Diagrams.ThreeD.Transform:aboutX from diagrams-lib-1.3.0.3, A

Percentage Accurate: 99.8% → 99.8%

Time: 4.2s

Alternatives: 9

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ x \cdot \cos y - z \cdot \sin y \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- (* x (cos y)) (* z (sin y))))

C

double code(double x, double y, double z) {
	return (x * cos(y)) - (z * sin(y));
}

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(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x * cos(y)) - (z * sin(y))
end function

Java

public static double code(double x, double y, double z) {
	return (x * Math.cos(y)) - (z * Math.sin(y));
}

Python

def code(x, y, z):
	return (x * math.cos(y)) - (z * math.sin(y))

Julia

function code(x, y, z)
	return Float64(Float64(x * cos(y)) - Float64(z * sin(y)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x * cos(y)) - (z * sin(y));
end

Wolfram

code[x_, y_, z_] := N[(N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision] - N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \cos y - z \cdot \sin y \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- (* x (cos y)) (* z (sin y))))

C

double code(double x, double y, double z) {
	return (x * cos(y)) - (z * sin(y));
}

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(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x * cos(y)) - (z * sin(y))
end function

Java

public static double code(double x, double y, double z) {
	return (x * Math.cos(y)) - (z * Math.sin(y));
}

Python

def code(x, y, z):
	return (x * math.cos(y)) - (z * math.sin(y))

Julia

function code(x, y, z)
	return Float64(Float64(x * cos(y)) - Float64(z * sin(y)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x * cos(y)) - (z * sin(y));
end

Wolfram

code[x_, y_, z_] := N[(N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision] - N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\cos y, x, \left(-z\right) \cdot \sin y\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fma (cos y) x (* (- z) (sin y))))

C

double code(double x, double y, double z) {
	return fma(cos(y), x, (-z * sin(y)));
}

Julia

function code(x, y, z)
	return fma(cos(y), x, Float64(Float64(-z) * sin(y)))
end

Wolfram

code[x_, y_, z_] := N[(N[Cos[y], $MachinePrecision] * x + N[((-z) * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[x \cdot \cos y - z \cdot \sin y \]
2. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \color{blue}{x \cdot \cos y - z \cdot \sin y} \]

   2. lift-*.f64 N/A

      \[\leadsto \color{blue}{x \cdot \cos y} - z \cdot \sin y \]

   3. lift-cos.f64 N/A

      \[\leadsto x \cdot \color{blue}{\cos y} - z \cdot \sin y \]

   4. lift-*.f64 N/A

      \[\leadsto x \cdot \cos y - \color{blue}{z \cdot \sin y} \]

   5. lift-sin.f64 N/A

      \[\leadsto x \cdot \cos y - z \cdot \color{blue}{\sin y} \]

   6. fp-cancel-sub-sign-inv N/A

      \[\leadsto \color{blue}{x \cdot \cos y + \left(\mathsf{neg}\left(z\right)\right) \cdot \sin y} \]

   7. *-commutative N/A

      \[\leadsto \color{blue}{\cos y \cdot x} + \left(\mathsf{neg}\left(z\right)\right) \cdot \sin y \]

   8. distribute-lft-neg-in N/A

      \[\leadsto \cos y \cdot x + \color{blue}{\left(\mathsf{neg}\left(z \cdot \sin y\right)\right)} \]

   9. mul-1-neg N/A

      \[\leadsto \cos y \cdot x + \color{blue}{-1 \cdot \left(z \cdot \sin y\right)} \]

   10. lower-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(\cos y, x, -1 \cdot \left(z \cdot \sin y\right)\right)} \]

   11. lift-cos.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\cos y}, x, -1 \cdot \left(z \cdot \sin y\right)\right) \]

   12. mul-1-neg N/A

       \[\leadsto \mathsf{fma}\left(\cos y, x, \color{blue}{\mathsf{neg}\left(z \cdot \sin y\right)}\right) \]

   13. distribute-lft-neg-in N/A

       \[\leadsto \mathsf{fma}\left(\cos y, x, \color{blue}{\left(\mathsf{neg}\left(z\right)\right) \cdot \sin y}\right) \]

   14. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\cos y, x, \color{blue}{\left(\mathsf{neg}\left(z\right)\right) \cdot \sin y}\right) \]

   15. lower-neg.f64 N/A

       \[\leadsto \mathsf{fma}\left(\cos y, x, \color{blue}{\left(-z\right)} \cdot \sin y\right) \]

   16. lift-sin.f64 99.8

       \[\leadsto \mathsf{fma}\left(\cos y, x, \left(-z\right) \cdot \color{blue}{\sin y}\right) \]

3. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\cos y, x, \left(-z\right) \cdot \sin y\right)} \]
4. Add Preprocessing

Alternative 2: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \cos y - z \cdot \sin y \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- (* x (cos y)) (* z (sin y))))

C

double code(double x, double y, double z) {
	return (x * cos(y)) - (z * sin(y));
}

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(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x * cos(y)) - (z * sin(y))
end function

Java

public static double code(double x, double y, double z) {
	return (x * Math.cos(y)) - (z * Math.sin(y));
}

Python

def code(x, y, z):
	return (x * math.cos(y)) - (z * math.sin(y))

Julia

function code(x, y, z)
	return Float64(Float64(x * cos(y)) - Float64(z * sin(y)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x * cos(y)) - (z * sin(y));
end

Wolfram

code[x_, y_, z_] := N[(N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision] - N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[x \cdot \cos y - z \cdot \sin y \]
2. Add Preprocessing

Alternative 3: 86.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \sin y \cdot z\\ \mathbf{if}\;z \leq -1.75 \cdot 10^{-39}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{-80}:\\ \;\;\;\;\cos y \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- x (* (sin y) z))))
   (if (<= z -1.75e-39) t_0 (if (<= z 7.5e-80) (* (cos y) x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x - (sin(y) * z);
	double tmp;
	if (z <= -1.75e-39) {
		tmp = t_0;
	} else if (z <= 7.5e-80) {
		tmp = cos(y) * x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

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(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x - (sin(y) * z)
    if (z <= (-1.75d-39)) then
        tmp = t_0
    else if (z <= 7.5d-80) then
        tmp = cos(y) * x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x - (Math.sin(y) * z);
	double tmp;
	if (z <= -1.75e-39) {
		tmp = t_0;
	} else if (z <= 7.5e-80) {
		tmp = Math.cos(y) * x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x - (math.sin(y) * z)
	tmp = 0
	if z <= -1.75e-39:
		tmp = t_0
	elif z <= 7.5e-80:
		tmp = math.cos(y) * x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x - Float64(sin(y) * z))
	tmp = 0.0
	if (z <= -1.75e-39)
		tmp = t_0;
	elseif (z <= 7.5e-80)
		tmp = Float64(cos(y) * x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x - (sin(y) * z);
	tmp = 0.0;
	if (z <= -1.75e-39)
		tmp = t_0;
	elseif (z <= 7.5e-80)
		tmp = cos(y) * x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x - N[(N[Sin[y], $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.75e-39], t$95$0, If[LessEqual[z, 7.5e-80], N[(N[Cos[y], $MachinePrecision] * x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.75e-39 or 7.49999999999999999e-80 < z

   1. Initial program 99.8%

      \[x \cdot \cos y - z \cdot \sin y \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x} - z \cdot \sin y \]
   3. Step-by-step derivation

   4. Applied rewrites 85.9%

      \[\leadsto \color{blue}{x} - z \cdot \sin y \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto x - \color{blue}{z \cdot \sin y} \]

      2. lift-sin.f64 N/A

         \[\leadsto x - z \cdot \color{blue}{\sin y} \]

      3. *-commutative N/A

         \[\leadsto x - \color{blue}{\sin y \cdot z} \]

      4. lower-*.f64 N/A

         \[\leadsto x - \color{blue}{\sin y \cdot z} \]

      5. lift-sin.f64 85.9

         \[\leadsto x - \color{blue}{\sin y} \cdot z \]

   6. Applied rewrites 85.9%

      \[\leadsto x - \color{blue}{\sin y \cdot z} \]

   if -1.75e-39 < z < 7.49999999999999999e-80

   1. Initial program 99.8%

      \[x \cdot \cos y - z \cdot \sin y \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \cos y} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos y \cdot \color{blue}{x} \]

      2. lower-*.f64 N/A

         \[\leadsto \cos y \cdot \color{blue}{x} \]

      3. lift-cos.f64 87.1

         \[\leadsto \cos y \cdot x \]

   4. Applied rewrites 87.1%

      \[\leadsto \color{blue}{\cos y \cdot x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 75.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-z\right) \cdot \sin y\\ \mathbf{if}\;y \leq -0.0056:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.0055:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, z \cdot y, -0.5 \cdot x\right) \cdot y - z, y, x\right)\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{+141}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\cos y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (- z) (sin y))))
   (if (<= y -0.0056)
     t_0
     (if (<= y 0.0055)
       (fma (- (* (fma 0.16666666666666666 (* z y) (* -0.5 x)) y) z) y x)
       (if (<= y 4.5e+141) t_0 (* (cos y) x))))))

