HairBSDF, Mp, upper

Percentage Accurate: 98.6% → 98.7%

Time: 17.5s

Alternatives: 12

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, 1.0× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (*
   (exp (/ (* sinTheta_O (- sinTheta_i)) v))
   (* (* cosTheta_O cosTheta_i) (/ 1.0 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_O * -sinTheta_i) / v)) * ((cosTheta_O * cosTheta_i) * (1.0f / 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_o * -sintheta_i) / v)) * ((costheta_o * costheta_i) * (1.0e0 / 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_O * Float32(-sinTheta_i)) / v)) * Float32(Float32(cosTheta_O * cosTheta_i) * Float32(Float32(1.0) / 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_O * -sinTheta_i) / v)) * ((cosTheta_O * cosTheta_i) * (single(1.0) / v))) / (v * (sinh((single(1.0) / v)) * single(2.0)));
end

Derivation

1. Initial program 98.4%

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

   1. div-inv 98.6%

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

   2. *-commutative 98.6%

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

4. Applied egg-rr 98.6%

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

5. Final simplification 98.6%

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

Alternative 2: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (expf((sinTheta_O * (-sinTheta_i / v))) / (v * 2.0f)) * (cosTheta_i * (cosTheta_O / (v * sinhf((1.0f / 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_o * (-sintheta_i / v))) / (v * 2.0e0)) * (costheta_i * (costheta_o / (v * sinh((1.0e0 / v)))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

   1. *-commutative 98.4%

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

   2. associate-*l* 98.4%

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

   3. times-frac 98.4%

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

   4. *-commutative 98.4%

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

   5. associate-*l/ 98.4%

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

   6. distribute-neg-frac 98.4%

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

   7. distribute-lft-neg-out 98.4%

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

   8. associate-*l/ 98.4%

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

   9. *-commutative 98.4%

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

3. Simplified 98.4%

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

   1. expm1-log1p-u 98.4%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{cosTheta\_O}{v} \cdot cosTheta\_i}{\sinh \left(\frac{1}{v}\right)}\right)\right)} \cdot \frac{e^{\frac{-sinTheta\_i}{v} \cdot sinTheta\_O}}{v \cdot 2} \]

   2. expm1-udef 52.3%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\frac{cosTheta\_O}{v} \cdot cosTheta\_i}{\sinh \left(\frac{1}{v}\right)}\right)} - 1\right)} \cdot \frac{e^{\frac{-sinTheta\_i}{v} \cdot sinTheta\_O}}{v \cdot 2} \]

6. Applied egg-rr 52.3%

   \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\frac{cosTheta\_O}{v} \cdot cosTheta\_i}{\sinh \left(\frac{1}{v}\right)}\right)} - 1\right)} \cdot \frac{e^{\frac{-sinTheta\_i}{v} \cdot sinTheta\_O}}{v \cdot 2} \]
7. Step-by-step derivation

   1. expm1-def 98.4%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{cosTheta\_O}{v} \cdot cosTheta\_i}{\sinh \left(\frac{1}{v}\right)}\right)\right)} \cdot \frac{e^{\frac{-sinTheta\_i}{v} \cdot sinTheta\_O}}{v \cdot 2} \]

   2. expm1-log1p 98.4%

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

   3. associate-/l* 97.9%

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

8. Simplified 97.9%

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

   1. associate-/r/ 98.5%

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

   2. associate-/l/ 98.5%

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

10. Applied egg-rr 98.5%

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

11. Final simplification 98.5%

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

Alternative 3: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (cosTheta_i * ((cosTheta_O / v) / sinhf((1.0f / v)))) * (expf((sinTheta_O * (-sinTheta_i / v))) / (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 * ((costheta_o / v) / sinh((1.0e0 / v)))) * (exp((sintheta_o * (-sintheta_i / v))) / (v * 2.0e0))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

