Toniolo and Linder, Equation (3b), real

Percentage Accurate: 93.8% → 99.5%

Time: 10.3s

Alternatives: 2

Speedup: N/A×

Specification

Math

\[\begin{array}{l} \\ \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (* (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) (sin th)))

C

double code(double kx, double ky, double th) {
	return (sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) * sin(th);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(kx, ky, th)
use fmin_fmax_functions
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    code = (sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) * sin(th)
end function

Java

public static double code(double kx, double ky, double th) {
	return (Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) * Math.sin(th);
}

Python

def code(kx, ky, th):
	return (math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) * math.sin(th)

Julia

function code(kx, ky, th)
	return Float64(Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) * sin(th))
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = (sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) * sin(th);
end

Wolfram

code[kx_, ky_, th_] := N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]

Initial Program: 93.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (* (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) (sin th)))

C

double code(double kx, double ky, double th) {
	return (sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) * sin(th);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(kx, ky, th)
use fmin_fmax_functions
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    code = (sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) * sin(th)
end function

Java

public static double code(double kx, double ky, double th) {
	return (Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) * Math.sin(th);
}

Python

def code(kx, ky, th):
	return (math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) * math.sin(th)

Julia

function code(kx, ky, th)
	return Float64(Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) * sin(th))
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = (sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) * sin(th);
end

Wolfram

code[kx_, ky_, th_] := N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.5% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \left({\left(\mathsf{hypot}\left(\sin kx, \sin ky\right)\right)}^{-1} \cdot \sin ky\right) \cdot \sin th \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (* (* (pow (hypot (sin kx) (sin ky)) -1.0) (sin ky)) (sin th)))

C

double code(double kx, double ky, double th) {
	return (pow(hypot(sin(kx), sin(ky)), -1.0) * sin(ky)) * sin(th);
}

Java

public static double code(double kx, double ky, double th) {
	return (Math.pow(Math.hypot(Math.sin(kx), Math.sin(ky)), -1.0) * Math.sin(ky)) * Math.sin(th);
}

Python

def code(kx, ky, th):
	return (math.pow(math.hypot(math.sin(kx), math.sin(ky)), -1.0) * math.sin(ky)) * math.sin(th)

Julia

function code(kx, ky, th)
	return Float64(Float64((hypot(sin(kx), sin(ky)) ^ -1.0) * sin(ky)) * sin(th))
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = ((hypot(sin(kx), sin(ky)) ^ -1.0) * sin(ky)) * sin(th);
end

Wolfram

code[kx_, ky_, th_] := N[(N[(N[Power[N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision], -1.0], $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 92.5%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
2. Add Preprocessing

3. Taylor expanded in kx around inf

   \[\leadsto \color{blue}{\left(\sin ky \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)} \cdot \sin th \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(\sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \color{blue}{\sin ky}\right) \cdot \sin th \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \color{blue}{\sin ky}\right) \cdot \sin th \]

   3. sqrt-div N/A

      \[\leadsto \left(\frac{\sqrt{1}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin \color{blue}{ky}\right) \cdot \sin th \]

   4. metadata-eval N/A

      \[\leadsto \left(\frac{1}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin ky\right) \cdot \sin th \]

   5. inv-pow N/A

      \[\leadsto \left({\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}^{-1} \cdot \sin \color{blue}{ky}\right) \cdot \sin th \]

   6. lower-pow.f64 N/A

      \[\leadsto \left({\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}^{-1} \cdot \sin \color{blue}{ky}\right) \cdot \sin th \]

   7. unpow2 N/A

      \[\leadsto \left({\left(\sqrt{\sin kx \cdot \sin kx + {\sin ky}^{2}}\right)}^{-1} \cdot \sin ky\right) \cdot \sin th \]

   8. unpow2 N/A

      \[\leadsto \left({\left(\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}\right)}^{-1} \cdot \sin ky\right) \cdot \sin th \]