C

double code(double x, double y, double z) {
	double t_0 = -z * sin(y);
	double tmp;
	if (y <= -0.0056) {
		tmp = t_0;
	} else if (y <= 0.0055) {
		tmp = fma(((fma(0.16666666666666666, (z * y), (-0.5 * x)) * y) - z), y, x);
	} else if (y <= 4.5e+141) {
		tmp = t_0;
	} else {
		tmp = cos(y) * x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(-z) * sin(y))
	tmp = 0.0
	if (y <= -0.0056)
		tmp = t_0;
	elseif (y <= 0.0055)
		tmp = fma(Float64(Float64(fma(0.16666666666666666, Float64(z * y), Float64(-0.5 * x)) * y) - z), y, x);
	elseif (y <= 4.5e+141)
		tmp = t_0;
	else
		tmp = Float64(cos(y) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[((-z) * N[Sin[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.0056], t$95$0, If[LessEqual[y, 0.0055], N[(N[(N[(N[(0.16666666666666666 * N[(z * y), $MachinePrecision] + N[(-0.5 * x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] - z), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[y, 4.5e+141], t$95$0, N[(N[Cos[y], $MachinePrecision] * x), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.00559999999999999994 or 0.0054999999999999997 < y < 4.5000000000000002e141

   1. Initial program 99.6%

      \[x \cdot \cos y - z \cdot \sin y \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-1 \cdot \left(z \cdot \sin y\right)} \]
   3. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(z \cdot \sin y\right) \]

      2. distribute-lft-neg-in N/A

         \[\leadsto \left(\mathsf{neg}\left(z\right)\right) \cdot \color{blue}{\sin y} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(z\right)\right) \cdot \color{blue}{\sin y} \]

      4. lower-neg.f64 N/A

         \[\leadsto \left(-z\right) \cdot \sin \color{blue}{y} \]

      5. lift-sin.f64 50.4

         \[\leadsto \left(-z\right) \cdot \sin y \]

   4. Applied rewrites 50.4%

      \[\leadsto \color{blue}{\left(-z\right) \cdot \sin y} \]

   if -0.00559999999999999994 < y < 0.0054999999999999997

   1. Initial program 100.0%

      \[x \cdot \cos y - z \cdot \sin y \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + y \cdot \left(y \cdot \left(\frac{-1}{2} \cdot x + \frac{1}{6} \cdot \left(y \cdot z\right)\right) - z\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto y \cdot \left(y \cdot \left(\frac{-1}{2} \cdot x + \frac{1}{6} \cdot \left(y \cdot z\right)\right) - z\right) + \color{blue}{x} \]

      2. *-commutative N/A

         \[\leadsto \left(y \cdot \left(\frac{-1}{2} \cdot x + \frac{1}{6} \cdot \left(y \cdot z\right)\right) - z\right) \cdot y + x \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(y \cdot \left(\frac{-1}{2} \cdot x + \frac{1}{6} \cdot \left(y \cdot z\right)\right) - z, \color{blue}{y}, x\right) \]

      4. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(y \cdot \left(\frac{-1}{2} \cdot x + \frac{1}{6} \cdot \left(y \cdot z\right)\right) - z, y, x\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{-1}{2} \cdot x + \frac{1}{6} \cdot \left(y \cdot z\right)\right) \cdot y - z, y, x\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{-1}{2} \cdot x + \frac{1}{6} \cdot \left(y \cdot z\right)\right) \cdot y - z, y, x\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{6} \cdot \left(y \cdot z\right) + \frac{-1}{2} \cdot x\right) \cdot y - z, y, x\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, y \cdot z, \frac{-1}{2} \cdot x\right) \cdot y - z, y, x\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, z \cdot y, \frac{-1}{2} \cdot x\right) \cdot y - z, y, x\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, z \cdot y, \frac{-1}{2} \cdot x\right) \cdot y - z, y, x\right) \]

      11. lower-*.f64 100.0

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, z \cdot y, -0.5 \cdot x\right) \cdot y - z, y, x\right) \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, z \cdot y, -0.5 \cdot x\right) \cdot y - z, y, x\right)} \]

   if 4.5000000000000002e141 < y

   1. Initial program 99.6%

      \[x \cdot \cos y - z \cdot \sin y \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \cos y} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos y \cdot \color{blue}{x} \]

      2. lower-*.f64 N/A

         \[\leadsto \cos y \cdot \color{blue}{x} \]