   1. *-commutative 98.4%

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

   2. associate-*l* 98.4%

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

   3. times-frac 98.4%

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

   4. *-commutative 98.4%

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

   5. associate-*l/ 98.4%

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

   6. distribute-neg-frac 98.4%

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

   7. distribute-lft-neg-out 98.4%

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

   8. associate-*l/ 98.4%

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

   9. *-commutative 98.4%

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

3. Simplified 98.4%

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

   1. expm1-log1p-u 98.4%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{cosTheta\_O}{v} \cdot cosTheta\_i}{\sinh \left(\frac{1}{v}\right)}\right)\right)} \cdot \frac{e^{\frac{-sinTheta\_i}{v} \cdot sinTheta\_O}}{v \cdot 2} \]

   2. expm1-udef 52.3%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\frac{cosTheta\_O}{v} \cdot cosTheta\_i}{\sinh \left(\frac{1}{v}\right)}\right)} - 1\right)} \cdot \frac{e^{\frac{-sinTheta\_i}{v} \cdot sinTheta\_O}}{v \cdot 2} \]

6. Applied egg-rr 52.3%

   \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\frac{cosTheta\_O}{v} \cdot cosTheta\_i}{\sinh \left(\frac{1}{v}\right)}\right)} - 1\right)} \cdot \frac{e^{\frac{-sinTheta\_i}{v} \cdot sinTheta\_O}}{v \cdot 2} \]
7. Step-by-step derivation

   1. expm1-def 98.4%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{cosTheta\_O}{v} \cdot cosTheta\_i}{\sinh \left(\frac{1}{v}\right)}\right)\right)} \cdot \frac{e^{\frac{-sinTheta\_i}{v} \cdot sinTheta\_O}}{v \cdot 2} \]

   2. expm1-log1p 98.4%

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

   3. associate-/l* 97.9%

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

8. Simplified 97.9%

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

   1. associate-/r/ 98.5%

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

   2. associate-/l/ 98.5%

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

10. Applied egg-rr 98.5%

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

    1. expm1-log1p-u 98.5%

       \[\leadsto \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{cosTheta\_O}{\sinh \left(\frac{1}{v}\right) \cdot v}\right)\right)} \cdot cosTheta\_i\right) \cdot \frac{e^{\frac{-sinTheta\_i}{v} \cdot sinTheta\_O}}{v \cdot 2} \]

    2. expm1-udef 58.8%

       \[\leadsto \left(\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{cosTheta\_O}{\sinh \left(\frac{1}{v}\right) \cdot v}\right)} - 1\right)} \cdot cosTheta\_i\right) \cdot \frac{e^{\frac{-sinTheta\_i}{v} \cdot sinTheta\_O}}{v \cdot 2} \]

    3. *-commutative 58.8%

       \[\leadsto \left(\left(e^{\mathsf{log1p}\left(\frac{cosTheta\_O}{\color{blue}{v \cdot \sinh \left(\frac{1}{v}\right)}}\right)} - 1\right) \cdot cosTheta\_i\right) \cdot \frac{e^{\frac{-sinTheta\_i}{v} \cdot sinTheta\_O}}{v \cdot 2} \]

12. Applied egg-rr 58.8%

    \[\leadsto \left(\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{cosTheta\_O}{v \cdot \sinh \left(\frac{1}{v}\right)}\right)} - 1\right)} \cdot cosTheta\_i\right) \cdot \frac{e^{\frac{-sinTheta\_i}{v} \cdot sinTheta\_O}}{v \cdot 2} \]
13. Step-by-step derivation

    1. expm1-def 98.5%

       \[\leadsto \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{cosTheta\_O}{v \cdot \sinh \left(\frac{1}{v}\right)}\right)\right)} \cdot cosTheta\_i\right) \cdot \frac{e^{\frac{-sinTheta\_i}{v} \cdot sinTheta\_O}}{v \cdot 2} \]

