HairBSDF, Mp, upper

Percentage Accurate: 98.6% → 98.7%

Time: 16.4s

Alternatives: 13

Speedup: 1.0×

Specification

\[\left(\left(\left(\left(\left(-1 \leq cosTheta\_i \land cosTheta\_i \leq 1\right) \land \left(-1 \leq cosTheta\_O \land cosTheta\_O \leq 1\right)\right) \land \left(-1 \leq sinTheta\_i \land sinTheta\_i \leq 1\right)\right) \land \left(-1 \leq sinTheta\_O \land sinTheta\_O \leq 1\right)\right) \land 0.1 < v\right) \land v \leq 1.5707964\]

Math

\[\begin{array}{l} \\ \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \end{array} \]

FPCore

(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)))

C

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);
}

Fortran

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

Julia

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

MATLAB

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

Initial Program: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \end{array} \]

FPCore

(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)))

C

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);
}

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \frac{cosTheta\_i}{\sinh \left(\frac{1}{v}\right)} \cdot \frac{cosTheta\_O}{\left(v \cdot \left(v \cdot 2\right)\right) \cdot {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  (/ cosTheta_i (sinh (/ 1.0 v)))
  (/ cosTheta_O (* (* v (* v 2.0)) (pow (exp sinTheta_i) (/ sinTheta_O v))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (cosTheta_i / sinhf((1.0f / v))) * (cosTheta_O / ((v * (v * 2.0f)) * powf(expf(sinTheta_i), (sinTheta_O / v))));
}

Fortran

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 / sinh((1.0e0 / v))) * (costheta_o / ((v * (v * 2.0e0)) * (exp(sintheta_i) ** (sintheta_o / v))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

3. Simplified 98.7%

   \[\leadsto \color{blue}{\frac{cosTheta\_i \cdot cosTheta\_O}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(\left(v \cdot 2\right) \cdot v\right)\right) \cdot {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-*l* 98.7%

      \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\color{blue}{\sinh \left(\frac{1}{v}\right) \cdot \left(\left(\left(v \cdot 2\right) \cdot v\right) \cdot {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}\right)}} \]

   2. times-frac 98.8%

      \[\leadsto \color{blue}{\frac{cosTheta\_i}{\sinh \left(\frac{1}{v}\right)} \cdot \frac{cosTheta\_O}{\left(\left(v \cdot 2\right) \cdot v\right) \cdot {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}}} \]

   3. *-commutative 98.8%

      \[\leadsto \frac{cosTheta\_i}{\sinh \left(\frac{1}{v}\right)} \cdot \frac{cosTheta\_O}{\color{blue}{\left(v \cdot \left(v \cdot 2\right)\right)} \cdot {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}} \]

6. Applied egg-rr 98.8%

   \[\leadsto \color{blue}{\frac{cosTheta\_i}{\sinh \left(\frac{1}{v}\right)} \cdot \frac{cosTheta\_O}{\left(v \cdot \left(v \cdot 2\right)\right) \cdot {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}}} \]

7. Final simplification 98.8%

   \[\leadsto \frac{cosTheta\_i}{\sinh \left(\frac{1}{v}\right)} \cdot \frac{cosTheta\_O}{\left(v \cdot \left(v \cdot 2\right)\right) \cdot {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}} \]
8. Add Preprocessing

Alternative 2: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{e^{\frac{sinTheta\_i \cdot \left(-sinTheta\_O\right)}{v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)\right)}{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (*
   (exp (/ (* sinTheta_i (- sinTheta_O)) v))
   (* (/ 1.0 v) (* cosTheta_i cosTheta_O)))
  (* v (* (sinh (/ 1.0 v)) 2.0))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (expf(((sinTheta_i * -sinTheta_O) / v)) * ((1.0f / v) * (cosTheta_i * cosTheta_O))) / (v * (sinhf((1.0f / v)) * 2.0f));
}

Fortran

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)) * ((1.0e0 / v) * (costheta_i * costheta_o))) / (v * (sinh((1.0e0 / v)) * 2.0e0))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. div-inv 98.7%

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \color{blue}{\left(\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \frac{1}{v}\right)}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

4. Applied egg-rr 98.7%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \color{blue}{\left(\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \frac{1}{v}\right)}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