      3. lift-cos.f64 50.2

         \[\leadsto \cos y \cdot x \]

   4. Applied rewrites 50.2%

      \[\leadsto \color{blue}{\cos y \cdot x} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 74.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos y \cdot x\\ \mathbf{if}\;y \leq -2.1 \cdot 10^{+21}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.17:\\ \;\;\;\;x - \mathsf{fma}\left(z \cdot \mathsf{fma}\left(y \cdot y, 0.008333333333333333, -0.16666666666666666\right), y \cdot y, z\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (cos y) x)))
   (if (<= y -2.1e+21)
     t_0
     (if (<= y 0.17)
       (-
        x
        (*
         (fma
          (* z (fma (* y y) 0.008333333333333333 -0.16666666666666666))
          (* y y)
          z)
         y))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = cos(y) * x;
	double tmp;
	if (y <= -2.1e+21) {
		tmp = t_0;
	} else if (y <= 0.17) {
		tmp = x - (fma((z * fma((y * y), 0.008333333333333333, -0.16666666666666666)), (y * y), z) * y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(cos(y) * x)
	tmp = 0.0
	if (y <= -2.1e+21)
		tmp = t_0;
	elseif (y <= 0.17)
		tmp = Float64(x - Float64(fma(Float64(z * fma(Float64(y * y), 0.008333333333333333, -0.16666666666666666)), Float64(y * y), z) * y));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Cos[y], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[y, -2.1e+21], t$95$0, If[LessEqual[y, 0.17], N[(x - N[(N[(N[(z * N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] * N[(y * y), $MachinePrecision] + z), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.1e21 or 0.170000000000000012 < y

   1. Initial program 99.6%

      \[x \cdot \cos y - z \cdot \sin y \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \cos y} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos y \cdot \color{blue}{x} \]

      2. lower-*.f64 N/A

         \[\leadsto \cos y \cdot \color{blue}{x} \]

      3. lift-cos.f64 50.3

         \[\leadsto \cos y \cdot x \]

   4. Applied rewrites 50.3%

      \[\leadsto \color{blue}{\cos y \cdot x} \]

   if -2.1e21 < y < 0.170000000000000012

   1. Initial program 100.0%

      \[x \cdot \cos y - z \cdot \sin y \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x} - z \cdot \sin y \]
   3. Step-by-step derivation

   4. Applied rewrites 98.2%

      \[\leadsto \color{blue}{x} - z \cdot \sin y \]

   5. Taylor expanded in y around 0

      \[\leadsto x - \color{blue}{y \cdot \left(z + {y}^{2} \cdot \left(\frac{-1}{6} \cdot z + \frac{1}{120} \cdot \left({y}^{2} \cdot z\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto x - \left(z + {y}^{2} \cdot \left(\frac{-1}{6} \cdot z + \frac{1}{120} \cdot \left({y}^{2} \cdot z\right)\right)\right) \cdot \color{blue}{y} \]

      2. lower-*.f64 N/A

         \[\leadsto x - \left(z + {y}^{2} \cdot \left(\frac{-1}{6} \cdot z + \frac{1}{120} \cdot \left({y}^{2} \cdot z\right)\right)\right) \cdot \color{blue}{y} \]

      3. +-commutative N/A

         \[\leadsto x - \left({y}^{2} \cdot \left(\frac{-1}{6} \cdot z + \frac{1}{120} \cdot \left({y}^{2} \cdot z\right)\right) + z\right) \cdot y \]

      4. *-commutative N/A

         \[\leadsto x - \left(\left(\frac{-1}{6} \cdot z + \frac{1}{120} \cdot \left({y}^{2} \cdot z\right)\right) \cdot {y}^{2} + z\right) \cdot y \]

      5. lower-fma.f64 N/A

         \[\leadsto x - \mathsf{fma}\left(\frac{-1}{6} \cdot z + \frac{1}{120} \cdot \left({y}^{2} \cdot z\right), {y}^{2}, z\right) \cdot y \]

      6. +-commutative N/A

         \[\leadsto x - \mathsf{fma}\left(\frac{1}{120} \cdot \left({y}^{2} \cdot z\right) + \frac{-1}{6} \cdot z, {y}^{2}, z\right) \cdot y \]

      7. associate-*r* N/A

         \[\leadsto x - \mathsf{fma}\left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot z + \frac{-1}{6} \cdot z, {y}^{2}, z\right) \cdot y \]

      8. lower-fma.f64 N/A

         \[\leadsto x - \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120} \cdot {y}^{2}, z, \frac{-1}{6} \cdot z\right), {y}^{2}, z\right) \cdot y \]

