(acos (+ (* (sin z0) (sin z1)) (* (* (cos (- z3 z2)) (cos z1)) (cos z0))))

Percentage Accurate: 73.9% → 93.7%

Time: 9.9s

Alternatives: 6

Speedup: 1.0×

Specification

Math

\[\cos^{-1} \left(\sin z0 \cdot \sin z1 + \left(\cos \left(z3 - z2\right) \cdot \cos z1\right) \cdot \cos z0\right) \]

FPCore

(FPCore (z0 z1 z3 z2)
  :precision binary64
  (acos
 (+ (* (sin z0) (sin z1)) (* (* (cos (- z3 z2)) (cos z1)) (cos z0)))))

C

double code(double z0, double z1, double z3, double z2) {
	return acos(((sin(z0) * sin(z1)) + ((cos((z3 - z2)) * cos(z1)) * cos(z0))));
}

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(z0, z1, z3, z2)
use fmin_fmax_functions
    real(8), intent (in) :: z0
    real(8), intent (in) :: z1
    real(8), intent (in) :: z3
    real(8), intent (in) :: z2
    code = acos(((sin(z0) * sin(z1)) + ((cos((z3 - z2)) * cos(z1)) * cos(z0))))
end function

Java

public static double code(double z0, double z1, double z3, double z2) {
	return Math.acos(((Math.sin(z0) * Math.sin(z1)) + ((Math.cos((z3 - z2)) * Math.cos(z1)) * Math.cos(z0))));
}

Python

def code(z0, z1, z3, z2):
	return math.acos(((math.sin(z0) * math.sin(z1)) + ((math.cos((z3 - z2)) * math.cos(z1)) * math.cos(z0))))

Julia

function code(z0, z1, z3, z2)
	return acos(Float64(Float64(sin(z0) * sin(z1)) + Float64(Float64(cos(Float64(z3 - z2)) * cos(z1)) * cos(z0))))
end

MATLAB

function tmp = code(z0, z1, z3, z2)
	tmp = acos(((sin(z0) * sin(z1)) + ((cos((z3 - z2)) * cos(z1)) * cos(z0))));
end

Wolfram

code[z0_, z1_, z3_, z2_] := N[ArcCos[N[(N[(N[Sin[z0], $MachinePrecision] * N[Sin[z1], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Cos[N[(z3 - z2), $MachinePrecision]], $MachinePrecision] * N[Cos[z1], $MachinePrecision]), $MachinePrecision] * N[Cos[z0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Initial Program: 73.9% accurate, 1.0× speedup

Math

\[\cos^{-1} \left(\sin z0 \cdot \sin z1 + \left(\cos \left(z3 - z2\right) \cdot \cos z1\right) \cdot \cos z0\right) \]

FPCore

(FPCore (z0 z1 z3 z2)
  :precision binary64
  (acos
 (+ (* (sin z0) (sin z1)) (* (* (cos (- z3 z2)) (cos z1)) (cos z0)))))

C

double code(double z0, double z1, double z3, double z2) {
	return acos(((sin(z0) * sin(z1)) + ((cos((z3 - z2)) * cos(z1)) * cos(z0))));
}

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(z0, z1, z3, z2)
use fmin_fmax_functions
    real(8), intent (in) :: z0
    real(8), intent (in) :: z1
    real(8), intent (in) :: z3
    real(8), intent (in) :: z2
    code = acos(((sin(z0) * sin(z1)) + ((cos((z3 - z2)) * cos(z1)) * cos(z0))))
end function

Java

public static double code(double z0, double z1, double z3, double z2) {
	return Math.acos(((Math.sin(z0) * Math.sin(z1)) + ((Math.cos((z3 - z2)) * Math.cos(z1)) * Math.cos(z0))));
}

Python

def code(z0, z1, z3, z2):
	return math.acos(((math.sin(z0) * math.sin(z1)) + ((math.cos((z3 - z2)) * math.cos(z1)) * math.cos(z0))))

Julia

function code(z0, z1, z3, z2)
	return acos(Float64(Float64(sin(z0) * sin(z1)) + Float64(Float64(cos(Float64(z3 - z2)) * cos(z1)) * cos(z0))))
end

MATLAB

function tmp = code(z0, z1, z3, z2)
	tmp = acos(((sin(z0) * sin(z1)) + ((cos((z3 - z2)) * cos(z1)) * cos(z0))));
end

Wolfram

code[z0_, z1_, z3_, z2_] := N[ArcCos[N[(N[(N[Sin[z0], $MachinePrecision] * N[Sin[z1], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Cos[N[(z3 - z2), $MachinePrecision]], $MachinePrecision] * N[Cos[z1], $MachinePrecision]), $MachinePrecision] * N[Cos[z0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Alternative 1: 93.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \cos z2 \cdot \cos z3\\ \cos^{-1} \left(\sin z0 \cdot \sin z1 + \left(\left(\left(1 + \frac{\sin z2 \cdot \sin z3}{t\_0}\right) \cdot t\_0\right) \cdot \cos z1\right) \cdot \cos z0\right) \end{array} \]

FPCore

(FPCore (z0 z1 z3 z2)
  :precision binary64
  (let* ((t_0 (* (cos z2) (cos z3))))
  (acos
   (+
    (* (sin z0) (sin z1))
    (*
     (* (* (+ 1 (/ (* (sin z2) (sin z3)) t_0)) t_0) (cos z1))
     (cos z0))))))

C

double code(double z0, double z1, double z3, double z2) {
	double t_0 = cos(z2) * cos(z3);
	return acos(((sin(z0) * sin(z1)) + ((((1.0 + ((sin(z2) * sin(z3)) / t_0)) * t_0) * cos(z1)) * cos(z0))));
}

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(z0, z1, z3, z2)
use fmin_fmax_functions
    real(8), intent (in) :: z0
    real(8), intent (in) :: z1
    real(8), intent (in) :: z3
    real(8), intent (in) :: z2
    real(8) :: t_0
    t_0 = cos(z2) * cos(z3)
    code = acos(((sin(z0) * sin(z1)) + ((((1.0d0 + ((sin(z2) * sin(z3)) / t_0)) * t_0) * cos(z1)) * cos(z0))))
end function