    2. expm1-log1p 98.5%

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

    3. associate-/r* 98.5%

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

14. Simplified 98.5%

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

15. Final simplification 98.5%

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

Alternative 4: 62.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{e^{\frac{sinTheta\_i}{\frac{v}{-sinTheta\_O}}}}{\frac{2 + \frac{0.3333333333333333}{{v}^{2}}}{\frac{cosTheta\_i}{\frac{v}{cosTheta\_O}}}} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (exp (/ sinTheta_i (/ v (- sinTheta_O))))
  (/
   (+ 2.0 (/ 0.3333333333333333 (pow v 2.0)))
   (/ cosTheta_i (/ v cosTheta_O)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return expf((sinTheta_i / (v / -sinTheta_O))) / ((2.0f + (0.3333333333333333f / powf(v, 2.0f))) / (cosTheta_i / (v / cosTheta_O)));
}

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 / (v / -sintheta_o))) / ((2.0e0 + (0.3333333333333333e0 / (v ** 2.0e0))) / (costheta_i / (v / costheta_o)))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(exp(Float32(sinTheta_i / Float32(v / Float32(-sinTheta_O)))) / Float32(Float32(Float32(2.0) + Float32(Float32(0.3333333333333333) / (v ^ Float32(2.0)))) / Float32(cosTheta_i / Float32(v / cosTheta_O))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = exp((sinTheta_i / (v / -sinTheta_O))) / ((single(2.0) + (single(0.3333333333333333) / (v ^ single(2.0)))) / (cosTheta_i / (v / cosTheta_O)));
end

Derivation

1. Initial program 98.4%

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

   1. associate-/l* 94.9%

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

   2. *-commutative 94.9%

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

   3. distribute-neg-frac 94.9%

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

   4. distribute-rgt-neg-out 94.9%

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

   5. associate-/l* 94.9%

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

   6. *-commutative 94.9%

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

   7. *-commutative 94.9%

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

   8. associate-/l* 94.9%

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

3. Simplified 94.9%

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

5. Taylor expanded in v around inf 62.7%

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

   1. associate-*r/ 62.7%

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

   2. metadata-eval 62.7%

      \[\leadsto \frac{e^{\frac{sinTheta\_i}{\frac{v}{-sinTheta\_O}}}}{\frac{2 + \frac{\color{blue}{0.3333333333333333}}{{v}^{2}}}{\frac{cosTheta\_i}{\frac{v}{cosTheta\_O}}}} \]

7. Simplified 62.7%

   \[\leadsto \frac{e^{\frac{sinTheta\_i}{\frac{v}{-sinTheta\_O}}}}{\frac{\color{blue}{2 + \frac{0.3333333333333333}{{v}^{2}}}}{\frac{cosTheta\_i}{\frac{v}{cosTheta\_O}}}} \]

8. Final simplification 62.7%

   \[\leadsto \frac{e^{\frac{sinTheta\_i}{\frac{v}{-sinTheta\_O}}}}{\frac{2 + \frac{0.3333333333333333}{{v}^{2}}}{\frac{cosTheta\_i}{\frac{v}{cosTheta\_O}}}} \]
9. Add Preprocessing

Alternative 5: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return ((cosTheta_O / expf((sinTheta_O * (sinTheta_i / v)))) * (cosTheta_i / v)) / (2.0f * (v * sinhf((1.0f / 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 = ((costheta_o / exp((sintheta_o * (sintheta_i / v)))) * (costheta_i / v)) / (2.0e0 * (v * sinh((1.0e0 / v))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

