HairBSDF, Mp, upper

Percentage Accurate: 98.6% → 98.5%

Time: 13.8s

Alternatives: 13

Speedup: N/A×

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: 98.5% accurate, 1.8× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (cosTheta_O / v) * (((0.5f / v) / sinhf((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 = (costheta_o / v) * (((0.5e0 / v) / sinh((1.0e0 / v))) * costheta_i)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

   1. lift-*.f32 N/A

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

   2. remove-double-div N/A

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

   3. lift-/.f32 N/A

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

   4. un-div-inv N/A

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

   5. lower-/.f32 98.9

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 98.9

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

4. Applied rewrites 98.9%

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

6. Applied rewrites 98.9%

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

7. Taylor expanded in sinTheta_i around 0

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

   1. associate-/r* N/A

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

   2. lower-/.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. lower--.f32 N/A

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

   5. lower-exp.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. rec-exp N/A

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

   8. lower-exp.f32 N/A

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

   9. distribute-neg-frac N/A

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

   10. metadata-eval N/A

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

   11. lower-/.f32 98.8

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

9. Applied rewrites 98.8%

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

11. Applied rewrites 98.9%

    \[\leadsto \frac{cosTheta\_O}{v} \cdot \left(\frac{\frac{0.5}{v}}{\color{blue}{\sinh \left(\frac{1}{v}\right)}} \cdot cosTheta\_i\right) \]
12. Add Preprocessing

Alternative 2: 98.5% accurate, 1.8× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (((0.5f / v) / sinhf((1.0f / v))) * (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 = (((0.5e0 / v) / sinh((1.0e0 / v))) * (costheta_o / v)) * costheta_i
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

   1. lift-*.f32 N/A

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

   2. remove-double-div N/A

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

   3. lift-/.f32 N/A

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

   4. un-div-inv N/A

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

   5. lower-/.f32 98.9

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 98.9

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

4. Applied rewrites 98.9%

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

6. Applied rewrites 98.9%

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

7. Taylor expanded in sinTheta_i around 0

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

   1. associate-/r* N/A

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

   2. lower-/.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. lower--.f32 N/A

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

   5. lower-exp.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. rec-exp N/A

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

   8. lower-exp.f32 N/A

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

   9. distribute-neg-frac N/A

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

   10. metadata-eval N/A

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

   11. lower-/.f32 98.8

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

9. Applied rewrites 98.8%

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

    1. lift-*.f32 N/A

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

    2. lift-*.f32 N/A

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

    3. associate-*r* N/A

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

    4. lower-*.f32 N/A

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

11. Applied rewrites 98.8%

    \[\leadsto \color{blue}{\left(\frac{\frac{0.5}{v}}{\sinh \left(\frac{1}{v}\right)} \cdot \frac{cosTheta\_O}{v}\right) \cdot cosTheta\_i} \]
12. Add Preprocessing

Alternative 3: 98.5% accurate, 1.8× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_O * ((((0.5f / v) / sinhf((1.0f / v))) * 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 = costheta_o * ((((0.5e0 / v) / sinh((1.0e0 / v))) * costheta_i) / v)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

   1. lift-*.f32 N/A

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

   2. remove-double-div N/A

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

   3. lift-/.f32 N/A

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

   4. un-div-inv N/A

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

   5. lower-/.f32 98.9

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 98.9

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

4. Applied rewrites 98.9%

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

6. Applied rewrites 98.9%

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

7. Taylor expanded in sinTheta_i around 0

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

   1. associate-/r* N/A

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

   2. lower-/.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. lower--.f32 N/A

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

   5. lower-exp.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. rec-exp N/A

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

   8. lower-exp.f32 N/A

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

   9. distribute-neg-frac N/A

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

   10. metadata-eval N/A

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

   11. lower-/.f32 98.8

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

9. Applied rewrites 98.8%

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

    1. lift-*.f32 N/A

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

    2. lift-/.f32 N/A

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

    3. associate-*l/ N/A

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

    4. associate-/l* N/A

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

    5. lower-*.f32 N/A

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

    6. lower-/.f32 98.8

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

11. Applied rewrites 98.8%

    \[\leadsto \color{blue}{cosTheta\_O \cdot \frac{\frac{\frac{0.5}{v}}{\sinh \left(\frac{1}{v}\right)} \cdot cosTheta\_i}{v}} \]
12. Add Preprocessing