Java

public static double code(double z0, double z1, double z3, double z2) {
	double t_0 = Math.cos(z2) * Math.cos(z3);
	return Math.acos(((Math.sin(z0) * Math.sin(z1)) + ((((1.0 + ((Math.sin(z2) * Math.sin(z3)) / t_0)) * t_0) * Math.cos(z1)) * Math.cos(z0))));
}

Python

def code(z0, z1, z3, z2):
	t_0 = math.cos(z2) * math.cos(z3)
	return math.acos(((math.sin(z0) * math.sin(z1)) + ((((1.0 + ((math.sin(z2) * math.sin(z3)) / t_0)) * t_0) * math.cos(z1)) * math.cos(z0))))

Julia

function code(z0, z1, z3, z2)
	t_0 = Float64(cos(z2) * cos(z3))
	return acos(Float64(Float64(sin(z0) * sin(z1)) + Float64(Float64(Float64(Float64(1.0 + Float64(Float64(sin(z2) * sin(z3)) / t_0)) * t_0) * cos(z1)) * cos(z0))))
end

MATLAB

function tmp = code(z0, z1, z3, z2)
	t_0 = cos(z2) * cos(z3);
	tmp = acos(((sin(z0) * sin(z1)) + ((((1.0 + ((sin(z2) * sin(z3)) / t_0)) * t_0) * cos(z1)) * cos(z0))));
end

Wolfram

code[z0_, z1_, z3_, z2_] := Block[{t$95$0 = N[(N[Cos[z2], $MachinePrecision] * N[Cos[z3], $MachinePrecision]), $MachinePrecision]}, N[ArcCos[N[(N[(N[Sin[z0], $MachinePrecision] * N[Sin[z1], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(1 + N[(N[(N[Sin[z2], $MachinePrecision] * N[Sin[z3], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Cos[z1], $MachinePrecision]), $MachinePrecision] * N[Cos[z0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Initial program 73.9%

   \[\cos^{-1} \left(\sin z0 \cdot \sin z1 + \left(\cos \left(z3 - z2\right) \cdot \cos z1\right) \cdot \cos z0\right) \]
2. Step-by-step derivation

   1. lift-cos.f64 N/A

      \[\leadsto \cos^{-1} \left(\sin z0 \cdot \sin z1 + \left(\color{blue}{\cos \left(z3 - z2\right)} \cdot \cos z1\right) \cdot \cos z0\right) \]

   2. lift--.f64 N/A

      \[\leadsto \cos^{-1} \left(\sin z0 \cdot \sin z1 + \left(\cos \color{blue}{\left(z3 - z2\right)} \cdot \cos z1\right) \cdot \cos z0\right) \]

   3. cos-diff N/A

      \[\leadsto \cos^{-1} \left(\sin z0 \cdot \sin z1 + \left(\color{blue}{\left(\cos z3 \cdot \cos z2 + \sin z3 \cdot \sin z2\right)} \cdot \cos z1\right) \cdot \cos z0\right) \]

   4. sum-to-mult N/A

      \[\leadsto \cos^{-1} \left(\sin z0 \cdot \sin z1 + \left(\color{blue}{\left(\left(1 + \frac{\sin z3 \cdot \sin z2}{\cos z3 \cdot \cos z2}\right) \cdot \left(\cos z3 \cdot \cos z2\right)\right)} \cdot \cos z1\right) \cdot \cos z0\right) \]

   5. lower-unsound-*.f64 N/A

      \[\leadsto \cos^{-1} \left(\sin z0 \cdot \sin z1 + \left(\color{blue}{\left(\left(1 + \frac{\sin z3 \cdot \sin z2}{\cos z3 \cdot \cos z2}\right) \cdot \left(\cos z3 \cdot \cos z2\right)\right)} \cdot \cos z1\right) \cdot \cos z0\right) \]

   6. lower-unsound-+.f64 N/A

      \[\leadsto \cos^{-1} \left(\sin z0 \cdot \sin z1 + \left(\left(\color{blue}{\left(1 + \frac{\sin z3 \cdot \sin z2}{\cos z3 \cdot \cos z2}\right)} \cdot \left(\cos z3 \cdot \cos z2\right)\right) \cdot \cos z1\right) \cdot \cos z0\right) \]

   7. lower-unsound-/.f64 N/A

      \[\leadsto \cos^{-1} \left(\sin z0 \cdot \sin z1 + \left(\left(\left(1 + \color{blue}{\frac{\sin z3 \cdot \sin z2}{\cos z3 \cdot \cos z2}}\right) \cdot \left(\cos z3 \cdot \cos z2\right)\right) \cdot \cos z1\right) \cdot \cos z0\right) \]

   8. *-commutative N/A

      \[\leadsto \cos^{-1} \left(\sin z0 \cdot \sin z1 + \left(\left(\left(1 + \frac{\color{blue}{\sin z2 \cdot \sin z3}}{\cos z3 \cdot \cos z2}\right) \cdot \left(\cos z3 \cdot \cos z2\right)\right) \cdot \cos z1\right) \cdot \cos z0\right) \]

   9. lower-*.f64 N/A

      \[\leadsto \cos^{-1} \left(\sin z0 \cdot \sin z1 + \left(\left(\left(1 + \frac{\color{blue}{\sin z2 \cdot \sin z3}}{\cos z3 \cdot \cos z2}\right) \cdot \left(\cos z3 \cdot \cos z2\right)\right) \cdot \cos z1\right) \cdot \cos z0\right) \]

   10. lower-sin.f64 N/A

       \[\leadsto \cos^{-1} \left(\sin z0 \cdot \sin z1 + \left(\left(\left(1 + \frac{\color{blue}{\sin z2} \cdot \sin z3}{\cos z3 \cdot \cos z2}\right) \cdot \left(\cos z3 \cdot \cos z2\right)\right) \cdot \cos z1\right) \cdot \cos z0\right) \]