   1. div-inv 98.6%

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

   2. *-commutative 98.6%

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

4. Applied egg-rr 98.6%

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

5. Taylor expanded in cosTheta_O around 0 98.4%

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

   1. associate-/l* 98.4%

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

7. Simplified 98.4%

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

   1. div-inv 98.4%

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

   2. distribute-neg-frac 98.4%

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

   3. div-inv 98.3%

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

   4. clear-num 98.4%

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

   5. associate-*l* 98.4%

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

   6. *-commutative 98.4%

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

9. Applied egg-rr 98.4%

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

    1. associate-*r/ 98.4%

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

    2. *-rgt-identity 98.4%

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

    3. associate-*r* 98.4%

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

    4. distribute-frac-neg 98.4%

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

    5. *-commutative 98.4%

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

    6. exp-neg 98.4%

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

    7. associate-*l/ 98.4%

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

    8. *-lft-identity 98.4%

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

    9. *-commutative 98.4%

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

    10. associate-*l/ 98.4%

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

    11. *-commutative 98.4%

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

    12. *-commutative 98.4%

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

    13. associate-*l* 98.4%

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

11. Simplified 98.4%

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

12. Final simplification 98.4%

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

Alternative 6: 62.9% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

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 / (v / -sintheta_o))) / ((2.0e0 * (v / (costheta_o * costheta_i))) + (0.3333333333333333e0 * (1.0e0 / (costheta_o * (v * costheta_i)))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

   1. associate-/l* 94.9%

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

   2. *-commutative 94.9%

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

   3. distribute-neg-frac 94.9%

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

   4. distribute-rgt-neg-out 94.9%

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

   5. associate-/l* 94.9%

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

   6. *-commutative 94.9%

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

   7. *-commutative 94.9%

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

   8. associate-/l* 94.9%

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

3. Simplified 94.9%

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

5. Taylor expanded in v around inf 62.7%

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

6. Final simplification 62.7%

   \[\leadsto \frac{e^{\frac{sinTheta\_i}{\frac{v}{-sinTheta\_O}}}}{2 \cdot \frac{v}{cosTheta\_O \cdot cosTheta\_i} + 0.3333333333333333 \cdot \frac{1}{cosTheta\_O \cdot \left(v \cdot cosTheta\_i\right)}} \]
7. Add Preprocessing

Alternative 7: 58.0% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

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_o * (-sintheta_i / v))) / (v * 2.0e0)) * ((costheta_o / v) / (1.0e0 / (v * costheta_i)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

   1. *-commutative 98.4%

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

   2. associate-*l* 98.4%

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

   3. times-frac 98.4%

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

   4. *-commutative 98.4%

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

   5. associate-*l/ 98.4%

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

   6. distribute-neg-frac 98.4%

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

   7. distribute-lft-neg-out 98.4%

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

   8. associate-*l/ 98.4%

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

   9. *-commutative 98.4%

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

3. Simplified 98.4%

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

   1. expm1-log1p-u 98.4%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{cosTheta\_O}{v} \cdot cosTheta\_i}{\sinh \left(\frac{1}{v}\right)}\right)\right)} \cdot \frac{e^{\frac{-sinTheta\_i}{v} \cdot sinTheta\_O}}{v \cdot 2} \]

   2. expm1-udef 52.3%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\frac{cosTheta\_O}{v} \cdot cosTheta\_i}{\sinh \left(\frac{1}{v}\right)}\right)} - 1\right)} \cdot \frac{e^{\frac{-sinTheta\_i}{v} \cdot sinTheta\_O}}{v \cdot 2} \]

6. Applied egg-rr 52.3%

   \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\frac{cosTheta\_O}{v} \cdot cosTheta\_i}{\sinh \left(\frac{1}{v}\right)}\right)} - 1\right)} \cdot \frac{e^{\frac{-sinTheta\_i}{v} \cdot sinTheta\_O}}{v \cdot 2} \]
7. Step-by-step derivation

   1. expm1-def 98.4%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{cosTheta\_O}{v} \cdot cosTheta\_i}{\sinh \left(\frac{1}{v}\right)}\right)\right)} \cdot \frac{e^{\frac{-sinTheta\_i}{v} \cdot sinTheta\_O}}{v \cdot 2} \]

   2. expm1-log1p 98.4%

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

   3. associate-/l* 97.9%

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

8. Simplified 97.9%

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

9. Taylor expanded in v around inf 58.5%

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

10. Final simplification 58.5%

    \[\leadsto \frac{e^{sinTheta\_O \cdot \frac{-sinTheta\_i}{v}}}{v \cdot 2} \cdot \frac{\frac{cosTheta\_O}{v}}{\frac{1}{v \cdot cosTheta\_i}} \]
11. Add Preprocessing