5. Final simplification 98.7%

   \[\leadsto \frac{e^{\frac{sinTheta\_i \cdot \left(-sinTheta\_O\right)}{v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)\right)}{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \]
6. Add Preprocessing

Alternative 3: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{e^{\frac{sinTheta\_i \cdot \left(-sinTheta\_O\right)}{v}} \cdot \left(cosTheta\_O \cdot \frac{cosTheta\_i}{v}\right)}{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (* (exp (/ (* sinTheta_i (- sinTheta_O)) v)) (* cosTheta_O (/ cosTheta_i v)))
  (* v (* (sinh (/ 1.0 v)) 2.0))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (expf(((sinTheta_i * -sinTheta_O) / v)) * (cosTheta_O * (cosTheta_i / v))) / (v * (sinhf((1.0f / v)) * 2.0f));
}

Fortran

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_o * (costheta_i / v))) / (v * (sinh((1.0e0 / v)) * 2.0e0))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Add Preprocessing

3. Taylor expanded in cosTheta_i around 0 98.6%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \color{blue}{\frac{cosTheta\_O \cdot cosTheta\_i}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
4. Step-by-step derivation

   1. associate-*r/ 98.5%

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \color{blue}{\left(cosTheta\_O \cdot \frac{cosTheta\_i}{v}\right)}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

5. Simplified 98.5%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \color{blue}{\left(cosTheta\_O \cdot \frac{cosTheta\_i}{v}\right)}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

6. Final simplification 98.5%

   \[\leadsto \frac{e^{\frac{sinTheta\_i \cdot \left(-sinTheta\_O\right)}{v}} \cdot \left(cosTheta\_O \cdot \frac{cosTheta\_i}{v}\right)}{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \]
7. Add Preprocessing

Alternative 4: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{e^{\frac{sinTheta\_i \cdot \left(-sinTheta\_O\right)}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (* (exp (/ (* sinTheta_i (- sinTheta_O)) v)) (/ (* cosTheta_i cosTheta_O) v))
  (* v (* (sinh (/ 1.0 v)) 2.0))))

C

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)) / (v * (sinhf((1.0f / v)) * 2.0f));
}

Fortran

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)) / (v * (sinh((1.0e0 / v)) * 2.0e0))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Add Preprocessing

3. Final simplification 98.6%

   \[\leadsto \frac{e^{\frac{sinTheta\_i \cdot \left(-sinTheta\_O\right)}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \]
4. Add Preprocessing

Alternative 5: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{e^{sinTheta\_i \cdot \frac{sinTheta\_O}{-v}} \cdot \left(cosTheta\_i \cdot \frac{cosTheta\_O}{v}\right)}{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (* (exp (* sinTheta_i (/ sinTheta_O (- v)))) (* cosTheta_i (/ cosTheta_O v)))
  (* v (* (sinh (/ 1.0 v)) 2.0))))

C

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))) / (v * (sinhf((1.0f / v)) * 2.0f));
}

Fortran

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))) / (v * (sinh((1.0e0 / v)) * 2.0e0))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Step-by-step derivation

   1. associate-*r/ 98.6%

      \[\leadsto \frac{\color{blue}{\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   2. associate-/l/ 98.6%

      \[\leadsto \color{blue}{\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right) \cdot v}} \]

   3. remove-double-neg 98.6%

      \[\leadsto \frac{\color{blue}{-\left(-e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)\right)}}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right) \cdot v} \]

   4. distribute-rgt-neg-out 98.6%

      \[\leadsto \frac{-\color{blue}{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(-cosTheta\_i \cdot cosTheta\_O\right)}}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right) \cdot v} \]

   5. distribute-rgt-neg-out 98.6%

      \[\leadsto \frac{-e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \color{blue}{\left(cosTheta\_i \cdot \left(-cosTheta\_O\right)\right)}}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right) \cdot v} \]

   6. distribute-lft-neg-in 98.6%

      \[\leadsto \frac{\color{blue}{\left(-e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right) \cdot \left(cosTheta\_i \cdot \left(-cosTheta\_O\right)\right)}}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right) \cdot v} \]

   7. associate-*r/ 98.7%

      \[\leadsto \color{blue}{\left(-e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right) \cdot \frac{cosTheta\_i \cdot \left(-cosTheta\_O\right)}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right) \cdot v}} \]