      9. lower-*.f64 N/A

         \[\leadsto x - \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120} \cdot {y}^{2}, z, \frac{-1}{6} \cdot z\right), {y}^{2}, z\right) \cdot y \]

      10. unpow2 N/A

          \[\leadsto x - \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120} \cdot \left(y \cdot y\right), z, \frac{-1}{6} \cdot z\right), {y}^{2}, z\right) \cdot y \]

      11. lower-*.f64 N/A

          \[\leadsto x - \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120} \cdot \left(y \cdot y\right), z, \frac{-1}{6} \cdot z\right), {y}^{2}, z\right) \cdot y \]

      12. lower-*.f64 N/A

          \[\leadsto x - \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120} \cdot \left(y \cdot y\right), z, \frac{-1}{6} \cdot z\right), {y}^{2}, z\right) \cdot y \]

      13. unpow2 N/A

          \[\leadsto x - \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120} \cdot \left(y \cdot y\right), z, \frac{-1}{6} \cdot z\right), y \cdot y, z\right) \cdot y \]

      14. lower-*.f64 96.9

          \[\leadsto x - \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333 \cdot \left(y \cdot y\right), z, -0.16666666666666666 \cdot z\right), y \cdot y, z\right) \cdot y \]

   7. Applied rewrites 96.9%

      \[\leadsto x - \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333 \cdot \left(y \cdot y\right), z, -0.16666666666666666 \cdot z\right), y \cdot y, z\right) \cdot y} \]
   8. Step-by-step derivation

      1. *-commutative 96.9

         \[\leadsto x - \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333 \cdot \left(y \cdot y\right), z, -0.16666666666666666 \cdot z\right), y \cdot y, z\right) \cdot y \]

      2. *-commutative N/A

         \[\leadsto x - \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120} \cdot \left(y \cdot y\right), z, \mathsf{Rewrite=>}\left(lift-*.f64, \left(\frac{-1}{6} \cdot z\right)\right)\right), y \cdot y, z\right) \cdot y \]

      3. *-commutative N/A

         \[\leadsto x - \mathsf{fma}\left(\mathsf{Rewrite=>}\left(lift-fma.f64, \left(\left(\frac{1}{120} \cdot \left(y \cdot y\right)\right) \cdot z + \frac{-1}{6} \cdot z\right)\right), y \cdot y, z\right) \cdot y \]

      4. *-commutative N/A

         \[\leadsto x - \mathsf{fma}\left(\mathsf{Rewrite=>}\left(distribute-rgt-out, \left(z \cdot \left(\frac{1}{120} \cdot \left(y \cdot y\right) + \frac{-1}{6}\right)\right)\right), y \cdot y, z\right) \cdot y \]

      5. *-commutative N/A

         \[\leadsto x - \mathsf{fma}\left(\mathsf{Rewrite=>}\left(lower-*.f64, \left(z \cdot \left(\frac{1}{120} \cdot \left(y \cdot y\right) + \frac{-1}{6}\right)\right)\right), y \cdot y, z\right) \cdot y \]

      6. *-commutative N/A

         \[\leadsto x - \mathsf{fma}\left(z \cdot \left(\frac{1}{120} \cdot \mathsf{Rewrite=>}\left(lift-*.f64, \left(y \cdot y\right)\right) + \frac{-1}{6}\right), y \cdot y, z\right) \cdot y \]

      7. *-commutative N/A

         \[\leadsto x - \mathsf{fma}\left(z \cdot \left(\mathsf{Rewrite=>}\left(lift-*.f64, \left(\frac{1}{120} \cdot \left(y \cdot y\right)\right)\right) + \frac{-1}{6}\right), y \cdot y, z\right) \cdot y \]

      8. *-commutative N/A

         \[\leadsto x - \mathsf{fma}\left(z \cdot \left(\frac{1}{120} \cdot \mathsf{Rewrite=>}\left(pow2, \left({y}^{2}\right)\right) + \frac{-1}{6}\right), y \cdot y, z\right) \cdot y \]

      9. *-commutative N/A

         \[\leadsto x - \mathsf{fma}\left(z \cdot \left(\mathsf{Rewrite=>}\left(*-commutative, \left({y}^{2} \cdot \frac{1}{120}\right)\right) + \frac{-1}{6}\right), y \cdot y, z\right) \cdot y \]

      10. *-commutative N/A

          \[\leadsto x - \mathsf{fma}\left(z \cdot \mathsf{Rewrite=>}\left(lower-fma.f64, \left(\mathsf{fma}\left({y}^{2}, \frac{1}{120}, \frac{-1}{6}\right)\right)\right), y \cdot y, z\right) \cdot y \]