Alternative 8: 58.7% accurate, 1.9× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (exp (/ sinTheta_i (/ v (- sinTheta_O))))
  (* v (/ 1.0 (* cosTheta_O (* cosTheta_i 0.5))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return expf((sinTheta_i / (v / -sinTheta_O))) / (v * (1.0f / (cosTheta_O * (cosTheta_i * 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 = exp((sintheta_i / (v / -sintheta_o))) / (v * (1.0e0 / (costheta_o * (costheta_i * 0.5e0))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

   1. associate-/l* 94.9%

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

   2. *-commutative 94.9%

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

   3. distribute-neg-frac 94.9%

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

   4. distribute-rgt-neg-out 94.9%

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

   5. associate-/l* 94.9%

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

   6. *-commutative 94.9%

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

   7. *-commutative 94.9%

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

   8. associate-/l* 94.9%

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

3. Simplified 94.9%

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

   1. clear-num 95.0%

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

   2. inv-pow 95.0%

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

   3. associate-/l* 94.9%

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

   4. associate-*r/ 94.9%

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

   5. *-commutative 94.9%

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

   6. *-commutative 94.9%

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

   7. associate-*l* 94.9%

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

   8. *-commutative 94.9%

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

6. Applied egg-rr 94.9%

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

   1. unpow-1 94.9%

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

   2. times-frac 95.0%

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

   3. associate-/r* 94.8%

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

   4. *-commutative 94.8%

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

   5. associate-/r* 94.8%

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

8. Simplified 94.8%

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

9. Taylor expanded in v around inf 58.4%

   \[\leadsto \frac{e^{\frac{sinTheta\_i}{\frac{v}{-sinTheta\_O}}}}{\frac{\frac{1}{\frac{\frac{cosTheta\_O}{v}}{\color{blue}{\frac{1}{v}}}}}{\frac{\frac{cosTheta\_i}{2}}{v}}} \]
10. Step-by-step derivation

    1. expm1-log1p-u 28.3%

       \[\leadsto \frac{e^{\frac{sinTheta\_i}{\frac{v}{-sinTheta\_O}}}}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{1}{\frac{\frac{cosTheta\_O}{v}}{\frac{1}{v}}}}{\frac{\frac{cosTheta\_i}{2}}{v}}\right)\right)}} \]

    2. expm1-udef 28.3%

       \[\leadsto \frac{e^{\frac{sinTheta\_i}{\frac{v}{-sinTheta\_O}}}}{\color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{1}{\frac{\frac{cosTheta\_O}{v}}{\frac{1}{v}}}}{\frac{\frac{cosTheta\_i}{2}}{v}}\right)} - 1}} \]

    3. associate-/r/ 28.4%

       \[\leadsto \frac{e^{\frac{sinTheta\_i}{\frac{v}{-sinTheta\_O}}}}{e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{1}{\frac{\frac{cosTheta\_O}{v}}{\frac{1}{v}}}}{\frac{cosTheta\_i}{2}} \cdot v}\right)} - 1} \]

    4. associate-/l/ 28.4%

       \[\leadsto \frac{e^{\frac{sinTheta\_i}{\frac{v}{-sinTheta\_O}}}}{e^{\mathsf{log1p}\left(\frac{\frac{1}{\color{blue}{\frac{cosTheta\_O}{\frac{1}{v} \cdot v}}}}{\frac{cosTheta\_i}{2}} \cdot v\right)} - 1} \]

    5. inv-pow 28.4%

       \[\leadsto \frac{e^{\frac{sinTheta\_i}{\frac{v}{-sinTheta\_O}}}}{e^{\mathsf{log1p}\left(\frac{\frac{1}{\frac{cosTheta\_O}{\color{blue}{{v}^{-1}} \cdot v}}}{\frac{cosTheta\_i}{2}} \cdot v\right)} - 1} \]