Alternative 4: 98.3% accurate, 1.8× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* cosTheta_i (* cosTheta_O (/ 1.0 (* (* 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_i * (cosTheta_O * (1.0f / ((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_i * (costheta_o * (1.0e0 / ((v * (2.0e0 * v)) * sinh((1.0e0 / v)))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

   1. lift-/.f32 N/A

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

   2. div-inv N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lift-/.f32 N/A

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

   6. lift-exp.f32 N/A

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

   7. lift-neg.f32 N/A

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

   8. exp-neg N/A

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

   9. frac-times N/A

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

   10. frac-times N/A

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

   11. *-rgt-identity N/A

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

   12. *-rgt-identity N/A

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

   13. lower-/.f32 N/A

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

4. Applied rewrites 98.7%

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

5. Taylor expanded in sinTheta_i around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. lower-exp.f32 N/A

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

   5. lower-/.f32 N/A

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

   6. rec-exp N/A

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

   7. distribute-neg-frac N/A

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

   8. metadata-eval N/A

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

   9. lower-exp.f32 N/A

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

   10. lower-/.f32 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f32 98.6

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

7. Applied rewrites 98.6%

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

   1. lift-/.f32 N/A

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

   2. div-inv N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. lower-*.f32 N/A

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

   7. lower-*.f32 N/A

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

9. Applied rewrites 98.7%

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

Alternative 5: 71.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.4620000123977661:\\ \;\;\;\;\frac{cosTheta\_O \cdot cosTheta\_i}{\left(e^{\frac{1}{v}} - 1\right) \cdot \left(v \cdot v\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{cosTheta\_O \cdot cosTheta\_i}{\frac{\frac{0.3333333333333333}{v \cdot v} + 2}{v} \cdot \left(v \cdot v\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (<= v 0.4620000123977661)
   (/ (* cosTheta_O cosTheta_i) (* (- (exp (/ 1.0 v)) 1.0) (* v v)))
   (/
    (* cosTheta_O cosTheta_i)
    (* (/ (+ (/ 0.3333333333333333 (* v v)) 2.0) v) (* v v)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float tmp;
	if (v <= 0.4620000123977661f) {
		tmp = (cosTheta_O * cosTheta_i) / ((expf((1.0f / v)) - 1.0f) * (v * v));
	} else {
		tmp = (cosTheta_O * cosTheta_i) / ((((0.3333333333333333f / (v * v)) + 2.0f) / v) * (v * v));
	}
	return tmp;
}

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
    real(4) :: tmp
    if (v <= 0.4620000123977661e0) then
        tmp = (costheta_o * costheta_i) / ((exp((1.0e0 / v)) - 1.0e0) * (v * v))
    else
        tmp = (costheta_o * costheta_i) / ((((0.3333333333333333e0 / (v * v)) + 2.0e0) / v) * (v * v))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.4620000123977661))
		tmp = Float32(Float32(cosTheta_O * cosTheta_i) / Float32(Float32(exp(Float32(Float32(1.0) / v)) - Float32(1.0)) * Float32(v * v)));
	else
		tmp = Float32(Float32(cosTheta_O * cosTheta_i) / Float32(Float32(Float32(Float32(Float32(0.3333333333333333) / Float32(v * v)) + Float32(2.0)) / v) * Float32(v * v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.0);
	if (v <= single(0.4620000123977661))
		tmp = (cosTheta_O * cosTheta_i) / ((exp((single(1.0) / v)) - single(1.0)) * (v * v));
	else
		tmp = (cosTheta_O * cosTheta_i) / ((((single(0.3333333333333333) / (v * v)) + single(2.0)) / v) * (v * v));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.462000012

