Average Error: 0.5 → 0.4
Time: 19.0s
Precision: binary32
Cost: 10464
\[\left(\left(\left(\left(\left(-1 \leq cosTheta_i \land cosTheta_i \leq 1\right) \land \left(-1 \leq cosTheta_O \land cosTheta_O \leq 1\right)\right) \land \left(-1 \leq sinTheta_i \land sinTheta_i \leq 1\right)\right) \land \left(-1 \leq sinTheta_O \land sinTheta_O \leq 1\right)\right) \land 0.1 < v\right) \land v \leq 1.5707964\]
\[ \begin{array}{c}[cosTheta_i, cosTheta_O] = \mathsf{sort}([cosTheta_i, cosTheta_O])\\ \end{array} \]
\[\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
↓
\[\frac{\left(\frac{cosTheta_i}{v} \cdot \frac{1}{v}\right) \cdot cosTheta_O}{\left(0.5 \cdot \left(e^{\frac{1}{v}} - e^{\frac{-1}{v}}\right)\right) \cdot \left(e^{sinTheta_O \cdot \frac{sinTheta_i}{v}} \cdot 2\right)} \]
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (* (exp (- (/ (* sinTheta_i sinTheta_O) v))) (/ (* cosTheta_i cosTheta_O) v))
  (* (* (sinh (/ 1.0 v)) 2.0) v)))
↓
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (* (* (/ cosTheta_i v) (/ 1.0 v)) cosTheta_O)
  (*
   (* 0.5 (- (exp (/ 1.0 v)) (exp (/ -1.0 v))))
   (* (exp (* sinTheta_O (/ sinTheta_i v))) 2.0))))
float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (expf(-((sinTheta_i * sinTheta_O) / v)) * ((cosTheta_i * cosTheta_O) / v)) / ((sinhf((1.0f / v)) * 2.0f) * v);
}
↓
float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (((cosTheta_i / v) * (1.0f / v)) * cosTheta_O) / ((0.5f * (expf((1.0f / v)) - expf((-1.0f / v)))) * (expf((sinTheta_O * (sinTheta_i / v))) * 2.0f));
}

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = (exp(-((sintheta_i * sintheta_o) / v)) * ((costheta_i * costheta_o) / v)) / ((sinh((1.0e0 / v)) * 2.0e0) * v)
end function
↓
real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = (((costheta_i / v) * (1.0e0 / v)) * costheta_o) / ((0.5e0 * (exp((1.0e0 / v)) - exp(((-1.0e0) / v)))) * (exp((sintheta_o * (sintheta_i / v))) * 2.0e0))
end function

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(exp(Float32(-Float32(Float32(sinTheta_i * sinTheta_O) / v))) * Float32(Float32(cosTheta_i * cosTheta_O) / v)) / Float32(Float32(sinh(Float32(Float32(1.0) / v)) * Float32(2.0)) * v))
end
↓
function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(Float32(cosTheta_i / v) * Float32(Float32(1.0) / v)) * cosTheta_O) / Float32(Float32(Float32(0.5) * Float32(exp(Float32(Float32(1.0) / v)) - exp(Float32(Float32(-1.0) / v)))) * Float32(exp(Float32(sinTheta_O * Float32(sinTheta_i / v))) * Float32(2.0))))
end

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (exp(-((sinTheta_i * sinTheta_O) / v)) * ((cosTheta_i * cosTheta_O) / v)) / ((sinh((single(1.0) / v)) * single(2.0)) * v);
end
↓
function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (((cosTheta_i / v) * (single(1.0) / v)) * cosTheta_O) / ((single(0.5) * (exp((single(1.0) / v)) - exp((single(-1.0) / v)))) * (exp((sinTheta_O * (sinTheta_i / v))) * single(2.0)));
end