    6. pow-plus 28.4%

       \[\leadsto \frac{e^{\frac{sinTheta\_i}{\frac{v}{-sinTheta\_O}}}}{e^{\mathsf{log1p}\left(\frac{\frac{1}{\frac{cosTheta\_O}{\color{blue}{{v}^{\left(-1 + 1\right)}}}}}{\frac{cosTheta\_i}{2}} \cdot v\right)} - 1} \]

    7. metadata-eval 28.4%

       \[\leadsto \frac{e^{\frac{sinTheta\_i}{\frac{v}{-sinTheta\_O}}}}{e^{\mathsf{log1p}\left(\frac{\frac{1}{\frac{cosTheta\_O}{{v}^{\color{blue}{0}}}}}{\frac{cosTheta\_i}{2}} \cdot v\right)} - 1} \]

    8. metadata-eval 28.4%

       \[\leadsto \frac{e^{\frac{sinTheta\_i}{\frac{v}{-sinTheta\_O}}}}{e^{\mathsf{log1p}\left(\frac{\frac{1}{\frac{cosTheta\_O}{\color{blue}{1}}}}{\frac{cosTheta\_i}{2}} \cdot v\right)} - 1} \]

    9. div-inv 28.4%

       \[\leadsto \frac{e^{\frac{sinTheta\_i}{\frac{v}{-sinTheta\_O}}}}{e^{\mathsf{log1p}\left(\frac{\frac{1}{\frac{cosTheta\_O}{1}}}{\color{blue}{cosTheta\_i \cdot \frac{1}{2}}} \cdot v\right)} - 1} \]

    10. metadata-eval 28.4%

        \[\leadsto \frac{e^{\frac{sinTheta\_i}{\frac{v}{-sinTheta\_O}}}}{e^{\mathsf{log1p}\left(\frac{\frac{1}{\frac{cosTheta\_O}{1}}}{cosTheta\_i \cdot \color{blue}{0.5}} \cdot v\right)} - 1} \]

11. Applied egg-rr 28.4%

    \[\leadsto \frac{e^{\frac{sinTheta\_i}{\frac{v}{-sinTheta\_O}}}}{\color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{1}{\frac{cosTheta\_O}{1}}}{cosTheta\_i \cdot 0.5} \cdot v\right)} - 1}} \]
12. Step-by-step derivation

    1. expm1-def 28.4%

       \[\leadsto \frac{e^{\frac{sinTheta\_i}{\frac{v}{-sinTheta\_O}}}}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{1}{\frac{cosTheta\_O}{1}}}{cosTheta\_i \cdot 0.5} \cdot v\right)\right)}} \]

    2. expm1-log1p 58.5%

       \[\leadsto \frac{e^{\frac{sinTheta\_i}{\frac{v}{-sinTheta\_O}}}}{\color{blue}{\frac{\frac{1}{\frac{cosTheta\_O}{1}}}{cosTheta\_i \cdot 0.5} \cdot v}} \]

    3. *-commutative 58.5%

       \[\leadsto \frac{e^{\frac{sinTheta\_i}{\frac{v}{-sinTheta\_O}}}}{\color{blue}{v \cdot \frac{\frac{1}{\frac{cosTheta\_O}{1}}}{cosTheta\_i \cdot 0.5}}} \]

    4. /-rgt-identity 58.5%

       \[\leadsto \frac{e^{\frac{sinTheta\_i}{\frac{v}{-sinTheta\_O}}}}{v \cdot \frac{\frac{1}{\color{blue}{cosTheta\_O}}}{cosTheta\_i \cdot 0.5}} \]

    5. associate-/r* 58.5%

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

13. Simplified 58.5%

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

14. Final simplification 58.5%

    \[\leadsto \frac{e^{\frac{sinTheta\_i}{\frac{v}{-sinTheta\_O}}}}{v \cdot \frac{1}{cosTheta\_O \cdot \left(cosTheta\_i \cdot 0.5\right)}} \]
15. Add Preprocessing