      11. *-commutative N/A

          \[\leadsto x - \mathsf{fma}\left(z \cdot \mathsf{fma}\left(\mathsf{Rewrite<=}\left(pow2, \left(y \cdot y\right)\right), \frac{1}{120}, \frac{-1}{6}\right), y \cdot y, z\right) \cdot y \]

      12. *-commutative N/A

          \[\leadsto x - \mathsf{fma}\left(z \cdot \mathsf{fma}\left(\mathsf{Rewrite<=}\left(lift-*.f64, \left(y \cdot y\right)\right), \frac{1}{120}, \frac{-1}{6}\right), y \cdot y, z\right) \cdot y \]

   9. Applied rewrites 96.9%

      \[\leadsto \color{blue}{x - \mathsf{fma}\left(z \cdot \mathsf{fma}\left(y \cdot y, 0.008333333333333333, -0.16666666666666666\right), y \cdot y, z\right) \cdot y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 53.5% accurate, 10.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-z, y, x\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fma (- z) y x))

C

double code(double x, double y, double z) {
	return fma(-z, y, x);
}

Julia

function code(x, y, z)
	return fma(Float64(-z), y, x)
end

Wolfram

code[x_, y_, z_] := N[((-z) * y + x), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[x \cdot \cos y - z \cdot \sin y \]

2. Taylor expanded in y around 0

   \[\leadsto \color{blue}{x + y \cdot \left(y \cdot \left(\frac{-1}{2} \cdot x + \frac{1}{6} \cdot \left(y \cdot z\right)\right) - z\right)} \]
3. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto y \cdot \left(y \cdot \left(\frac{-1}{2} \cdot x + \frac{1}{6} \cdot \left(y \cdot z\right)\right) - z\right) + \color{blue}{x} \]

   2. *-commutative N/A

      \[\leadsto \left(y \cdot \left(\frac{-1}{2} \cdot x + \frac{1}{6} \cdot \left(y \cdot z\right)\right) - z\right) \cdot y + x \]

   3. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(y \cdot \left(\frac{-1}{2} \cdot x + \frac{1}{6} \cdot \left(y \cdot z\right)\right) - z, \color{blue}{y}, x\right) \]

   4. lower--.f64 N/A

      \[\leadsto \mathsf{fma}\left(y \cdot \left(\frac{-1}{2} \cdot x + \frac{1}{6} \cdot \left(y \cdot z\right)\right) - z, y, x\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\left(\frac{-1}{2} \cdot x + \frac{1}{6} \cdot \left(y \cdot z\right)\right) \cdot y - z, y, x\right) \]

   6. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\left(\frac{-1}{2} \cdot x + \frac{1}{6} \cdot \left(y \cdot z\right)\right) \cdot y - z, y, x\right) \]

   7. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(\left(\frac{1}{6} \cdot \left(y \cdot z\right) + \frac{-1}{2} \cdot x\right) \cdot y - z, y, x\right) \]

   8. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, y \cdot z, \frac{-1}{2} \cdot x\right) \cdot y - z, y, x\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, z \cdot y, \frac{-1}{2} \cdot x\right) \cdot y - z, y, x\right) \]

   10. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, z \cdot y, \frac{-1}{2} \cdot x\right) \cdot y - z, y, x\right) \]

   11. lower-*.f64 52.3

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, z \cdot y, -0.5 \cdot x\right) \cdot y - z, y, x\right) \]

4. Applied rewrites 52.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, z \cdot y, -0.5 \cdot x\right) \cdot y - z, y, x\right)} \]

5. Taylor expanded in y around 0

   \[\leadsto \mathsf{fma}\left(-1 \cdot z, y, x\right) \]
6. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(z\right), y, x\right) \]

   2. lift-neg.f64 53.5

      \[\leadsto \mathsf{fma}\left(-z, y, x\right) \]

7. Applied rewrites 53.5%

   \[\leadsto \mathsf{fma}\left(-z, y, x\right) \]
8. Add Preprocessing

Alternative 7: 53.5% accurate, 11.1× speedup

Math

\[\begin{array}{l} \\ x - z \cdot y \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- x (* z y)))

C

double code(double x, double y, double z) {
	return x - (z * y);
}

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(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x - (z * y)
end function

Java

public static double code(double x, double y, double z) {
	return x - (z * y);
}

Python

def code(x, y, z):
	return x - (z * y)