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

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.5

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

  2. Simplified0.5

    \[\leadsto \color{blue}{\frac{\frac{0.5}{{\left(e^{sinTheta_O}\right)}^{\left(\frac{sinTheta_i}{v}\right)}}}{\sinh \left(\frac{1}{v}\right)} \cdot \left(\frac{cosTheta_i}{v \cdot v} \cdot cosTheta_O\right)} \]
    Proof
    (*.f32 (/.f32 (/.f32 1/2 (pow.f32 (exp.f32 sinTheta_O) (/.f32 sinTheta_i v))) (sinh.f32 (/.f32 1 v))) (*.f32 (/.f32 cosTheta_i (*.f32 v v)) cosTheta_O)): 0 points increase in error, 0 points decrease in error
    (*.f32 (/.f32 (/.f32 (Rewrite<= metadata-eval (/.f32 1 2)) (pow.f32 (exp.f32 sinTheta_O) (/.f32 sinTheta_i v))) (sinh.f32 (/.f32 1 v))) (*.f32 (/.f32 cosTheta_i (*.f32 v v)) cosTheta_O)): 0 points increase in error, 0 points decrease in error
    (*.f32 (/.f32 (/.f32 (/.f32 1 2) (Rewrite<= exp-prod_binary32 (exp.f32 (*.f32 sinTheta_O (/.f32 sinTheta_i v))))) (sinh.f32 (/.f32 1 v))) (*.f32 (/.f32 cosTheta_i (*.f32 v v)) cosTheta_O)): 0 points increase in error, 1 points decrease in error
    (*.f32 (/.f32 (/.f32 (/.f32 1 2) (exp.f32 (Rewrite<= *-commutative_binary32 (*.f32 (/.f32 sinTheta_i v) sinTheta_O)))) (sinh.f32 (/.f32 1 v))) (*.f32 (/.f32 cosTheta_i (*.f32 v v)) cosTheta_O)): 0 points increase in error, 0 points decrease in error
    (*.f32 (/.f32 (/.f32 (/.f32 1 2) (exp.f32 (Rewrite=> associate-*l/_binary32 (/.f32 (*.f32 sinTheta_i sinTheta_O) v)))) (sinh.f32 (/.f32 1 v))) (*.f32 (/.f32 cosTheta_i (*.f32 v v)) cosTheta_O)): 0 points increase in error, 0 points decrease in error
    (*.f32 (/.f32 (Rewrite<= associate-/r*_binary32 (/.f32 1 (*.f32 2 (exp.f32 (/.f32 (*.f32 sinTheta_i sinTheta_O) v))))) (sinh.f32 (/.f32 1 v))) (*.f32 (/.f32 cosTheta_i (*.f32 v v)) cosTheta_O)): 0 points increase in error, 0 points decrease in error
    (*.f32 (/.f32 (Rewrite<= associate-/l/_binary32 (/.f32 (/.f32 1 (exp.f32 (/.f32 (*.f32 sinTheta_i sinTheta_O) v))) 2)) (sinh.f32 (/.f32 1 v))) (*.f32 (/.f32 cosTheta_i (*.f32 v v)) cosTheta_O)): 0 points increase in error, 0 points decrease in error
    (*.f32 (/.f32 (/.f32 (Rewrite<= exp-neg_binary32 (exp.f32 (neg.f32 (/.f32 (*.f32 sinTheta_i sinTheta_O) v)))) 2) (sinh.f32 (/.f32 1 v))) (*.f32 (/.f32 cosTheta_i (*.f32 v v)) cosTheta_O)): 1 points increase in error, 1 points decrease in error
    (*.f32 (Rewrite=> associate-/l/_binary32 (/.f32 (exp.f32 (neg.f32 (/.f32 (*.f32 sinTheta_i sinTheta_O) v))) (*.f32 (sinh.f32 (/.f32 1 v)) 2))) (*.f32 (/.f32 cosTheta_i (*.f32 v v)) cosTheta_O)): 0 points increase in error, 0 points decrease in error
    (*.f32 (/.f32 (exp.f32 (neg.f32 (/.f32 (*.f32 sinTheta_i sinTheta_O) v))) (*.f32 (sinh.f32 (/.f32 1 v)) 2)) (Rewrite<= associate-/r/_binary32 (/.f32 cosTheta_i (/.f32 (*.f32 v v) cosTheta_O)))): 25 points increase in error, 12 points decrease in error
    (*.f32 (/.f32 (exp.f32 (neg.f32 (/.f32 (*.f32 sinTheta_i sinTheta_O) v))) (*.f32 (sinh.f32 (/.f32 1 v)) 2)) (Rewrite<= associate-/l*_binary32 (/.f32 (*.f32 cosTheta_i cosTheta_O) (*.f32 v v)))): 21 points increase in error, 21 points decrease in error
    (*.f32 (/.f32 (exp.f32 (neg.f32 (/.f32 (*.f32 sinTheta_i sinTheta_O) v))) (*.f32 (sinh.f32 (/.f32 1 v)) 2)) (Rewrite<= associate-/l/_binary32 (/.f32 (/.f32 (*.f32 cosTheta_i cosTheta_O) v) v))): 17 points increase in error, 15 points decrease in error
    (Rewrite<= times-frac_binary32 (/.f32 (*.f32 (exp.f32 (neg.f32 (/.f32 (*.f32 sinTheta_i sinTheta_O) v))) (/.f32 (*.f32 cosTheta_i cosTheta_O) v)) (*.f32 (*.f32 (sinh.f32 (/.f32 1 v)) 2) v))): 20 points increase in error, 33 points decrease in error

  3. Applied egg-rr0.5

    \[\leadsto \color{blue}{\frac{\frac{\frac{cosTheta_i}{v}}{v} \cdot cosTheta_O}{\sinh \left(\frac{1}{v}\right) \cdot \left(e^{sinTheta_O \cdot \frac{sinTheta_i}{v}} \cdot 2\right)}} \]