   11. lower-sin.f64 N/A

       \[\leadsto \cos^{-1} \left(\sin z0 \cdot \sin z1 + \left(\left(\left(1 + \frac{\sin z2 \cdot \color{blue}{\sin z3}}{\cos z3 \cdot \cos z2}\right) \cdot \left(\cos z3 \cdot \cos z2\right)\right) \cdot \cos z1\right) \cdot \cos z0\right) \]

   12. *-commutative N/A

       \[\leadsto \cos^{-1} \left(\sin z0 \cdot \sin z1 + \left(\left(\left(1 + \frac{\sin z2 \cdot \sin z3}{\color{blue}{\cos z2 \cdot \cos z3}}\right) \cdot \left(\cos z3 \cdot \cos z2\right)\right) \cdot \cos z1\right) \cdot \cos z0\right) \]

   13. lower-*.f64 N/A

       \[\leadsto \cos^{-1} \left(\sin z0 \cdot \sin z1 + \left(\left(\left(1 + \frac{\sin z2 \cdot \sin z3}{\color{blue}{\cos z2 \cdot \cos z3}}\right) \cdot \left(\cos z3 \cdot \cos z2\right)\right) \cdot \cos z1\right) \cdot \cos z0\right) \]

   14. lower-cos.f64 N/A

       \[\leadsto \cos^{-1} \left(\sin z0 \cdot \sin z1 + \left(\left(\left(1 + \frac{\sin z2 \cdot \sin z3}{\color{blue}{\cos z2} \cdot \cos z3}\right) \cdot \left(\cos z3 \cdot \cos z2\right)\right) \cdot \cos z1\right) \cdot \cos z0\right) \]

   15. lower-cos.f64 N/A

       \[\leadsto \cos^{-1} \left(\sin z0 \cdot \sin z1 + \left(\left(\left(1 + \frac{\sin z2 \cdot \sin z3}{\cos z2 \cdot \color{blue}{\cos z3}}\right) \cdot \left(\cos z3 \cdot \cos z2\right)\right) \cdot \cos z1\right) \cdot \cos z0\right) \]

   16. *-commutative N/A

       \[\leadsto \cos^{-1} \left(\sin z0 \cdot \sin z1 + \left(\left(\left(1 + \frac{\sin z2 \cdot \sin z3}{\cos z2 \cdot \cos z3}\right) \cdot \color{blue}{\left(\cos z2 \cdot \cos z3\right)}\right) \cdot \cos z1\right) \cdot \cos z0\right) \]

   17. lower-*.f64 N/A

       \[\leadsto \cos^{-1} \left(\sin z0 \cdot \sin z1 + \left(\left(\left(1 + \frac{\sin z2 \cdot \sin z3}{\cos z2 \cdot \cos z3}\right) \cdot \color{blue}{\left(\cos z2 \cdot \cos z3\right)}\right) \cdot \cos z1\right) \cdot \cos z0\right) \]

   18. lower-cos.f64 N/A

       \[\leadsto \cos^{-1} \left(\sin z0 \cdot \sin z1 + \left(\left(\left(1 + \frac{\sin z2 \cdot \sin z3}{\cos z2 \cdot \cos z3}\right) \cdot \left(\color{blue}{\cos z2} \cdot \cos z3\right)\right) \cdot \cos z1\right) \cdot \cos z0\right) \]

   19. lower-cos.f64 93.7%

       \[\leadsto \cos^{-1} \left(\sin z0 \cdot \sin z1 + \left(\left(\left(1 + \frac{\sin z2 \cdot \sin z3}{\cos z2 \cdot \cos z3}\right) \cdot \left(\cos z2 \cdot \color{blue}{\cos z3}\right)\right) \cdot \cos z1\right) \cdot \cos z0\right) \]

3. Applied rewrites 93.7%

   \[\leadsto \cos^{-1} \left(\sin z0 \cdot \sin z1 + \left(\color{blue}{\left(\left(1 + \frac{\sin z2 \cdot \sin z3}{\cos z2 \cdot \cos z3}\right) \cdot \left(\cos z2 \cdot \cos z3\right)\right)} \cdot \cos z1\right) \cdot \cos z0\right) \]
4. Add Preprocessing

Alternative 2: 93.7% accurate, 0.7× speedup

Math

\[\cos^{-1} \left(\sin z0 \cdot \sin z1 + \left(\left(\left(\tan z3 \cdot \tan z2 - -1\right) \cdot \left(\cos z3 \cdot \cos z2\right)\right) \cdot \cos z1\right) \cdot \cos z0\right) \]

FPCore

(FPCore (z0 z1 z3 z2)
  :precision binary64
  (acos
 (+
  (* (sin z0) (sin z1))
  (*
   (* (* (- (* (tan z3) (tan z2)) -1) (* (cos z3) (cos z2))) (cos z1))
   (cos z0)))))

C

double code(double z0, double z1, double z3, double z2) {
	return acos(((sin(z0) * sin(z1)) + (((((tan(z3) * tan(z2)) - -1.0) * (cos(z3) * cos(z2))) * cos(z1)) * cos(z0))));
}

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(z0, z1, z3, z2)
use fmin_fmax_functions
    real(8), intent (in) :: z0
    real(8), intent (in) :: z1
    real(8), intent (in) :: z3
    real(8), intent (in) :: z2
    code = acos(((sin(z0) * sin(z1)) + (((((tan(z3) * tan(z2)) - (-1.0d0)) * (cos(z3) * cos(z2))) * cos(z1)) * cos(z0))))
end function

Java

public static double code(double z0, double z1, double z3, double z2) {
	return Math.acos(((Math.sin(z0) * Math.sin(z1)) + (((((Math.tan(z3) * Math.tan(z2)) - -1.0) * (Math.cos(z3) * Math.cos(z2))) * Math.cos(z1)) * Math.cos(z0))));
}

Python

def code(z0, z1, z3, z2):
	return math.acos(((math.sin(z0) * math.sin(z1)) + (((((math.tan(z3) * math.tan(z2)) - -1.0) * (math.cos(z3) * math.cos(z2))) * math.cos(z1)) * math.cos(z0))))