   8. associate-/l/ 98.6%

      \[\leadsto \left(-e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right) \cdot \color{blue}{\frac{\frac{cosTheta\_i \cdot \left(-cosTheta\_O\right)}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v}} \]

   9. associate-*r/ 98.6%

      \[\leadsto \color{blue}{\frac{\left(-e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right) \cdot \frac{cosTheta\_i \cdot \left(-cosTheta\_O\right)}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v}} \]

3. Simplified 98.6%

   \[\leadsto \color{blue}{\frac{e^{sinTheta\_i \cdot \frac{sinTheta\_O}{-v}} \cdot \left(cosTheta\_i \cdot \frac{cosTheta\_O}{v}\right)}{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)}} \]
4. Add Preprocessing

5. Final simplification 98.6%

   \[\leadsto \frac{e^{sinTheta\_i \cdot \frac{sinTheta\_O}{-v}} \cdot \left(cosTheta\_i \cdot \frac{cosTheta\_O}{v}\right)}{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \]
6. Add Preprocessing

Alternative 6: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{cosTheta\_O}{\sinh \left(\frac{1}{v}\right)} \cdot \frac{cosTheta\_i \cdot 0.5}{{v}^{2}} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* (/ cosTheta_O (sinh (/ 1.0 v))) (/ (* cosTheta_i 0.5) (pow v 2.0))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (cosTheta_O / sinhf((1.0f / v))) * ((cosTheta_i * 0.5f) / powf(v, 2.0f));
}

Fortran

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_o / sinh((1.0e0 / v))) * ((costheta_i * 0.5e0) / (v ** 2.0e0))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

3. Simplified 98.7%

   \[\leadsto \color{blue}{\frac{cosTheta\_i \cdot cosTheta\_O}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(\left(v \cdot 2\right) \cdot v\right)\right) \cdot {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. *-commutative 98.7%

      \[\leadsto \frac{\color{blue}{cosTheta\_O \cdot cosTheta\_i}}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(\left(v \cdot 2\right) \cdot v\right)\right) \cdot {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}} \]

   2. associate-*l* 98.7%

      \[\leadsto \frac{cosTheta\_O \cdot cosTheta\_i}{\color{blue}{\sinh \left(\frac{1}{v}\right) \cdot \left(\left(\left(v \cdot 2\right) \cdot v\right) \cdot {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}\right)}} \]

   3. times-frac 98.7%

      \[\leadsto \color{blue}{\frac{cosTheta\_O}{\sinh \left(\frac{1}{v}\right)} \cdot \frac{cosTheta\_i}{\left(\left(v \cdot 2\right) \cdot v\right) \cdot {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}}} \]

   4. *-commutative 98.7%

      \[\leadsto \frac{cosTheta\_O}{\sinh \left(\frac{1}{v}\right)} \cdot \frac{cosTheta\_i}{\color{blue}{\left(v \cdot \left(v \cdot 2\right)\right)} \cdot {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}} \]

6. Applied egg-rr 98.7%

   \[\leadsto \color{blue}{\frac{cosTheta\_O}{\sinh \left(\frac{1}{v}\right)} \cdot \frac{cosTheta\_i}{\left(v \cdot \left(v \cdot 2\right)\right) \cdot {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}}} \]

7. Taylor expanded in v around inf 98.3%

   \[\leadsto \frac{cosTheta\_O}{\sinh \left(\frac{1}{v}\right)} \cdot \color{blue}{\left(0.5 \cdot \frac{cosTheta\_i}{{v}^{2}}\right)} \]
8. Step-by-step derivation

   1. associate-*r/ 98.3%

      \[\leadsto \frac{cosTheta\_O}{\sinh \left(\frac{1}{v}\right)} \cdot \color{blue}{\frac{0.5 \cdot cosTheta\_i}{{v}^{2}}} \]

9. Simplified 98.3%

   \[\leadsto \frac{cosTheta\_O}{\sinh \left(\frac{1}{v}\right)} \cdot \color{blue}{\frac{0.5 \cdot cosTheta\_i}{{v}^{2}}} \]

10. Final simplification 98.3%

    \[\leadsto \frac{cosTheta\_O}{\sinh \left(\frac{1}{v}\right)} \cdot \frac{cosTheta\_i \cdot 0.5}{{v}^{2}} \]
11. Add Preprocessing