   9. lower-hypot.f64 N/A

      \[\leadsto \left({\left(\mathsf{hypot}\left(\sin kx, \sin ky\right)\right)}^{-1} \cdot \sin ky\right) \cdot \sin th \]

   10. lift-sin.f64 N/A

       \[\leadsto \left({\left(\mathsf{hypot}\left(\sin kx, \sin ky\right)\right)}^{-1} \cdot \sin ky\right) \cdot \sin th \]

   11. lift-sin.f64 N/A

       \[\leadsto \left({\left(\mathsf{hypot}\left(\sin kx, \sin ky\right)\right)}^{-1} \cdot \sin ky\right) \cdot \sin th \]

   12. lift-sin.f64 99.6

       \[\leadsto \left({\left(\mathsf{hypot}\left(\sin kx, \sin ky\right)\right)}^{-1} \cdot \sin ky\right) \cdot \sin th \]

5. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\left({\left(\mathsf{hypot}\left(\sin kx, \sin ky\right)\right)}^{-1} \cdot \sin ky\right)} \cdot \sin th \]
6. Add Preprocessing

Alternative 2: 99.4% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \left({\left(\mathsf{hypot}\left(\sin kx, \sin ky\right)\right)}^{-1} \cdot \sin th\right) \cdot \sin ky \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (* (* (pow (hypot (sin kx) (sin ky)) -1.0) (sin th)) (sin ky)))

C

double code(double kx, double ky, double th) {
	return (pow(hypot(sin(kx), sin(ky)), -1.0) * sin(th)) * sin(ky);
}

Java

public static double code(double kx, double ky, double th) {
	return (Math.pow(Math.hypot(Math.sin(kx), Math.sin(ky)), -1.0) * Math.sin(th)) * Math.sin(ky);
}

Python

def code(kx, ky, th):
	return (math.pow(math.hypot(math.sin(kx), math.sin(ky)), -1.0) * math.sin(th)) * math.sin(ky)

Julia

function code(kx, ky, th)
	return Float64(Float64((hypot(sin(kx), sin(ky)) ^ -1.0) * sin(th)) * sin(ky))
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = ((hypot(sin(kx), sin(ky)) ^ -1.0) * sin(th)) * sin(ky);
end

Wolfram

code[kx_, ky_, th_] := N[(N[(N[Power[N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision], -1.0], $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 92.5%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
2. Add Preprocessing

3. Taylor expanded in kx around inf

   \[\leadsto \color{blue}{\left(\sin ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \color{blue}{\left(\sin ky \cdot \sin th\right)} \]

   2. lower-*.f64 N/A

      \[\leadsto \sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \color{blue}{\left(\sin ky \cdot \sin th\right)} \]

   3. sqrt-div N/A

      \[\leadsto \frac{\sqrt{1}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \left(\color{blue}{\sin ky} \cdot \sin th\right) \]

   4. metadata-eval N/A

      \[\leadsto \frac{1}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \left(\sin \color{blue}{ky} \cdot \sin th\right) \]

   5. inv-pow N/A

      \[\leadsto {\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}^{-1} \cdot \left(\color{blue}{\sin ky} \cdot \sin th\right) \]

   6. lower-pow.f64 N/A

      \[\leadsto {\left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}^{-1} \cdot \left(\color{blue}{\sin ky} \cdot \sin th\right) \]

   7. unpow2 N/A

      \[\leadsto {\left(\sqrt{\sin kx \cdot \sin kx + {\sin ky}^{2}}\right)}^{-1} \cdot \left(\sin ky \cdot \sin th\right) \]

   8. unpow2 N/A

      \[\leadsto {\left(\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}\right)}^{-1} \cdot \left(\sin ky \cdot \sin th\right) \]

   9. lower-hypot.f64 N/A

      \[\leadsto {\left(\mathsf{hypot}\left(\sin kx, \sin ky\right)\right)}^{-1} \cdot \left(\sin \color{blue}{ky} \cdot \sin th\right) \]