Julia

function code(z0, z1, z3, z2)
	return acos(Float64(Float64(sin(z0) * sin(z1)) + Float64(Float64(Float64(Float64(Float64(tan(z3) * tan(z2)) - -1.0) * Float64(cos(z3) * cos(z2))) * cos(z1)) * cos(z0))))
end

MATLAB

function tmp = code(z0, z1, z3, z2)
	tmp = acos(((sin(z0) * sin(z1)) + (((((tan(z3) * tan(z2)) - -1.0) * (cos(z3) * cos(z2))) * cos(z1)) * cos(z0))));
end

Wolfram

code[z0_, z1_, z3_, z2_] := N[ArcCos[N[(N[(N[Sin[z0], $MachinePrecision] * N[Sin[z1], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(N[(N[Tan[z3], $MachinePrecision] * N[Tan[z2], $MachinePrecision]), $MachinePrecision] - -1), $MachinePrecision] * N[(N[Cos[z3], $MachinePrecision] * N[Cos[z2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[z1], $MachinePrecision]), $MachinePrecision] * N[Cos[z0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 73.9%

   \[\cos^{-1} \left(\sin z0 \cdot \sin z1 + \left(\cos \left(z3 - z2\right) \cdot \cos z1\right) \cdot \cos z0\right) \]
2. Step-by-step derivation

   1. lift-cos.f64 N/A

      \[\leadsto \cos^{-1} \left(\sin z0 \cdot \sin z1 + \left(\color{blue}{\cos \left(z3 - z2\right)} \cdot \cos z1\right) \cdot \cos z0\right) \]

   2. lift--.f64 N/A

      \[\leadsto \cos^{-1} \left(\sin z0 \cdot \sin z1 + \left(\cos \color{blue}{\left(z3 - z2\right)} \cdot \cos z1\right) \cdot \cos z0\right) \]

   3. cos-diff N/A

      \[\leadsto \cos^{-1} \left(\sin z0 \cdot \sin z1 + \left(\color{blue}{\left(\cos z3 \cdot \cos z2 + \sin z3 \cdot \sin z2\right)} \cdot \cos z1\right) \cdot \cos z0\right) \]

   4. sum-to-mult N/A

      \[\leadsto \cos^{-1} \left(\sin z0 \cdot \sin z1 + \left(\color{blue}{\left(\left(1 + \frac{\sin z3 \cdot \sin z2}{\cos z3 \cdot \cos z2}\right) \cdot \left(\cos z3 \cdot \cos z2\right)\right)} \cdot \cos z1\right) \cdot \cos z0\right) \]

   5. lower-unsound-*.f64 N/A

      \[\leadsto \cos^{-1} \left(\sin z0 \cdot \sin z1 + \left(\color{blue}{\left(\left(1 + \frac{\sin z3 \cdot \sin z2}{\cos z3 \cdot \cos z2}\right) \cdot \left(\cos z3 \cdot \cos z2\right)\right)} \cdot \cos z1\right) \cdot \cos z0\right) \]

   6. lower-unsound-+.f64 N/A

      \[\leadsto \cos^{-1} \left(\sin z0 \cdot \sin z1 + \left(\left(\color{blue}{\left(1 + \frac{\sin z3 \cdot \sin z2}{\cos z3 \cdot \cos z2}\right)} \cdot \left(\cos z3 \cdot \cos z2\right)\right) \cdot \cos z1\right) \cdot \cos z0\right) \]

   7. lower-unsound-/.f64 N/A

      \[\leadsto \cos^{-1} \left(\sin z0 \cdot \sin z1 + \left(\left(\left(1 + \color{blue}{\frac{\sin z3 \cdot \sin z2}{\cos z3 \cdot \cos z2}}\right) \cdot \left(\cos z3 \cdot \cos z2\right)\right) \cdot \cos z1\right) \cdot \cos z0\right) \]

   8. *-commutative N/A

      \[\leadsto \cos^{-1} \left(\sin z0 \cdot \sin z1 + \left(\left(\left(1 + \frac{\color{blue}{\sin z2 \cdot \sin z3}}{\cos z3 \cdot \cos z2}\right) \cdot \left(\cos z3 \cdot \cos z2\right)\right) \cdot \cos z1\right) \cdot \cos z0\right) \]

   9. lower-*.f64 N/A

      \[\leadsto \cos^{-1} \left(\sin z0 \cdot \sin z1 + \left(\left(\left(1 + \frac{\color{blue}{\sin z2 \cdot \sin z3}}{\cos z3 \cdot \cos z2}\right) \cdot \left(\cos z3 \cdot \cos z2\right)\right) \cdot \cos z1\right) \cdot \cos z0\right) \]

   10. lower-sin.f64 N/A

       \[\leadsto \cos^{-1} \left(\sin z0 \cdot \sin z1 + \left(\left(\left(1 + \frac{\color{blue}{\sin z2} \cdot \sin z3}{\cos z3 \cdot \cos z2}\right) \cdot \left(\cos z3 \cdot \cos z2\right)\right) \cdot \cos z1\right) \cdot \cos z0\right) \]

   11. lower-sin.f64 N/A

       \[\leadsto \cos^{-1} \left(\sin z0 \cdot \sin z1 + \left(\left(\left(1 + \frac{\sin z2 \cdot \color{blue}{\sin z3}}{\cos z3 \cdot \cos z2}\right) \cdot \left(\cos z3 \cdot \cos z2\right)\right) \cdot \cos z1\right) \cdot \cos z0\right) \]

   12. *-commutative N/A

       \[\leadsto \cos^{-1} \left(\sin z0 \cdot \sin z1 + \left(\left(\left(1 + \frac{\sin z2 \cdot \sin z3}{\color{blue}{\cos z2 \cdot \cos z3}}\right) \cdot \left(\cos z3 \cdot \cos z2\right)\right) \cdot \cos z1\right) \cdot \cos z0\right) \]