Alternative 7: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{cosTheta\_i}{\sinh \left(\frac{1}{v}\right)} \cdot \frac{cosTheta\_O \cdot 0.5}{{v}^{2}} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* (/ cosTheta_i (sinh (/ 1.0 v))) (/ (* cosTheta_O 0.5) (pow v 2.0))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (cosTheta_i / sinhf((1.0f / v))) * ((cosTheta_O * 0.5f) / powf(v, 2.0f));
}

Fortran

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 / sinh((1.0e0 / v))) * ((costheta_o * 0.5e0) / (v ** 2.0e0))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

3. Simplified 98.7%

   \[\leadsto \color{blue}{\frac{cosTheta\_i \cdot cosTheta\_O}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(\left(v \cdot 2\right) \cdot v\right)\right) \cdot {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-*l* 98.7%

      \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\color{blue}{\sinh \left(\frac{1}{v}\right) \cdot \left(\left(\left(v \cdot 2\right) \cdot v\right) \cdot {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}\right)}} \]

   2. times-frac 98.8%

      \[\leadsto \color{blue}{\frac{cosTheta\_i}{\sinh \left(\frac{1}{v}\right)} \cdot \frac{cosTheta\_O}{\left(\left(v \cdot 2\right) \cdot v\right) \cdot {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}}} \]

   3. *-commutative 98.8%

      \[\leadsto \frac{cosTheta\_i}{\sinh \left(\frac{1}{v}\right)} \cdot \frac{cosTheta\_O}{\color{blue}{\left(v \cdot \left(v \cdot 2\right)\right)} \cdot {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}} \]

6. Applied egg-rr 98.8%

   \[\leadsto \color{blue}{\frac{cosTheta\_i}{\sinh \left(\frac{1}{v}\right)} \cdot \frac{cosTheta\_O}{\left(v \cdot \left(v \cdot 2\right)\right) \cdot {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}}} \]

7. Taylor expanded in v around inf 98.3%

   \[\leadsto \frac{cosTheta\_i}{\sinh \left(\frac{1}{v}\right)} \cdot \color{blue}{\left(0.5 \cdot \frac{cosTheta\_O}{{v}^{2}}\right)} \]
8. Step-by-step derivation

   1. associate-*r/ 98.3%

      \[\leadsto \frac{cosTheta\_i}{\sinh \left(\frac{1}{v}\right)} \cdot \color{blue}{\frac{0.5 \cdot cosTheta\_O}{{v}^{2}}} \]

9. Simplified 98.3%

   \[\leadsto \frac{cosTheta\_i}{\sinh \left(\frac{1}{v}\right)} \cdot \color{blue}{\frac{0.5 \cdot cosTheta\_O}{{v}^{2}}} \]

10. Final simplification 98.3%

    \[\leadsto \frac{cosTheta\_i}{\sinh \left(\frac{1}{v}\right)} \cdot \frac{cosTheta\_O \cdot 0.5}{{v}^{2}} \]
11. Add Preprocessing

Alternative 8: 71.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ cosTheta\_O \cdot \frac{cosTheta\_i}{0.016666666666666666 \cdot \frac{1}{{v}^{3}} + \left(\frac{1}{v} \cdot 0.3333333333333333 + v \cdot 2\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O
  (/
   cosTheta_i
   (+
    (* 0.016666666666666666 (/ 1.0 (pow v 3.0)))
    (+ (* (/ 1.0 v) 0.3333333333333333) (* v 2.0))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_O * (cosTheta_i / ((0.016666666666666666f * (1.0f / powf(v, 3.0f))) + (((1.0f / v) * 0.3333333333333333f) + (v * 2.0f))));
}

Fortran

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_o * (costheta_i / ((0.016666666666666666e0 * (1.0e0 / (v ** 3.0e0))) + (((1.0e0 / v) * 0.3333333333333333e0) + (v * 2.0e0))))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_O * Float32(cosTheta_i / Float32(Float32(Float32(0.016666666666666666) * Float32(Float32(1.0) / (v ^ Float32(3.0)))) + Float32(Float32(Float32(Float32(1.0) / v) * Float32(0.3333333333333333)) + Float32(v * Float32(2.0))))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_O * (cosTheta_i / ((single(0.016666666666666666) * (single(1.0) / (v ^ single(3.0)))) + (((single(1.0) / v) * single(0.3333333333333333)) + (v * single(2.0)))));
end