   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. lift-/.f32 N/A

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

      2. div-inv N/A

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

      3. lift-*.f32 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f32 N/A

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

      6. lift-exp.f32 N/A

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

      7. lift-neg.f32 N/A

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

      8. exp-neg N/A

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

      9. frac-times N/A

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

      10. frac-times N/A

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

      11. *-rgt-identity N/A

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

      12. *-rgt-identity N/A

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

      13. lower-/.f32 N/A

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

   4. Applied rewrites 98.4%

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

   5. Taylor expanded in sinTheta_i around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. lower-exp.f32 N/A

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

      5. lower-/.f32 N/A

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

      6. rec-exp N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. lower-exp.f32 N/A

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

      10. lower-/.f32 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f32 98.5

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

   7. Applied rewrites 98.5%

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

   8. Taylor expanded in v around inf

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

   10. Applied rewrites 74.0%

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

   if 0.462000012 < v

   1. Initial program 99.0%

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

      1. lift-/.f32 N/A

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

      2. div-inv N/A

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

      3. lift-*.f32 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f32 N/A

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

      6. lift-exp.f32 N/A

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

      7. lift-neg.f32 N/A

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

      8. exp-neg N/A

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

      9. frac-times N/A

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

      10. frac-times N/A

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

      11. *-rgt-identity N/A

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

      12. *-rgt-identity N/A

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

      13. lower-/.f32 N/A

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in sinTheta_i around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. lower-exp.f32 N/A

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

      5. lower-/.f32 N/A

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

      6. rec-exp N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. lower-exp.f32 N/A

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

      10. lower-/.f32 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f32 98.9

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

   7. Applied rewrites 98.9%

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

   8. Taylor expanded in v around inf

      \[\leadsto \frac{cosTheta\_O \cdot cosTheta\_i}{\frac{2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}}{v} \cdot \left(\color{blue}{v} \cdot v\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 72.7%

       \[\leadsto \frac{cosTheta\_O \cdot cosTheta\_i}{\frac{\frac{0.3333333333333333}{v \cdot v} + 2}{v} \cdot \left(\color{blue}{v} \cdot v\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 98.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

   1. lift-/.f32 N/A

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

   2. div-inv N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lift-/.f32 N/A

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

   6. lift-exp.f32 N/A

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

   7. lift-neg.f32 N/A

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

   8. exp-neg N/A

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

   9. frac-times N/A

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

   10. frac-times N/A

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

   11. *-rgt-identity N/A

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

   12. *-rgt-identity N/A

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

   13. lower-/.f32 N/A

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

4. Applied rewrites 98.7%

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

5. Taylor expanded in sinTheta_i around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. lower-exp.f32 N/A

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

   5. lower-/.f32 N/A

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

   6. rec-exp N/A

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

   7. distribute-neg-frac N/A

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

   8. metadata-eval N/A

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

   9. lower-exp.f32 N/A

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

   10. lower-/.f32 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f32 98.6

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

7. Applied rewrites 98.6%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

9. Applied rewrites 98.7%

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

Alternative 7: 98.4% accurate, 1.9× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* cosTheta_i (/ cosTheta_O (* (* 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_i * (cosTheta_O / ((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_i * (costheta_o / ((v * (2.0e0 * v)) * sinh((1.0e0 / v))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