   13. lower-*.f64 N/A

       \[\leadsto \cos^{-1} \left(\sin z0 \cdot \sin z1 + \left(\left(\left(1 + \frac{\sin z2 \cdot \sin z3}{\color{blue}{\cos z2 \cdot \cos z3}}\right) \cdot \left(\cos z3 \cdot \cos z2\right)\right) \cdot \cos z1\right) \cdot \cos z0\right) \]

   14. lower-cos.f64 N/A

       \[\leadsto \cos^{-1} \left(\sin z0 \cdot \sin z1 + \left(\left(\left(1 + \frac{\sin z2 \cdot \sin z3}{\color{blue}{\cos z2} \cdot \cos z3}\right) \cdot \left(\cos z3 \cdot \cos z2\right)\right) \cdot \cos z1\right) \cdot \cos z0\right) \]

   15. lower-cos.f64 N/A

       \[\leadsto \cos^{-1} \left(\sin z0 \cdot \sin z1 + \left(\left(\left(1 + \frac{\sin z2 \cdot \sin z3}{\cos z2 \cdot \color{blue}{\cos z3}}\right) \cdot \left(\cos z3 \cdot \cos z2\right)\right) \cdot \cos z1\right) \cdot \cos z0\right) \]

   16. *-commutative N/A

       \[\leadsto \cos^{-1} \left(\sin z0 \cdot \sin z1 + \left(\left(\left(1 + \frac{\sin z2 \cdot \sin z3}{\cos z2 \cdot \cos z3}\right) \cdot \color{blue}{\left(\cos z2 \cdot \cos z3\right)}\right) \cdot \cos z1\right) \cdot \cos z0\right) \]

   17. lower-*.f64 N/A

       \[\leadsto \cos^{-1} \left(\sin z0 \cdot \sin z1 + \left(\left(\left(1 + \frac{\sin z2 \cdot \sin z3}{\cos z2 \cdot \cos z3}\right) \cdot \color{blue}{\left(\cos z2 \cdot \cos z3\right)}\right) \cdot \cos z1\right) \cdot \cos z0\right) \]

   18. lower-cos.f64 N/A

       \[\leadsto \cos^{-1} \left(\sin z0 \cdot \sin z1 + \left(\left(\left(1 + \frac{\sin z2 \cdot \sin z3}{\cos z2 \cdot \cos z3}\right) \cdot \left(\color{blue}{\cos z2} \cdot \cos z3\right)\right) \cdot \cos z1\right) \cdot \cos z0\right) \]

   19. lower-cos.f64 93.7%

       \[\leadsto \cos^{-1} \left(\sin z0 \cdot \sin z1 + \left(\left(\left(1 + \frac{\sin z2 \cdot \sin z3}{\cos z2 \cdot \cos z3}\right) \cdot \left(\cos z2 \cdot \color{blue}{\cos z3}\right)\right) \cdot \cos z1\right) \cdot \cos z0\right) \]

3. Applied rewrites 93.7%

   \[\leadsto \cos^{-1} \left(\sin z0 \cdot \sin z1 + \left(\color{blue}{\left(\left(1 + \frac{\sin z2 \cdot \sin z3}{\cos z2 \cdot \cos z3}\right) \cdot \left(\cos z2 \cdot \cos z3\right)\right)} \cdot \cos z1\right) \cdot \cos z0\right) \]

5. Applied rewrites 93.7%

   \[\leadsto \cos^{-1} \left(\sin z0 \cdot \sin z1 + \left(\color{blue}{\left(\left(\tan z3 \cdot \tan z2 - -1\right) \cdot \left(\cos z3 \cdot \cos z2\right)\right)} \cdot \cos z1\right) \cdot \cos z0\right) \]
6. Add Preprocessing

Alternative 3: 93.7% accurate, 0.7× speedup

Math

\[\cos^{-1} \left(\sin z0 \cdot \sin z1 + \left(\left(\sin z2 \cdot \sin z3 + \cos z2 \cdot \cos z3\right) \cdot \cos z1\right) \cdot \cos z0\right) \]

FPCore

(FPCore (z0 z1 z3 z2)
  :precision binary64
  (acos
 (+
  (* (sin z0) (sin z1))
  (*
   (* (+ (* (sin z2) (sin z3)) (* (cos z2) (cos z3))) (cos z1))
   (cos z0)))))

C

double code(double z0, double z1, double z3, double z2) {
	return acos(((sin(z0) * sin(z1)) + ((((sin(z2) * sin(z3)) + (cos(z2) * cos(z3))) * cos(z1)) * cos(z0))));
}

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(z0, z1, z3, z2)
use fmin_fmax_functions
    real(8), intent (in) :: z0
    real(8), intent (in) :: z1
    real(8), intent (in) :: z3
    real(8), intent (in) :: z2
    code = acos(((sin(z0) * sin(z1)) + ((((sin(z2) * sin(z3)) + (cos(z2) * cos(z3))) * cos(z1)) * cos(z0))))
end function

Java

public static double code(double z0, double z1, double z3, double z2) {
	return Math.acos(((Math.sin(z0) * Math.sin(z1)) + ((((Math.sin(z2) * Math.sin(z3)) + (Math.cos(z2) * Math.cos(z3))) * Math.cos(z1)) * Math.cos(z0))));
}

Python

def code(z0, z1, z3, z2):
	return math.acos(((math.sin(z0) * math.sin(z1)) + ((((math.sin(z2) * math.sin(z3)) + (math.cos(z2) * math.cos(z3))) * math.cos(z1)) * math.cos(z0))))

Julia

function code(z0, z1, z3, z2)
	return acos(Float64(Float64(sin(z0) * sin(z1)) + Float64(Float64(Float64(Float64(sin(z2) * sin(z3)) + Float64(cos(z2) * cos(z3))) * cos(z1)) * cos(z0))))
end

MATLAB

function tmp = code(z0, z1, z3, z2)
	tmp = acos(((sin(z0) * sin(z1)) + ((((sin(z2) * sin(z3)) + (cos(z2) * cos(z3))) * cos(z1)) * cos(z0))));
end