Derivation

1. Initial program 98.6%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

3. Simplified 98.7%

   \[\leadsto \color{blue}{\frac{cosTheta\_i \cdot cosTheta\_O}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(\left(v \cdot 2\right) \cdot v\right)\right) \cdot {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-*l* 98.7%

      \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\color{blue}{\sinh \left(\frac{1}{v}\right) \cdot \left(\left(\left(v \cdot 2\right) \cdot v\right) \cdot {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}\right)}} \]

   2. times-frac 98.8%

      \[\leadsto \color{blue}{\frac{cosTheta\_i}{\sinh \left(\frac{1}{v}\right)} \cdot \frac{cosTheta\_O}{\left(\left(v \cdot 2\right) \cdot v\right) \cdot {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}}} \]

   3. *-commutative 98.8%

      \[\leadsto \frac{cosTheta\_i}{\sinh \left(\frac{1}{v}\right)} \cdot \frac{cosTheta\_O}{\color{blue}{\left(v \cdot \left(v \cdot 2\right)\right)} \cdot {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}} \]

6. Applied egg-rr 98.8%

   \[\leadsto \color{blue}{\frac{cosTheta\_i}{\sinh \left(\frac{1}{v}\right)} \cdot \frac{cosTheta\_O}{\left(v \cdot \left(v \cdot 2\right)\right) \cdot {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}}} \]

7. Taylor expanded in sinTheta_i around 0 98.2%

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

   1. associate-/l* 98.2%

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

   2. rec-exp 98.3%

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

9. Simplified 98.3%

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

10. Taylor expanded in v around inf 71.3%

    \[\leadsto cosTheta\_O \cdot \frac{cosTheta\_i}{\color{blue}{0.016666666666666666 \cdot \frac{1}{{v}^{3}} + \left(0.3333333333333333 \cdot \frac{1}{v} + 2 \cdot v\right)}} \]

11. Final simplification 71.3%

    \[\leadsto cosTheta\_O \cdot \frac{cosTheta\_i}{0.016666666666666666 \cdot \frac{1}{{v}^{3}} + \left(\frac{1}{v} \cdot 0.3333333333333333 + v \cdot 2\right)} \]
12. Add Preprocessing

Alternative 9: 64.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ cosTheta\_O \cdot \frac{cosTheta\_i}{\mathsf{fma}\left(v, 2, \frac{0.3333333333333333}{v}\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* cosTheta_O (/ cosTheta_i (fma v 2.0 (/ 0.3333333333333333 v)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_O * (cosTheta_i / fmaf(v, 2.0f, (0.3333333333333333f / v)));
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_O * Float32(cosTheta_i / fma(v, Float32(2.0), Float32(Float32(0.3333333333333333) / v))))
end

Derivation

1. Initial program 98.6%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

3. Simplified 98.7%

   \[\leadsto \color{blue}{\frac{cosTheta\_i \cdot cosTheta\_O}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(\left(v \cdot 2\right) \cdot v\right)\right) \cdot {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-*l* 98.7%

      \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\color{blue}{\sinh \left(\frac{1}{v}\right) \cdot \left(\left(\left(v \cdot 2\right) \cdot v\right) \cdot {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}\right)}} \]

   2. times-frac 98.8%

      \[\leadsto \color{blue}{\frac{cosTheta\_i}{\sinh \left(\frac{1}{v}\right)} \cdot \frac{cosTheta\_O}{\left(\left(v \cdot 2\right) \cdot v\right) \cdot {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}}} \]

   3. *-commutative 98.8%

      \[\leadsto \frac{cosTheta\_i}{\sinh \left(\frac{1}{v}\right)} \cdot \frac{cosTheta\_O}{\color{blue}{\left(v \cdot \left(v \cdot 2\right)\right)} \cdot {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}} \]

6. Applied egg-rr 98.8%

   \[\leadsto \color{blue}{\frac{cosTheta\_i}{\sinh \left(\frac{1}{v}\right)} \cdot \frac{cosTheta\_O}{\left(v \cdot \left(v \cdot 2\right)\right) \cdot {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}}} \]