   1. lift-/.f32 N/A

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

   2. div-inv N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lift-/.f32 N/A

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

   6. lift-exp.f32 N/A

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

   7. lift-neg.f32 N/A

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

   8. exp-neg N/A

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

   9. frac-times N/A

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

   10. frac-times N/A

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

   11. *-rgt-identity N/A

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

   12. *-rgt-identity N/A

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

   13. lower-/.f32 N/A

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

4. Applied rewrites 98.7%

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

5. Taylor expanded in sinTheta_i around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. lower-exp.f32 N/A

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

   5. lower-/.f32 N/A

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

   6. rec-exp N/A

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

   7. distribute-neg-frac N/A

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

   8. metadata-eval N/A

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

   9. lower-exp.f32 N/A

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

   10. lower-/.f32 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f32 98.6

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

7. Applied rewrites 98.6%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. lower-*.f32 N/A

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

   6. lower-/.f32 98.7

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

9. Applied rewrites 98.8%

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

Alternative 8: 63.8% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\frac{2 \cdot \frac{\frac{0.16666666666666666}{v \cdot v} + 1}{v}}{\frac{1}{v}}} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (* 1.0 (/ (* cosTheta_i cosTheta_O) v))
  (/ (* 2.0 (/ (+ (/ 0.16666666666666666 (* v v)) 1.0) v)) (/ 1.0 v))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (1.0f * ((cosTheta_i * cosTheta_O) / v)) / ((2.0f * (((0.16666666666666666f / (v * v)) + 1.0f) / v)) / (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 = (1.0e0 * ((costheta_i * costheta_o) / v)) / ((2.0e0 * (((0.16666666666666666e0 / (v * v)) + 1.0e0) / v)) / (1.0e0 / v))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(1.0) * Float32(Float32(cosTheta_i * cosTheta_O) / v)) / Float32(Float32(Float32(2.0) * Float32(Float32(Float32(Float32(0.16666666666666666) / Float32(v * v)) + Float32(1.0)) / v)) / Float32(Float32(1.0) / v)))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (single(1.0) * ((cosTheta_i * cosTheta_O) / v)) / ((single(2.0) * (((single(0.16666666666666666) / (v * v)) + single(1.0)) / v)) / (single(1.0) / v));
end

Derivation

1. Initial program 98.7%

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

   1. lift-*.f32 N/A

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

   2. remove-double-div N/A

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

   3. lift-/.f32 N/A

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

   4. un-div-inv N/A

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

   5. lower-/.f32 98.9

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 98.9

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

4. Applied rewrites 98.9%

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

5. Taylor expanded in sinTheta_i around 0

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

7. Applied rewrites 98.8%

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

8. Taylor expanded in v around inf

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

   1. lower-/.f32 N/A

      \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\frac{2 \cdot \color{blue}{\frac{1 + \frac{1}{6} \cdot \frac{1}{{v}^{2}}}{v}}}{\frac{1}{v}}} \]

   2. +-commutative N/A

      \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\frac{2 \cdot \frac{\color{blue}{\frac{1}{6} \cdot \frac{1}{{v}^{2}} + 1}}{v}}{\frac{1}{v}}} \]

   3. lower-+.f32 N/A

      \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\frac{2 \cdot \frac{\color{blue}{\frac{1}{6} \cdot \frac{1}{{v}^{2}} + 1}}{v}}{\frac{1}{v}}} \]

   4. associate-*r/ N/A

      \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\frac{2 \cdot \frac{\color{blue}{\frac{\frac{1}{6} \cdot 1}{{v}^{2}}} + 1}{v}}{\frac{1}{v}}} \]

   5. metadata-eval N/A

      \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\frac{2 \cdot \frac{\frac{\color{blue}{\frac{1}{6}}}{{v}^{2}} + 1}{v}}{\frac{1}{v}}} \]

   6. lower-/.f32 N/A

      \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\frac{2 \cdot \frac{\color{blue}{\frac{\frac{1}{6}}{{v}^{2}}} + 1}{v}}{\frac{1}{v}}} \]

   7. unpow2 N/A

      \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\frac{2 \cdot \frac{\frac{\frac{1}{6}}{\color{blue}{v \cdot v}} + 1}{v}}{\frac{1}{v}}} \]

   8. lower-*.f32 66.3

      \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\frac{2 \cdot \frac{\frac{0.16666666666666666}{\color{blue}{v \cdot v}} + 1}{v}}{\frac{1}{v}}} \]

10. Applied rewrites 66.3%

    \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\frac{2 \cdot \color{blue}{\frac{\frac{0.16666666666666666}{v \cdot v} + 1}{v}}}{\frac{1}{v}}} \]
11. Add Preprocessing