Wolfram

code[z0_, z1_, z3_, z2_] := N[ArcCos[N[(N[(N[Sin[z0], $MachinePrecision] * N[Sin[z1], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(N[Sin[z2], $MachinePrecision] * N[Sin[z3], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[z2], $MachinePrecision] * N[Cos[z3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[z1], $MachinePrecision]), $MachinePrecision] * N[Cos[z0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 73.9%

   \[\cos^{-1} \left(\sin z0 \cdot \sin z1 + \left(\cos \left(z3 - z2\right) \cdot \cos z1\right) \cdot \cos z0\right) \]
2. Step-by-step derivation

   1. lift-cos.f64 N/A

      \[\leadsto \cos^{-1} \left(\sin z0 \cdot \sin z1 + \left(\color{blue}{\cos \left(z3 - z2\right)} \cdot \cos z1\right) \cdot \cos z0\right) \]

   2. lift--.f64 N/A

      \[\leadsto \cos^{-1} \left(\sin z0 \cdot \sin z1 + \left(\cos \color{blue}{\left(z3 - z2\right)} \cdot \cos z1\right) \cdot \cos z0\right) \]

   3. cos-diff N/A

      \[\leadsto \cos^{-1} \left(\sin z0 \cdot \sin z1 + \left(\color{blue}{\left(\cos z3 \cdot \cos z2 + \sin z3 \cdot \sin z2\right)} \cdot \cos z1\right) \cdot \cos z0\right) \]

   4. +-commutative N/A

      \[\leadsto \cos^{-1} \left(\sin z0 \cdot \sin z1 + \left(\color{blue}{\left(\sin z3 \cdot \sin z2 + \cos z3 \cdot \cos z2\right)} \cdot \cos z1\right) \cdot \cos z0\right) \]

   5. lower-+.f64 N/A

      \[\leadsto \cos^{-1} \left(\sin z0 \cdot \sin z1 + \left(\color{blue}{\left(\sin z3 \cdot \sin z2 + \cos z3 \cdot \cos z2\right)} \cdot \cos z1\right) \cdot \cos z0\right) \]

   6. *-commutative N/A

      \[\leadsto \cos^{-1} \left(\sin z0 \cdot \sin z1 + \left(\left(\color{blue}{\sin z2 \cdot \sin z3} + \cos z3 \cdot \cos z2\right) \cdot \cos z1\right) \cdot \cos z0\right) \]

   7. lower-*.f64 N/A

      \[\leadsto \cos^{-1} \left(\sin z0 \cdot \sin z1 + \left(\left(\color{blue}{\sin z2 \cdot \sin z3} + \cos z3 \cdot \cos z2\right) \cdot \cos z1\right) \cdot \cos z0\right) \]

   8. lower-sin.f64 N/A

      \[\leadsto \cos^{-1} \left(\sin z0 \cdot \sin z1 + \left(\left(\color{blue}{\sin z2} \cdot \sin z3 + \cos z3 \cdot \cos z2\right) \cdot \cos z1\right) \cdot \cos z0\right) \]

   9. lower-sin.f64 N/A

      \[\leadsto \cos^{-1} \left(\sin z0 \cdot \sin z1 + \left(\left(\sin z2 \cdot \color{blue}{\sin z3} + \cos z3 \cdot \cos z2\right) \cdot \cos z1\right) \cdot \cos z0\right) \]

   10. *-commutative N/A

       \[\leadsto \cos^{-1} \left(\sin z0 \cdot \sin z1 + \left(\left(\sin z2 \cdot \sin z3 + \color{blue}{\cos z2 \cdot \cos z3}\right) \cdot \cos z1\right) \cdot \cos z0\right) \]

   11. lower-*.f64 N/A

       \[\leadsto \cos^{-1} \left(\sin z0 \cdot \sin z1 + \left(\left(\sin z2 \cdot \sin z3 + \color{blue}{\cos z2 \cdot \cos z3}\right) \cdot \cos z1\right) \cdot \cos z0\right) \]

   12. lower-cos.f64 N/A

       \[\leadsto \cos^{-1} \left(\sin z0 \cdot \sin z1 + \left(\left(\sin z2 \cdot \sin z3 + \color{blue}{\cos z2} \cdot \cos z3\right) \cdot \cos z1\right) \cdot \cos z0\right) \]

   13. lower-cos.f64 93.7%

       \[\leadsto \cos^{-1} \left(\sin z0 \cdot \sin z1 + \left(\left(\sin z2 \cdot \sin z3 + \cos z2 \cdot \color{blue}{\cos z3}\right) \cdot \cos z1\right) \cdot \cos z0\right) \]

3. Applied rewrites 93.7%

   \[\leadsto \cos^{-1} \left(\sin z0 \cdot \sin z1 + \left(\color{blue}{\left(\sin z2 \cdot \sin z3 + \cos z2 \cdot \cos z3\right)} \cdot \cos z1\right) \cdot \cos z0\right) \]
4. Add Preprocessing

Alternative 4: 73.9% accurate, 1.0× speedup

Math

\[\pi \cdot \frac{1}{2} - \left(\pi \cdot \frac{1}{2} - \cos^{-1} \left(\left(\cos \left(z3 - z2\right) \cdot \cos z1\right) \cdot \cos z0 + \sin z1 \cdot \sin z0\right)\right) \]

FPCore

(FPCore (z0 z1 z3 z2)
  :precision binary64
  (-
 (* PI 1/2)
 (-
  (* PI 1/2)
  (acos
   (+
    (* (* (cos (- z3 z2)) (cos z1)) (cos z0))
    (* (sin z1) (sin z0)))))))

C

double code(double z0, double z1, double z3, double z2) {
	return (((double) M_PI) * 0.5) - ((((double) M_PI) * 0.5) - acos((((cos((z3 - z2)) * cos(z1)) * cos(z0)) + (sin(z1) * sin(z0)))));
}