7. Taylor expanded in sinTheta_i around 0 98.2%

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

   1. associate-/l* 98.2%

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

   2. rec-exp 98.3%

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

9. Simplified 98.3%

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

10. Taylor expanded in v around inf 65.2%

    \[\leadsto cosTheta\_O \cdot \frac{cosTheta\_i}{\color{blue}{2 \cdot v + 0.3333333333333333 \cdot \frac{1}{v}}} \]
11. Step-by-step derivation

    1. *-commutative 65.2%

       \[\leadsto cosTheta\_O \cdot \frac{cosTheta\_i}{\color{blue}{v \cdot 2} + 0.3333333333333333 \cdot \frac{1}{v}} \]

    2. fma-define 65.2%

       \[\leadsto cosTheta\_O \cdot \frac{cosTheta\_i}{\color{blue}{\mathsf{fma}\left(v, 2, 0.3333333333333333 \cdot \frac{1}{v}\right)}} \]

    3. associate-*r/ 65.2%

       \[\leadsto cosTheta\_O \cdot \frac{cosTheta\_i}{\mathsf{fma}\left(v, 2, \color{blue}{\frac{0.3333333333333333 \cdot 1}{v}}\right)} \]

    4. metadata-eval 65.2%

       \[\leadsto cosTheta\_O \cdot \frac{cosTheta\_i}{\mathsf{fma}\left(v, 2, \frac{\color{blue}{0.3333333333333333}}{v}\right)} \]

12. Simplified 65.2%

    \[\leadsto cosTheta\_O \cdot \frac{cosTheta\_i}{\color{blue}{\mathsf{fma}\left(v, 2, \frac{0.3333333333333333}{v}\right)}} \]

13. Final simplification 65.2%

    \[\leadsto cosTheta\_O \cdot \frac{cosTheta\_i}{\mathsf{fma}\left(v, 2, \frac{0.3333333333333333}{v}\right)} \]
14. Add Preprocessing

Alternative 10: 64.8% accurate, 16.9× speedup

Math

\[\begin{array}{l} \\ cosTheta\_O \cdot \frac{cosTheta\_i}{\frac{1}{v} \cdot 0.3333333333333333 + v \cdot 2} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* cosTheta_O (/ cosTheta_i (+ (* (/ 1.0 v) 0.3333333333333333) (* v 2.0)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_O * (cosTheta_i / (((1.0f / v) * 0.3333333333333333f) + (v * 2.0f)));
}

Fortran

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_o * (costheta_i / (((1.0e0 / v) * 0.3333333333333333e0) + (v * 2.0e0)))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_O * Float32(cosTheta_i / Float32(Float32(Float32(Float32(1.0) / v) * Float32(0.3333333333333333)) + Float32(v * Float32(2.0)))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_O * (cosTheta_i / (((single(1.0) / v) * single(0.3333333333333333)) + (v * single(2.0))));
end

Derivation

1. Initial program 98.6%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

3. Simplified 98.7%

   \[\leadsto \color{blue}{\frac{cosTheta\_i \cdot cosTheta\_O}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(\left(v \cdot 2\right) \cdot v\right)\right) \cdot {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-*l* 98.7%

      \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\color{blue}{\sinh \left(\frac{1}{v}\right) \cdot \left(\left(\left(v \cdot 2\right) \cdot v\right) \cdot {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}\right)}} \]

   2. times-frac 98.8%

      \[\leadsto \color{blue}{\frac{cosTheta\_i}{\sinh \left(\frac{1}{v}\right)} \cdot \frac{cosTheta\_O}{\left(\left(v \cdot 2\right) \cdot v\right) \cdot {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}}} \]

   3. *-commutative 98.8%

      \[\leadsto \frac{cosTheta\_i}{\sinh \left(\frac{1}{v}\right)} \cdot \frac{cosTheta\_O}{\color{blue}{\left(v \cdot \left(v \cdot 2\right)\right)} \cdot {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}} \]

6. Applied egg-rr 98.8%

   \[\leadsto \color{blue}{\frac{cosTheta\_i}{\sinh \left(\frac{1}{v}\right)} \cdot \frac{cosTheta\_O}{\left(v \cdot \left(v \cdot 2\right)\right) \cdot {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}}} \]