Alternative 9: 58.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{e^{\frac{sinTheta\_i}{\frac{v}{-sinTheta\_O}}}}{v \cdot \frac{\frac{1}{cosTheta\_O}}{cosTheta\_i \cdot 0.5}} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (exp (/ sinTheta_i (/ v (- sinTheta_O))))
  (* v (/ (/ 1.0 cosTheta_O) (* cosTheta_i 0.5)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return expf((sinTheta_i / (v / -sinTheta_O))) / (v * ((1.0f / cosTheta_O) / (cosTheta_i * 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 = exp((sintheta_i / (v / -sintheta_o))) / (v * ((1.0e0 / costheta_o) / (costheta_i * 0.5e0)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

   1. associate-/l* 94.9%

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

   2. *-commutative 94.9%

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

   3. distribute-neg-frac 94.9%

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

   4. distribute-rgt-neg-out 94.9%

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

   5. associate-/l* 94.9%

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

   6. *-commutative 94.9%

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

   7. *-commutative 94.9%

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

   8. associate-/l* 94.9%

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

3. Simplified 94.9%

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

   1. clear-num 95.0%

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

   2. inv-pow 95.0%

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

   3. associate-/l* 94.9%

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

   4. associate-*r/ 94.9%

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

   5. *-commutative 94.9%

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

   6. *-commutative 94.9%

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

   7. associate-*l* 94.9%

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

   8. *-commutative 94.9%

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

6. Applied egg-rr 94.9%

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

   1. unpow-1 94.9%

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

   2. times-frac 95.0%

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

   3. associate-/r* 94.8%

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

   4. *-commutative 94.8%

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

   5. associate-/r* 94.8%

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

8. Simplified 94.8%

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

9. Taylor expanded in v around inf 58.4%

   \[\leadsto \frac{e^{\frac{sinTheta\_i}{\frac{v}{-sinTheta\_O}}}}{\frac{\frac{1}{\frac{\frac{cosTheta\_O}{v}}{\color{blue}{\frac{1}{v}}}}}{\frac{\frac{cosTheta\_i}{2}}{v}}} \]
10. Step-by-step derivation

    1. associate-/r/ 58.5%

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

    2. associate-/l/ 58.5%

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

    3. inv-pow 58.5%

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

    4. pow-plus 58.5%

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

    5. metadata-eval 58.5%

       \[\leadsto \frac{e^{\frac{sinTheta\_i}{\frac{v}{-sinTheta\_O}}}}{\frac{\frac{1}{\frac{cosTheta\_O}{{v}^{\color{blue}{0}}}}}{\frac{cosTheta\_i}{2}} \cdot v} \]

    6. metadata-eval 58.5%

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

    7. div-inv 58.5%

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

    8. metadata-eval 58.5%

       \[\leadsto \frac{e^{\frac{sinTheta\_i}{\frac{v}{-sinTheta\_O}}}}{\frac{\frac{1}{\frac{cosTheta\_O}{1}}}{cosTheta\_i \cdot \color{blue}{0.5}} \cdot v} \]

11. Applied egg-rr 58.5%

    \[\leadsto \frac{e^{\frac{sinTheta\_i}{\frac{v}{-sinTheta\_O}}}}{\color{blue}{\frac{\frac{1}{\frac{cosTheta\_O}{1}}}{cosTheta\_i \cdot 0.5} \cdot v}} \]

12. Final simplification 58.5%

    \[\leadsto \frac{e^{\frac{sinTheta\_i}{\frac{v}{-sinTheta\_O}}}}{v \cdot \frac{\frac{1}{cosTheta\_O}}{cosTheta\_i \cdot 0.5}} \]
13. Add Preprocessing