Java

public static double code(double z0, double z1, double z3, double z2) {
	return (Math.PI * 0.5) - ((Math.PI * 0.5) - Math.acos((((Math.cos((z3 - z2)) * Math.cos(z1)) * Math.cos(z0)) + (Math.sin(z1) * Math.sin(z0)))));
}

Python

def code(z0, z1, z3, z2):
	return (math.pi * 0.5) - ((math.pi * 0.5) - math.acos((((math.cos((z3 - z2)) * math.cos(z1)) * math.cos(z0)) + (math.sin(z1) * math.sin(z0)))))

Julia

function code(z0, z1, z3, z2)
	return Float64(Float64(pi * 0.5) - Float64(Float64(pi * 0.5) - acos(Float64(Float64(Float64(cos(Float64(z3 - z2)) * cos(z1)) * cos(z0)) + Float64(sin(z1) * sin(z0))))))
end

MATLAB

function tmp = code(z0, z1, z3, z2)
	tmp = (pi * 0.5) - ((pi * 0.5) - acos((((cos((z3 - z2)) * cos(z1)) * cos(z0)) + (sin(z1) * sin(z0)))));
end

Wolfram

code[z0_, z1_, z3_, z2_] := N[(N[(Pi * 1/2), $MachinePrecision] - N[(N[(Pi * 1/2), $MachinePrecision] - N[ArcCos[N[(N[(N[(N[Cos[N[(z3 - z2), $MachinePrecision]], $MachinePrecision] * N[Cos[z1], $MachinePrecision]), $MachinePrecision] * N[Cos[z0], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[z1], $MachinePrecision] * N[Sin[z0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 73.9%

   \[\cos^{-1} \left(\sin z0 \cdot \sin z1 + \left(\cos \left(z3 - z2\right) \cdot \cos z1\right) \cdot \cos z0\right) \]
2. Step-by-step derivation

   1. lift-acos.f64 N/A

      \[\leadsto \color{blue}{\cos^{-1} \left(\sin z0 \cdot \sin z1 + \left(\cos \left(z3 - z2\right) \cdot \cos z1\right) \cdot \cos z0\right)} \]

   2. acos-asin N/A

      \[\leadsto \color{blue}{\frac{\mathsf{PI}\left(\right)}{2} - \sin^{-1} \left(\sin z0 \cdot \sin z1 + \left(\cos \left(z3 - z2\right) \cdot \cos z1\right) \cdot \cos z0\right)} \]

   3. lower--.f64 N/A

      \[\leadsto \color{blue}{\frac{\mathsf{PI}\left(\right)}{2} - \sin^{-1} \left(\sin z0 \cdot \sin z1 + \left(\cos \left(z3 - z2\right) \cdot \cos z1\right) \cdot \cos z0\right)} \]

   4. mult-flip N/A

      \[\leadsto \color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{2}} - \sin^{-1} \left(\sin z0 \cdot \sin z1 + \left(\cos \left(z3 - z2\right) \cdot \cos z1\right) \cdot \cos z0\right) \]

   5. lower-*.f64 N/A

      \[\leadsto \color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{2}} - \sin^{-1} \left(\sin z0 \cdot \sin z1 + \left(\cos \left(z3 - z2\right) \cdot \cos z1\right) \cdot \cos z0\right) \]

   6. lower-PI.f64 N/A

      \[\leadsto \color{blue}{\pi} \cdot \frac{1}{2} - \sin^{-1} \left(\sin z0 \cdot \sin z1 + \left(\cos \left(z3 - z2\right) \cdot \cos z1\right) \cdot \cos z0\right) \]

   7. metadata-eval N/A

      \[\leadsto \pi \cdot \color{blue}{\frac{1}{2}} - \sin^{-1} \left(\sin z0 \cdot \sin z1 + \left(\cos \left(z3 - z2\right) \cdot \cos z1\right) \cdot \cos z0\right) \]

   8. lower-asin.f64 73.7%

      \[\leadsto \pi \cdot \frac{1}{2} - \color{blue}{\sin^{-1} \left(\sin z0 \cdot \sin z1 + \left(\cos \left(z3 - z2\right) \cdot \cos z1\right) \cdot \cos z0\right)} \]

   9. lift-+.f64 N/A

      \[\leadsto \pi \cdot \frac{1}{2} - \sin^{-1} \color{blue}{\left(\sin z0 \cdot \sin z1 + \left(\cos \left(z3 - z2\right) \cdot \cos z1\right) \cdot \cos z0\right)} \]

   10. +-commutative N/A

       \[\leadsto \pi \cdot \frac{1}{2} - \sin^{-1} \color{blue}{\left(\left(\cos \left(z3 - z2\right) \cdot \cos z1\right) \cdot \cos z0 + \sin z0 \cdot \sin z1\right)} \]

   11. lower-+.f64 73.7%

       \[\leadsto \pi \cdot \frac{1}{2} - \sin^{-1} \color{blue}{\left(\left(\cos \left(z3 - z2\right) \cdot \cos z1\right) \cdot \cos z0 + \sin z0 \cdot \sin z1\right)} \]

3. Applied rewrites 73.7%

   \[\leadsto \color{blue}{\pi \cdot \frac{1}{2} - \sin^{-1} \left(\cos z0 \cdot \left(\cos z1 \cdot \cos \left(z2 - z3\right)\right) + \sin z1 \cdot \sin z0\right)} \]

5. Applied rewrites 73.9%

   \[\leadsto \pi \cdot \frac{1}{2} - \color{blue}{\left(\pi \cdot \frac{1}{2} - \cos^{-1} \left(\left(\cos \left(z3 - z2\right) \cdot \cos z1\right) \cdot \cos z0 + \sin z1 \cdot \sin z0\right)\right)} \]
6. Add Preprocessing

Alternative 5: 73.9% accurate, 1.0× speedup

Math

\[\cos^{-1} \left(\sin z0 \cdot \sin z1 + \left(\cos z0 \cdot \cos z1\right) \cdot \cos \left(z2 - z3\right)\right) \]

FPCore

(FPCore (z0 z1 z3 z2)
  :precision binary64
  (acos
 (+ (* (sin z0) (sin z1)) (* (* (cos z0) (cos z1)) (cos (- z2 z3))))))