Alternative 9: 63.8% accurate, 5.2× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (1.0f * ((cosTheta_i * cosTheta_O) / v)) / ((0.3333333333333333f / (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 = (1.0e0 * ((costheta_i * costheta_o) / v)) / ((0.3333333333333333e0 / (v * v)) + 2.0e0)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

   1. lift-*.f32 N/A

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

   2. remove-double-div N/A

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

   3. lift-/.f32 N/A

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

   4. un-div-inv N/A

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

   5. lower-/.f32 98.9

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 98.9

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

4. Applied rewrites 98.9%

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

5. Taylor expanded in sinTheta_i around 0

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

7. Applied rewrites 98.8%

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

8. Taylor expanded in v around inf

   \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}}} \]
9. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{\frac{1}{3} \cdot \frac{1}{{v}^{2}} + 2}} \]

   2. lower-+.f32 N/A

      \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{\frac{1}{3} \cdot \frac{1}{{v}^{2}} + 2}} \]

   3. associate-*r/ N/A

      \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{\frac{\frac{1}{3} \cdot 1}{{v}^{2}}} + 2} \]

   4. metadata-eval N/A

      \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\frac{\color{blue}{\frac{1}{3}}}{{v}^{2}} + 2} \]

   5. lower-/.f32 N/A

      \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{\frac{\frac{1}{3}}{{v}^{2}}} + 2} \]

   6. unpow2 N/A

      \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\frac{\frac{1}{3}}{\color{blue}{v \cdot v}} + 2} \]

   7. lower-*.f32 66.3

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

10. Applied rewrites 66.3%

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

Alternative 10: 63.8% accurate, 6.6× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (cosTheta_O * cosTheta_i) / (((0.3333333333333333f / (v * 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 = (costheta_o * costheta_i) / (((0.3333333333333333e0 / (v * v)) + 2.0e0) * v)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

   1. lift-/.f32 N/A

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

   2. div-inv N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lift-/.f32 N/A

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

   6. lift-exp.f32 N/A

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

   7. lift-neg.f32 N/A

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

   8. exp-neg N/A

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

   9. frac-times N/A

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

   10. frac-times N/A

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

   11. *-rgt-identity N/A

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

   12. *-rgt-identity N/A

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

   13. lower-/.f32 N/A

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

4. Applied rewrites 98.7%

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

5. Taylor expanded in sinTheta_i around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. lower-exp.f32 N/A

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

   5. lower-/.f32 N/A

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

   6. rec-exp N/A

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

   7. distribute-neg-frac N/A

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

   8. metadata-eval N/A

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

   9. lower-exp.f32 N/A

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

   10. lower-/.f32 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f32 98.6

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

7. Applied rewrites 98.6%

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

8. Taylor expanded in v around inf

   \[\leadsto \frac{cosTheta\_O \cdot cosTheta\_i}{v \cdot \color{blue}{\left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)}} \]
9. Step-by-step derivation

10. Applied rewrites 66.3%

    \[\leadsto \frac{cosTheta\_O \cdot cosTheta\_i}{\left(\frac{0.3333333333333333}{v \cdot v} + 2\right) \cdot \color{blue}{v}} \]
11. Add Preprocessing

Alternative 11: 58.6% accurate, 9.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in v around inf

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

   1. lower-*.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower-*.f32 60.7

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

5. Applied rewrites 60.7%

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

7. Applied rewrites 61.5%

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

Alternative 12: 58.1% accurate, 12.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in v around inf

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

   1. lower-*.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower-*.f32 60.7

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

5. Applied rewrites 60.7%

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

7. Applied rewrites 60.7%

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

9. Applied rewrites 60.7%

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

11. Applied rewrites 60.8%

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

Alternative 13: 58.1% accurate, 12.4× speedup

Math

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

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(Float32(cosTheta_O * 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 98.7%

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

3. Taylor expanded in v around inf

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

   1. lower-*.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower-*.f32 60.7

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

5. Applied rewrites 60.7%

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

Reproduce

herbie shell --seed 2024318 
(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)))