  4. Applied egg-rr0.4

    \[\leadsto \frac{\color{blue}{\left(\frac{cosTheta_i}{v} \cdot \frac{1}{v}\right)} \cdot cosTheta_O}{\sinh \left(\frac{1}{v}\right) \cdot \left(e^{sinTheta_O \cdot \frac{sinTheta_i}{v}} \cdot 2\right)} \]

  5. Taylor expanded in v around 0 0.4

    \[\leadsto \frac{\left(\frac{cosTheta_i}{v} \cdot \frac{1}{v}\right) \cdot cosTheta_O}{\color{blue}{\left(0.5 \cdot \left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right)\right)} \cdot \left(e^{sinTheta_O \cdot \frac{sinTheta_i}{v}} \cdot 2\right)} \]

  6. Simplified0.4

    \[\leadsto \frac{\left(\frac{cosTheta_i}{v} \cdot \frac{1}{v}\right) \cdot cosTheta_O}{\color{blue}{\left(0.5 \cdot \left(e^{\frac{1}{v}} - e^{\frac{-1}{v}}\right)\right)} \cdot \left(e^{sinTheta_O \cdot \frac{sinTheta_i}{v}} \cdot 2\right)} \]
    Proof
    (*.f32 1/2 (-.f32 (exp.f32 (/.f32 1 v)) (exp.f32 (/.f32 -1 v)))): 0 points increase in error, 0 points decrease in error
    (*.f32 1/2 (-.f32 (exp.f32 (/.f32 1 v)) (exp.f32 (/.f32 (Rewrite<= metadata-eval (neg.f32 1)) v)))): 0 points increase in error, 0 points decrease in error
    (*.f32 1/2 (-.f32 (exp.f32 (/.f32 1 v)) (exp.f32 (Rewrite<= distribute-neg-frac_binary32 (neg.f32 (/.f32 1 v)))))): 0 points increase in error, 0 points decrease in error
    (*.f32 1/2 (-.f32 (exp.f32 (/.f32 1 v)) (Rewrite<= rec-exp_binary32 (/.f32 1 (exp.f32 (/.f32 1 v)))))): 2 points increase in error, 2 points decrease in error

  7. Final simplification0.4

    \[\leadsto \frac{\left(\frac{cosTheta_i}{v} \cdot \frac{1}{v}\right) \cdot cosTheta_O}{\left(0.5 \cdot \left(e^{\frac{1}{v}} - e^{\frac{-1}{v}}\right)\right) \cdot \left(e^{sinTheta_O \cdot \frac{sinTheta_i}{v}} \cdot 2\right)} \]

Alternatives

Alternative 1
Error0.4
Cost7072
\[\frac{cosTheta_O \cdot \frac{\frac{1}{v}}{\frac{v}{cosTheta_i}}}{\left(e^{sinTheta_O \cdot \frac{sinTheta_i}{v}} \cdot 2\right) \cdot \sinh \left(\frac{1}{v}\right)} \]
Alternative 2
Error0.4
Cost7072
\[\frac{\left(\frac{cosTheta_i}{v} \cdot \frac{1}{v}\right) \cdot cosTheta_O}{\left(e^{sinTheta_O \cdot \frac{sinTheta_i}{v}} \cdot 2\right) \cdot \sinh \left(\frac{1}{v}\right)} \]
Alternative 3
Error0.5
Cost7008
\[\frac{cosTheta_O \cdot \frac{cosTheta_i}{v \cdot v}}{\left(e^{sinTheta_O \cdot \frac{sinTheta_i}{v}} \cdot 2\right) \cdot \sinh \left(\frac{1}{v}\right)} \]
Alternative 4
Error0.5
Cost6880
\[cosTheta_i \cdot \frac{\frac{cosTheta_O}{v \cdot v}}{e^{\frac{1}{v}} - e^{\frac{-1}{v}}} \]
Alternative 5
Error0.5
Cost6880
\[cosTheta_O \cdot \frac{\frac{cosTheta_i}{v \cdot v}}{e^{\frac{1}{v}} - e^{\frac{-1}{v}}} \]
Alternative 6
Error0.5
Cost3680
\[\left(cosTheta_O \cdot \frac{cosTheta_i}{v \cdot v}\right) \cdot \frac{1}{2 \cdot \sinh \left(\frac{1}{v}\right)} \]
Alternative 7
Error11.6
Cost3552
\[cosTheta_O \cdot \frac{cosTheta_i}{\mathsf{fma}\left(0.3333333333333333, \frac{1}{v}, v \cdot 2\right)} \]
Alternative 8
Error13.3
Cost224
\[\frac{0.5}{\frac{v}{cosTheta_i \cdot cosTheta_O}} \]

Error

Reproduce

herbie shell --seed 2022297 
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
  :name "HairBSDF, Mp, upper"
  :precision binary32
  :pre (and (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))) (< 0.1 v)) (<= v 1.5707964))
  (/ (* (exp (- (/ (* sinTheta_i sinTheta_O) v))) (/ (* cosTheta_i cosTheta_O) v)) (* (* (sinh (/ 1.0 v)) 2.0) v)))