7. Taylor expanded in sinTheta_i around 0 98.2%

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

   1. associate-/l* 98.2%

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

   2. rec-exp 98.3%

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

9. Simplified 98.3%

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

10. Taylor expanded in v around inf 65.2%

    \[\leadsto cosTheta\_O \cdot \frac{cosTheta\_i}{\color{blue}{2 \cdot v + 0.3333333333333333 \cdot \frac{1}{v}}} \]

11. Final simplification 65.2%

    \[\leadsto cosTheta\_O \cdot \frac{cosTheta\_i}{\frac{1}{v} \cdot 0.3333333333333333 + v \cdot 2} \]
12. Add Preprocessing

Alternative 11: 59.1% accurate, 31.4× speedup

Math

\[\begin{array}{l} \\ \left(cosTheta\_i \cdot \frac{cosTheta\_O}{v}\right) \cdot 0.5 \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* (* cosTheta_i (/ cosTheta_O v)) 0.5))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (cosTheta_i * (cosTheta_O / v)) * 0.5f;
}

Fortran

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 * (costheta_o / v)) * 0.5e0
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(cosTheta_i * Float32(cosTheta_O / v)) * Float32(0.5))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (cosTheta_i * (cosTheta_O / v)) * single(0.5);
end

Derivation

1. Initial program 98.6%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

3. Simplified 98.7%

   \[\leadsto \color{blue}{\frac{cosTheta\_i \cdot cosTheta\_O}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(\left(v \cdot 2\right) \cdot v\right)\right) \cdot {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-*l* 98.7%

      \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\color{blue}{\sinh \left(\frac{1}{v}\right) \cdot \left(\left(\left(v \cdot 2\right) \cdot v\right) \cdot {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}\right)}} \]

   2. times-frac 98.8%

      \[\leadsto \color{blue}{\frac{cosTheta\_i}{\sinh \left(\frac{1}{v}\right)} \cdot \frac{cosTheta\_O}{\left(\left(v \cdot 2\right) \cdot v\right) \cdot {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}}} \]

   3. *-commutative 98.8%

      \[\leadsto \frac{cosTheta\_i}{\sinh \left(\frac{1}{v}\right)} \cdot \frac{cosTheta\_O}{\color{blue}{\left(v \cdot \left(v \cdot 2\right)\right)} \cdot {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}} \]

6. Applied egg-rr 98.8%

   \[\leadsto \color{blue}{\frac{cosTheta\_i}{\sinh \left(\frac{1}{v}\right)} \cdot \frac{cosTheta\_O}{\left(v \cdot \left(v \cdot 2\right)\right) \cdot {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}}} \]

7. Taylor expanded in sinTheta_i around 0 98.2%

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

   1. associate-/l* 98.2%

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

   2. rec-exp 98.3%

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

9. Simplified 98.3%

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

10. Taylor expanded in v around inf 59.4%

    \[\leadsto cosTheta\_O \cdot \frac{cosTheta\_i}{\color{blue}{2 \cdot v}} \]
11. Step-by-step derivation

    1. *-commutative 59.4%

       \[\leadsto cosTheta\_O \cdot \frac{cosTheta\_i}{\color{blue}{v \cdot 2}} \]

12. Simplified 59.4%

    \[\leadsto cosTheta\_O \cdot \frac{cosTheta\_i}{\color{blue}{v \cdot 2}} \]

13. Taylor expanded in cosTheta_O around 0 59.4%

    \[\leadsto \color{blue}{0.5 \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]
14. Step-by-step derivation

    1. *-commutative 59.4%

       \[\leadsto 0.5 \cdot \frac{\color{blue}{cosTheta\_i \cdot cosTheta\_O}}{v} \]

    2. associate-/l* 59.4%

       \[\leadsto 0.5 \cdot \color{blue}{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v}\right)} \]

15. Simplified 59.4%

    \[\leadsto \color{blue}{0.5 \cdot \left(cosTheta\_i \cdot \frac{cosTheta\_O}{v}\right)} \]

16. Final simplification 59.4%

    \[\leadsto \left(cosTheta\_i \cdot \frac{cosTheta\_O}{v}\right) \cdot 0.5 \]
17. Add Preprocessing

Alternative 12: 59.1% accurate, 31.4× speedup

Math

\[\begin{array}{l} \\ \frac{cosTheta\_i \cdot cosTheta\_O}{v} \cdot 0.5 \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* (/ (* cosTheta_i cosTheta_O) v) 0.5))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return ((cosTheta_i * cosTheta_O) / v) * 0.5f;
}