Julia

function code(x, y, z)
	return Float64(x - Float64(z * y))
end

MATLAB

function tmp = code(x, y, z)
	tmp = x - (z * y);
end

Wolfram

code[x_, y_, z_] := N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[x \cdot \cos y - z \cdot \sin y \]

2. Taylor expanded in y around 0

   \[\leadsto \color{blue}{x + -1 \cdot \left(y \cdot z\right)} \]
3. Step-by-step derivation

   1. fp-cancel-sign-sub-inv N/A

      \[\leadsto x - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right) \cdot \left(y \cdot z\right)} \]

   2. metadata-eval N/A

      \[\leadsto x - 1 \cdot \left(\color{blue}{y} \cdot z\right) \]

   3. *-lft-identity N/A

      \[\leadsto x - y \cdot \color{blue}{z} \]

   4. lower--.f64 N/A

      \[\leadsto x - \color{blue}{y \cdot z} \]

   5. *-commutative N/A

      \[\leadsto x - z \cdot \color{blue}{y} \]

   6. lower-*.f64 53.5

      \[\leadsto x - z \cdot \color{blue}{y} \]

4. Applied rewrites 53.5%

   \[\leadsto \color{blue}{x - z \cdot y} \]
5. Add Preprocessing

Alternative 8: 39.6% accurate, 8.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.9 \cdot 10^{-77}:\\ \;\;\;\;\left(-y\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= y -2.9e-77) (* (- y) z) x))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.9e-77) {
		tmp = -y * z;
	} else {
		tmp = x;
	}
	return tmp;
}

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(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-2.9d-77)) then
        tmp = -y * z
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.9e-77) {
		tmp = -y * z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -2.9e-77:
		tmp = -y * z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2.9e-77)
		tmp = Float64(Float64(-y) * z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -2.9e-77)
		tmp = -y * z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2.9e-77], N[((-y) * z), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if y < -2.8999999999999999e-77

   1. Initial program 99.7%

      \[x \cdot \cos y - z \cdot \sin y \]

   2. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(-1 \cdot \cos y + \frac{z \cdot \sin y}{x}\right)\right)} \]
   3. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(x \cdot \left(-1 \cdot \cos y + \frac{z \cdot \sin y}{x}\right)\right) \]

      2. distribute-lft-neg-in N/A

         \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot \color{blue}{\left(-1 \cdot \cos y + \frac{z \cdot \sin y}{x}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot \color{blue}{\left(-1 \cdot \cos y + \frac{z \cdot \sin y}{x}\right)} \]

      4. lower-neg.f64 N/A

         \[\leadsto \left(-x\right) \cdot \left(\color{blue}{-1 \cdot \cos y} + \frac{z \cdot \sin y}{x}\right) \]

      5. +-commutative N/A

         \[\leadsto \left(-x\right) \cdot \left(\frac{z \cdot \sin y}{x} + \color{blue}{-1 \cdot \cos y}\right) \]

      6. associate-/l* N/A

         \[\leadsto \left(-x\right) \cdot \left(z \cdot \frac{\sin y}{x} + \color{blue}{-1} \cdot \cos y\right) \]

      7. *-commutative N/A

         \[\leadsto \left(-x\right) \cdot \left(\frac{\sin y}{x} \cdot z + \color{blue}{-1} \cdot \cos y\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(-x\right) \cdot \mathsf{fma}\left(\frac{\sin y}{x}, \color{blue}{z}, -1 \cdot \cos y\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \left(-x\right) \cdot \mathsf{fma}\left(\frac{\sin y}{x}, z, -1 \cdot \cos y\right) \]

      10. lift-sin.f64 N/A

          \[\leadsto \left(-x\right) \cdot \mathsf{fma}\left(\frac{\sin y}{x}, z, -1 \cdot \cos y\right) \]

      11. mul-1-neg N/A

          \[\leadsto \left(-x\right) \cdot \mathsf{fma}\left(\frac{\sin y}{x}, z, \mathsf{neg}\left(\cos y\right)\right) \]

      12. lower-neg.f64 N/A

          \[\leadsto \left(-x\right) \cdot \mathsf{fma}\left(\frac{\sin y}{x}, z, -\cos y\right) \]

      13. lift-cos.f64 88.3

          \[\leadsto \left(-x\right) \cdot \mathsf{fma}\left(\frac{\sin y}{x}, z, -\cos y\right) \]

   4. Applied rewrites 88.3%

      \[\leadsto \color{blue}{\left(-x\right) \cdot \mathsf{fma}\left(\frac{\sin y}{x}, z, -\cos y\right)} \]