   10. lift-sin.f64 N/A

       \[\leadsto {\left(\mathsf{hypot}\left(\sin kx, \sin ky\right)\right)}^{-1} \cdot \left(\sin ky \cdot \sin th\right) \]

   11. lift-sin.f64 N/A

       \[\leadsto {\left(\mathsf{hypot}\left(\sin kx, \sin ky\right)\right)}^{-1} \cdot \left(\sin ky \cdot \sin th\right) \]

   12. *-commutative N/A

       \[\leadsto {\left(\mathsf{hypot}\left(\sin kx, \sin ky\right)\right)}^{-1} \cdot \left(\sin th \cdot \color{blue}{\sin ky}\right) \]

   13. lower-*.f64 N/A

       \[\leadsto {\left(\mathsf{hypot}\left(\sin kx, \sin ky\right)\right)}^{-1} \cdot \left(\sin th \cdot \color{blue}{\sin ky}\right) \]

   14. lift-sin.f64 N/A

       \[\leadsto {\left(\mathsf{hypot}\left(\sin kx, \sin ky\right)\right)}^{-1} \cdot \left(\sin th \cdot \sin \color{blue}{ky}\right) \]

   15. lift-sin.f64 94.3

       \[\leadsto {\left(\mathsf{hypot}\left(\sin kx, \sin ky\right)\right)}^{-1} \cdot \left(\sin th \cdot \sin ky\right) \]

5. Applied rewrites 94.3%

   \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(\sin kx, \sin ky\right)\right)}^{-1} \cdot \left(\sin th \cdot \sin ky\right)} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto {\left(\mathsf{hypot}\left(\sin kx, \sin ky\right)\right)}^{-1} \cdot \color{blue}{\left(\sin th \cdot \sin ky\right)} \]

   2. lift-pow.f64 N/A

      \[\leadsto {\left(\mathsf{hypot}\left(\sin kx, \sin ky\right)\right)}^{-1} \cdot \left(\color{blue}{\sin th} \cdot \sin ky\right) \]

   3. lift-sin.f64 N/A

      \[\leadsto {\left(\mathsf{hypot}\left(\sin kx, \sin ky\right)\right)}^{-1} \cdot \left(\sin th \cdot \sin ky\right) \]

   4. lift-sin.f64 N/A

      \[\leadsto {\left(\mathsf{hypot}\left(\sin kx, \sin ky\right)\right)}^{-1} \cdot \left(\sin th \cdot \sin ky\right) \]

   5. lift-hypot.f64 N/A

      \[\leadsto {\left(\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}\right)}^{-1} \cdot \left(\sin \color{blue}{th} \cdot \sin ky\right) \]

   6. lift-*.f64 N/A

      \[\leadsto {\left(\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}\right)}^{-1} \cdot \left(\sin th \cdot \color{blue}{\sin ky}\right) \]

   7. lift-sin.f64 N/A

      \[\leadsto {\left(\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}\right)}^{-1} \cdot \left(\sin th \cdot \sin \color{blue}{ky}\right) \]

   8. lift-sin.f64 N/A

      \[\leadsto {\left(\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}\right)}^{-1} \cdot \left(\sin th \cdot \sin ky\right) \]

   9. associate-*r* N/A

      \[\leadsto \left({\left(\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}\right)}^{-1} \cdot \sin th\right) \cdot \color{blue}{\sin ky} \]

   10. lower-*.f64 N/A

       \[\leadsto \left({\left(\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}\right)}^{-1} \cdot \sin th\right) \cdot \color{blue}{\sin ky} \]

7. Applied rewrites 99.4%

   \[\leadsto \left({\left(\mathsf{hypot}\left(\sin kx, \sin ky\right)\right)}^{-1} \cdot \sin th\right) \cdot \color{blue}{\sin ky} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025057 
(FPCore (kx ky th)
  :name "Toniolo and Linder, Equation (3b), real"
  :precision binary64
  (* (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) (sin th)))