Alternative 10: 58.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (0.5f * ((cosTheta_O * cosTheta_i) * expf((-sinTheta_O / (v / sinTheta_i))))) / 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 = (0.5e0 * ((costheta_o * costheta_i) * exp((-sintheta_o / (v / sintheta_i))))) / v
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

   1. associate-/l* 94.9%

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

   2. *-commutative 94.9%

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

   3. distribute-neg-frac 94.9%

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

   4. distribute-rgt-neg-out 94.9%

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

   5. associate-/l* 94.9%

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

   6. *-commutative 94.9%

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

   7. *-commutative 94.9%

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

   8. associate-/l* 94.9%

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

3. Simplified 94.9%

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

   1. clear-num 95.0%

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

   2. inv-pow 95.0%

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

   3. associate-/l* 94.9%

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

   4. associate-*r/ 94.9%

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

   5. *-commutative 94.9%

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

   6. *-commutative 94.9%

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

   7. associate-*l* 94.9%

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

   8. *-commutative 94.9%

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

6. Applied egg-rr 94.9%

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

   1. unpow-1 94.9%

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

   2. times-frac 95.0%

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

   3. associate-/r* 94.8%

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

   4. *-commutative 94.8%

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

   5. associate-/r* 94.8%

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

8. Simplified 94.8%

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

9. Taylor expanded in v around inf 58.4%

   \[\leadsto \frac{e^{\frac{sinTheta\_i}{\frac{v}{-sinTheta\_O}}}}{\frac{\frac{1}{\frac{\frac{cosTheta\_O}{v}}{\color{blue}{\frac{1}{v}}}}}{\frac{\frac{cosTheta\_i}{2}}{v}}} \]

10. Taylor expanded in sinTheta_i around inf 58.4%

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

    1. associate-*r/ 58.5%

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

    2. associate-*r* 58.5%

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

    3. mul-1-neg 58.5%

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

    4. associate-/l* 58.5%

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

12. Simplified 58.5%

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

13. Final simplification 58.5%

    \[\leadsto \frac{0.5 \cdot \left(\left(cosTheta\_O \cdot cosTheta\_i\right) \cdot e^{\frac{-sinTheta\_O}{\frac{v}{sinTheta\_i}}}\right)}{v} \]
14. Add Preprocessing

Alternative 11: 58.0% accurate, 31.4× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return 0.5f * (cosTheta_O * (cosTheta_i / 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 = 0.5e0 * (costheta_o * (costheta_i / v))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

   1. *-commutative 98.4%

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

   2. associate-*l* 98.4%

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

   3. times-frac 98.4%

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

   4. *-commutative 98.4%

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

   5. associate-*l/ 98.4%

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

   6. distribute-neg-frac 98.4%

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

   7. distribute-lft-neg-out 98.4%

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

   8. associate-*l/ 98.4%

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

   9. *-commutative 98.4%

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

3. Simplified 98.4%

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

5. Taylor expanded in cosTheta_O around 0 98.4%

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

   1. associate-*r/ 98.5%

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

7. Simplified 98.5%

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

8. Taylor expanded in v around inf 58.4%

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

   1. associate-*r/ 58.5%

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

10. Simplified 58.5%

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

11. Final simplification 58.5%

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

Alternative 12: 58.0% accurate, 31.4× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return ((cosTheta_O * cosTheta_i) * 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_o * costheta_i) * 0.5e0) / v
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

   1. *-commutative 98.4%

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

   2. associate-*l* 98.4%

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

   3. times-frac 98.4%

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

   4. *-commutative 98.4%

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

   5. associate-*l/ 98.4%

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

   6. distribute-neg-frac 98.4%

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

   7. distribute-lft-neg-out 98.4%

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

   8. associate-*l/ 98.4%

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

   9. *-commutative 98.4%

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

3. Simplified 98.4%

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

5. Taylor expanded in v around inf 58.4%

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

   1. associate-*r/ 58.5%

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

7. Simplified 58.5%

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

8. Final simplification 58.5%

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

Reproduce

herbie shell --seed 2024026 
(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)))