   5. Taylor expanded in z around inf

      \[\leadsto \left(-x\right) \cdot \left(z \cdot \color{blue}{\left(-1 \cdot \frac{\cos y}{z} + \frac{\sin y}{x}\right)}\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(-x\right) \cdot \left(\left(-1 \cdot \frac{\cos y}{z} + \frac{\sin y}{x}\right) \cdot z\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(-x\right) \cdot \left(\left(-1 \cdot \frac{\cos y}{z} + \frac{\sin y}{x}\right) \cdot z\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \left(-x\right) \cdot \left(\left(-1 \cdot \frac{\cos y}{z} + \frac{\sin y}{x}\right) \cdot z\right) \]

      4. associate-*r/ N/A

         \[\leadsto \left(-x\right) \cdot \left(\left(\frac{-1 \cdot \cos y}{z} + \frac{\sin y}{x}\right) \cdot z\right) \]

      5. mul-1-neg N/A

         \[\leadsto \left(-x\right) \cdot \left(\left(\frac{\mathsf{neg}\left(\cos y\right)}{z} + \frac{\sin y}{x}\right) \cdot z\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \left(-x\right) \cdot \left(\left(\frac{\mathsf{neg}\left(\cos y\right)}{z} + \frac{\sin y}{x}\right) \cdot z\right) \]

      7. lift-neg.f64 N/A

         \[\leadsto \left(-x\right) \cdot \left(\left(\frac{-\cos y}{z} + \frac{\sin y}{x}\right) \cdot z\right) \]

      8. lift-cos.f64 N/A

         \[\leadsto \left(-x\right) \cdot \left(\left(\frac{-\cos y}{z} + \frac{\sin y}{x}\right) \cdot z\right) \]

      9. lift-sin.f64 N/A

         \[\leadsto \left(-x\right) \cdot \left(\left(\frac{-\cos y}{z} + \frac{\sin y}{x}\right) \cdot z\right) \]

      10. lift-/.f64 88.2

          \[\leadsto \left(-x\right) \cdot \left(\left(\frac{-\cos y}{z} + \frac{\sin y}{x}\right) \cdot z\right) \]

   7. Applied rewrites 88.2%

      \[\leadsto \left(-x\right) \cdot \left(\left(\frac{-\cos y}{z} + \frac{\sin y}{x}\right) \cdot \color{blue}{z}\right) \]

   8. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + -1 \cdot \left(y \cdot z\right)} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto -1 \cdot \left(y \cdot z\right) + \color{blue}{x} \]

      2. associate-*r* N/A

         \[\leadsto \left(-1 \cdot y\right) \cdot z + x \]

      3. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(y\right)\right) \cdot z + x \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(y\right), \color{blue}{z}, x\right) \]

      5. lower-neg.f64 24.3

         \[\leadsto \mathsf{fma}\left(-y, z, x\right) \]

   10. Applied rewrites 24.3%

       \[\leadsto \color{blue}{\mathsf{fma}\left(-y, z, x\right)} \]

   11. Taylor expanded in x around 0

       \[\leadsto -1 \cdot \color{blue}{\left(y \cdot z\right)} \]
   12. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(y \cdot z\right) \]

       2. distribute-lft-neg-out N/A

          \[\leadsto \left(\mathsf{neg}\left(y\right)\right) \cdot z \]

       3. lift-neg.f64 N/A

          \[\leadsto \left(-y\right) \cdot z \]

       4. lower-*.f64 12.2

          \[\leadsto \left(-y\right) \cdot z \]

   13. Applied rewrites 12.2%

       \[\leadsto \left(-y\right) \cdot \color{blue}{z} \]

   if -2.8999999999999999e-77 < y

   1. Initial program 99.9%

      \[x \cdot \cos y - z \cdot \sin y \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x} \]
   3. Step-by-step derivation

   4. Applied rewrites 49.6%

      \[\leadsto \color{blue}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 38.4% accurate, 73.2× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 x)

C

double code(double x, double y, double z) {
	return x;
}

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(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x
end function

Java

public static double code(double x, double y, double z) {
	return x;
}

Python

def code(x, y, z):
	return x

Julia

function code(x, y, z)
	return x
end

MATLAB

function tmp = code(x, y, z)
	tmp = x;
end

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 99.8%

   \[x \cdot \cos y - z \cdot \sin y \]

2. Taylor expanded in y around 0

   \[\leadsto \color{blue}{x} \]
3. Step-by-step derivation

4. Applied rewrites 39.6%

   \[\leadsto \color{blue}{x} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025112 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Transform:aboutX from diagrams-lib-1.3.0.3, A"
  :precision binary64
  (- (* x (cos y)) (* z (sin y))))