cosTheta_O) v) (/.f32 (*.f32 sinTheta_i sinTheta_O) v)) (/.f32 1 v)) 6931/10000))) (/.f32 1/2 v)): 0 points increase in error, 0 points decrease in error
    (*.f32 (exp.f32 (+.f32 (-.f32 (-.f32 (/.f32 (*.f32 cosTheta_i cosTheta_O) v) (/.f32 (*.f32 sinTheta_i sinTheta_O) v)) (/.f32 1 v)) 6931/10000)) (/.f32 (Rewrite<= metadata-eval (/.f32 1 2)) v)): 0 points increase in error, 0 points decrease in error
    (*.f32 (exp.f32 (+.f32 (-.f32 (-.f32 (/.f32 (*.f32 cosTheta_i cosTheta_O) v) (/.f32 (*.f32 sinTheta_i sinTheta_O) v)) (/.f32 1 v)) 6931/10000)) (Rewrite<= associate-/r*_binary32 (/.f32 1 (*.f32 2 v)))): 0 points increase in error, 0 points decrease in error
    (*.f32 (exp.f32 (+.f32 (-.f32 (-.f32 (/.f32 (*.f32 cosTheta_i cosTheta_O) v) (/.f32 (*.f32 sinTheta_i sinTheta_O) v)) (/.f32 1 v)) 6931/10000)) (Rewrite<= rem-exp-log_binary32 (exp.f32 (log.f32 (/.f32 1 (*.f32 2 v)))))): 3 points increase in error, 3 points decrease in error
    (Rewrite<= exp-sum_binary32 (exp.f32 (+.f32 (+.f32 (-.f32 (-.f32 (/.f32 (*.f32 cosTheta_i cosTheta_O) v) (/.f32 (*.f32 sinTheta_i sinTheta_O) v)) (/.f32 1 v)) 6931/10000) (log.f32 (/.f32 1 (*.f32 2 v)))))): 3 points increase in error, 3 points decrease in error

  3. Taylor expanded in cosTheta_i around 0 0.1

    \[\leadsto \color{blue}{0.5 \cdot \frac{e^{0.6931 - \left(\frac{1}{v} + \frac{sinTheta_i \cdot sinTheta_O}{v}\right)}}{v}} \]

  4. Applied egg-rr0.1

    \[\leadsto 0.5 \cdot \color{blue}{e^{\left(0.6931 - \frac{1}{v} \cdot \mathsf{fma}\left(sinTheta_i, sinTheta_O, 1\right)\right) - \log v}} \]

  5. Applied egg-rr0.1

    \[\leadsto 0.5 \cdot \color{blue}{{e}^{\left(0.6931 - \left(\frac{\mathsf{fma}\left(sinTheta_i, sinTheta_O, 1\right)}{v} + \log v\right)\right)}} \]

  6. Taylor expanded in v around 0 0.1

    \[\leadsto 0.5 \cdot {e}^{\color{blue}{\left(\left(0.6931 + -1 \cdot \frac{1 + sinTheta_i \cdot sinTheta_O}{v}\right) - \log v\right)}} \]

  7. Final simplification0.1

    \[\leadsto 0.5 \cdot {e}^{\left(\left(0.6931 + \frac{-1 - sinTheta_i \cdot sinTheta_O}{v}\right) - \log v\right)} \]

Alternatives

Alternative 1
Error0.1
Cost10016
\[0.5 \cdot e^{\left(0.6931 + \mathsf{fma}\left(sinTheta_i, sinTheta_O, 1\right) \cdot \frac{-1}{v}\right) - \log v} \]
Alternative 2
Error0.1
Cost3680
\[0.5 \cdot \frac{e^{0.6931 + \left(\frac{-1}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right)}}{v} \]
Alternative 3
Error0.1
Cost3488
\[0.5 \cdot \frac{e^{0.6931 + \frac{-1}{v}}}{v} \]
Alternative 4
Error0.7
Cost3360
\[0.5 \cdot e^{\frac{-1}{v}} \]

Error

Reproduce

herbie shell --seed 2022338 
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
  :name "HairBSDF, Mp, lower"
  :precision binary32
  :pre (and (and (and (and (and (<= -1.0 cosTheta_i) (<= cosTheta_i 1.0)) (and (<= -1.0 cosTheta_O) (<= cosTheta_O 1.0))) (and (<= -1.0 sinTheta_i) (<= sinTheta_i 1.0))) (and (<= -1.0 sinTheta_O) (<= sinTheta_O 1.0))) (and (<= -1.5707964 v) (<= v 0.1)))
  (exp (+ (+ (- (- (/ (* cosTheta_i cosTheta_O) v) (/ (* sinTheta_i sinTheta_O) v)) (/ 1.0 v)) 0.6931) (log (/ 1.0 (* 2.0 v))))))