C

double code(double z0, double z1, double z3, double z2) {
	return acos(((sin(z0) * sin(z1)) + ((cos(z0) * cos(z1)) * cos((z2 - z3)))));
}

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(z0, z1, z3, z2)
use fmin_fmax_functions
    real(8), intent (in) :: z0
    real(8), intent (in) :: z1
    real(8), intent (in) :: z3
    real(8), intent (in) :: z2
    code = acos(((sin(z0) * sin(z1)) + ((cos(z0) * cos(z1)) * cos((z2 - z3)))))
end function

Java

public static double code(double z0, double z1, double z3, double z2) {
	return Math.acos(((Math.sin(z0) * Math.sin(z1)) + ((Math.cos(z0) * Math.cos(z1)) * Math.cos((z2 - z3)))));
}

Python

def code(z0, z1, z3, z2):
	return math.acos(((math.sin(z0) * math.sin(z1)) + ((math.cos(z0) * math.cos(z1)) * math.cos((z2 - z3)))))

Julia

function code(z0, z1, z3, z2)
	return acos(Float64(Float64(sin(z0) * sin(z1)) + Float64(Float64(cos(z0) * cos(z1)) * cos(Float64(z2 - z3)))))
end

MATLAB

function tmp = code(z0, z1, z3, z2)
	tmp = acos(((sin(z0) * sin(z1)) + ((cos(z0) * cos(z1)) * cos((z2 - z3)))));
end

Wolfram

code[z0_, z1_, z3_, z2_] := N[ArcCos[N[(N[(N[Sin[z0], $MachinePrecision] * N[Sin[z1], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Cos[z0], $MachinePrecision] * N[Cos[z1], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(z2 - z3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 73.9%

   \[\cos^{-1} \left(\sin z0 \cdot \sin z1 + \left(\cos \left(z3 - z2\right) \cdot \cos z1\right) \cdot \cos z0\right) \]
2. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \cos^{-1} \left(\sin z0 \cdot \sin z1 + \color{blue}{\left(\cos \left(z3 - z2\right) \cdot \cos z1\right) \cdot \cos z0}\right) \]

   2. lift-*.f64 N/A

      \[\leadsto \cos^{-1} \left(\sin z0 \cdot \sin z1 + \color{blue}{\left(\cos \left(z3 - z2\right) \cdot \cos z1\right)} \cdot \cos z0\right) \]

   3. associate-*l* N/A

      \[\leadsto \cos^{-1} \left(\sin z0 \cdot \sin z1 + \color{blue}{\cos \left(z3 - z2\right) \cdot \left(\cos z1 \cdot \cos z0\right)}\right) \]

   4. *-commutative N/A

      \[\leadsto \cos^{-1} \left(\sin z0 \cdot \sin z1 + \color{blue}{\left(\cos z1 \cdot \cos z0\right) \cdot \cos \left(z3 - z2\right)}\right) \]

   5. lower-*.f64 N/A

      \[\leadsto \cos^{-1} \left(\sin z0 \cdot \sin z1 + \color{blue}{\left(\cos z1 \cdot \cos z0\right) \cdot \cos \left(z3 - z2\right)}\right) \]

   6. *-commutative N/A

      \[\leadsto \cos^{-1} \left(\sin z0 \cdot \sin z1 + \color{blue}{\left(\cos z0 \cdot \cos z1\right)} \cdot \cos \left(z3 - z2\right)\right) \]

   7. lower-*.f64 73.9%

      \[\leadsto \cos^{-1} \left(\sin z0 \cdot \sin z1 + \color{blue}{\left(\cos z0 \cdot \cos z1\right)} \cdot \cos \left(z3 - z2\right)\right) \]

   8. lift-cos.f64 N/A

      \[\leadsto \cos^{-1} \left(\sin z0 \cdot \sin z1 + \left(\cos z0 \cdot \cos z1\right) \cdot \color{blue}{\cos \left(z3 - z2\right)}\right) \]

   9. cos-neg-rev N/A

      \[\leadsto \cos^{-1} \left(\sin z0 \cdot \sin z1 + \left(\cos z0 \cdot \cos z1\right) \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\left(z3 - z2\right)\right)\right)}\right) \]

   10. lower-cos.f64 N/A

       \[\leadsto \cos^{-1} \left(\sin z0 \cdot \sin z1 + \left(\cos z0 \cdot \cos z1\right) \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\left(z3 - z2\right)\right)\right)}\right) \]

   11. lift--.f64 N/A

       \[\leadsto \cos^{-1} \left(\sin z0 \cdot \sin z1 + \left(\cos z0 \cdot \cos z1\right) \cdot \cos \left(\mathsf{neg}\left(\color{blue}{\left(z3 - z2\right)}\right)\right)\right) \]

   12. sub-negate-rev N/A

       \[\leadsto \cos^{-1} \left(\sin z0 \cdot \sin z1 + \left(\cos z0 \cdot \cos z1\right) \cdot \cos \color{blue}{\left(z2 - z3\right)}\right) \]

   13. lower--.f64 73.9%

       \[\leadsto \cos^{-1} \left(\sin z0 \cdot \sin z1 + \left(\cos z0 \cdot \cos z1\right) \cdot \cos \color{blue}{\left(z2 - z3\right)}\right) \]

3. Applied rewrites 73.9%

   \[\leadsto \cos^{-1} \left(\sin z0 \cdot \sin z1 + \color{blue}{\left(\cos z0 \cdot \cos z1\right) \cdot \cos \left(z2 - z3\right)}\right) \]
4. Add Preprocessing

Reproduce

herbie shell --seed 2025277 -o generate:taylor -o generate:evaluate
(FPCore (z0 z1 z3 z2)
  :name "(acos (+ (* (sin z0) (sin z1)) (* (* (cos (- z3 z2)) (cos z1)) (cos z0))))"
  :precision binary64
  (acos (+ (* (sin z0) (sin z1)) (* (* (cos (- z3 z2)) (cos z1)) (cos z0)))))