Fortran

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 * costheta_o) / v) * 0.5e0
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(cosTheta_i * cosTheta_O) / v) * Float32(0.5))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = ((cosTheta_i * cosTheta_O) / v) * single(0.5);
end

Derivation

1. Initial program 98.6%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

3. Simplified 98.7%

   \[\leadsto \color{blue}{\frac{cosTheta\_i \cdot cosTheta\_O}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(\left(v \cdot 2\right) \cdot v\right)\right) \cdot {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-*l* 98.7%

      \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\color{blue}{\sinh \left(\frac{1}{v}\right) \cdot \left(\left(\left(v \cdot 2\right) \cdot v\right) \cdot {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}\right)}} \]

   2. times-frac 98.8%

      \[\leadsto \color{blue}{\frac{cosTheta\_i}{\sinh \left(\frac{1}{v}\right)} \cdot \frac{cosTheta\_O}{\left(\left(v \cdot 2\right) \cdot v\right) \cdot {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}}} \]

   3. *-commutative 98.8%

      \[\leadsto \frac{cosTheta\_i}{\sinh \left(\frac{1}{v}\right)} \cdot \frac{cosTheta\_O}{\color{blue}{\left(v \cdot \left(v \cdot 2\right)\right)} \cdot {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}} \]

6. Applied egg-rr 98.8%

   \[\leadsto \color{blue}{\frac{cosTheta\_i}{\sinh \left(\frac{1}{v}\right)} \cdot \frac{cosTheta\_O}{\left(v \cdot \left(v \cdot 2\right)\right) \cdot {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}}} \]

7. Taylor expanded in v around inf 59.4%

   \[\leadsto \color{blue}{0.5 \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]

8. Final simplification 59.4%

   \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{v} \cdot 0.5 \]
9. Add Preprocessing

Alternative 13: 59.2% accurate, 31.4× speedup

Math

\[\begin{array}{l} \\ \frac{\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot 0.5}{v} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/ (* (* cosTheta_i cosTheta_O) 0.5) v))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return ((cosTheta_i * cosTheta_O) * 0.5f) / v;
}

Fortran

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 * costheta_o) * 0.5e0) / v
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(cosTheta_i * cosTheta_O) * Float32(0.5)) / v)
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = ((cosTheta_i * cosTheta_O) * single(0.5)) / v;
end

Derivation

1. Initial program 98.6%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

3. Simplified 98.7%

   \[\leadsto \color{blue}{\frac{cosTheta\_i \cdot cosTheta\_O}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(\left(v \cdot 2\right) \cdot v\right)\right) \cdot {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-*l* 98.7%

      \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\color{blue}{\sinh \left(\frac{1}{v}\right) \cdot \left(\left(\left(v \cdot 2\right) \cdot v\right) \cdot {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}\right)}} \]

   2. times-frac 98.8%

      \[\leadsto \color{blue}{\frac{cosTheta\_i}{\sinh \left(\frac{1}{v}\right)} \cdot \frac{cosTheta\_O}{\left(\left(v \cdot 2\right) \cdot v\right) \cdot {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}}} \]

   3. *-commutative 98.8%

      \[\leadsto \frac{cosTheta\_i}{\sinh \left(\frac{1}{v}\right)} \cdot \frac{cosTheta\_O}{\color{blue}{\left(v \cdot \left(v \cdot 2\right)\right)} \cdot {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}} \]

6. Applied egg-rr 98.8%

   \[\leadsto \color{blue}{\frac{cosTheta\_i}{\sinh \left(\frac{1}{v}\right)} \cdot \frac{cosTheta\_O}{\left(v \cdot \left(v \cdot 2\right)\right) \cdot {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}}} \]

7. Taylor expanded in v around inf 59.4%

   \[\leadsto \color{blue}{0.5 \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]
8. Step-by-step derivation

   1. associate-*r/ 59.5%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)}{v}} \]

9. Simplified 59.5%

   \[\leadsto \color{blue}{\frac{0.5 \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)}{v}} \]

10. Final simplification 59.5%

    \[\leadsto \frac{\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot 0.5}{v} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024040 
(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)))