Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTicks from plot-0.2.3.4, A

Percentage Accurate: 85.2% → 96.8%

Time: 9.1s

Alternatives: 10

Speedup: 0.7×

Specification

Math

\[\begin{array}{l} \\ x + \frac{y \cdot \left(z - t\right)}{z - a} \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (+ x (/ (* y (- z t)) (- z a))))

C

double code(double x, double y, double z, double t, double a) {
	return x + ((y * (z - t)) / (z - a));
}

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

Java

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

Python

def code(x, y, z, t, a):
	return x + ((y * (z - t)) / (z - a))

Julia

function code(x, y, z, t, a)
	return Float64(x + Float64(Float64(y * Float64(z - t)) / Float64(z - a)))
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = x + ((y * (z - t)) / (z - a));
end

Wolfram

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

Initial Program: 85.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x + \frac{y \cdot \left(z - t\right)}{z - a} \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (+ x (/ (* y (- z t)) (- z a))))

C

double code(double x, double y, double z, double t, double a) {
	return x + ((y * (z - t)) / (z - a));
}

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

Java

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

Python

def code(x, y, z, t, a):
	return x + ((y * (z - t)) / (z - a))

Julia

function code(x, y, z, t, a)
	return Float64(x + Float64(Float64(y * Float64(z - t)) / Float64(z - a)))
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = x + ((y * (z - t)) / (z - a));
end

Wolfram

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

Alternative 1: 96.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{z - a} \cdot \left(z - t\right)\\ t_2 := x + \frac{y \cdot \left(z - t\right)}{z - a}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+307}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (/ y (- z a)) (- z t))) (t_2 (+ x (/ (* y (- z t)) (- z a)))))
   (if (<= t_2 (- INFINITY)) t_1 (if (<= t_2 2e+307) t_2 t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y / (z - a)) * (z - t);
	double t_2 = x + ((y * (z - t)) / (z - a));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_2 <= 2e+307) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (y / (z - a)) * (z - t);
	double t_2 = x + ((y * (z - t)) / (z - a));
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else if (t_2 <= 2e+307) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (y / (z - a)) * (z - t)
	t_2 = x + ((y * (z - t)) / (z - a))
	tmp = 0
	if t_2 <= -math.inf:
		tmp = t_1
	elif t_2 <= 2e+307:
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y / Float64(z - a)) * Float64(z - t))
	t_2 = Float64(x + Float64(Float64(y * Float64(z - t)) / Float64(z - a)))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_2 <= 2e+307)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (y / (z - a)) * (z - t);
	t_2 = x + ((y * (z - t)) / (z - a));
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = t_1;
	elseif (t_2 <= 2e+307)
		tmp = t_2;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y / N[(z - a), $MachinePrecision]), $MachinePrecision] * N[(z - t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x + N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$1, If[LessEqual[t$95$2, 2e+307], t$95$2, t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.f64 (*.f64 y (-.f64 z t)) (-.f64 z a))) < -inf.0 or 1.99999999999999997e307 < (+.f64 x (/.f64 (*.f64 y (-.f64 z t)) (-.f64 z a)))

   1. Initial program 39.3%

      \[x + \frac{y \cdot \left(z - t\right)}{z - a} \]

   2. Taylor expanded in x around 0

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

      1. distribute-rgt-out-- N/A

         \[\leadsto \frac{z \cdot y - t \cdot y}{\color{blue}{z} - a} \]

      2. *-commutative N/A

         \[\leadsto \frac{y \cdot z - t \cdot y}{z - a} \]

      3. *-lft-identity N/A

         \[\leadsto \frac{y \cdot z - \left(1 \cdot t\right) \cdot y}{z - a} \]

      4. metadata-eval N/A

         \[\leadsto \frac{y \cdot z - \left(\left(\mathsf{neg}\left(-1\right)\right) \cdot t\right) \cdot y}{z - a} \]

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

         \[\leadsto \frac{y \cdot z - \left(\mathsf{neg}\left(-1 \cdot t\right)\right) \cdot y}{z - a} \]

      6. mul-1-neg N/A

         \[\leadsto \frac{y \cdot z - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(t\right)\right)\right)\right) \cdot y}{z - a} \]

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

         \[\leadsto \frac{y \cdot z + \left(\mathsf{neg}\left(t\right)\right) \cdot y}{\color{blue}{z} - a} \]

      8. mul-1-neg N/A

         \[\leadsto \frac{y \cdot z + \left(-1 \cdot t\right) \cdot y}{z - a} \]

      9. associate-*r* N/A

         \[\leadsto \frac{y \cdot z + -1 \cdot \left(t \cdot y\right)}{z - a} \]

      10. +-commutative N/A

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

      11. div-add-rev N/A

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

      12. associate-*r* N/A

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

      13. mul-1-neg N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(t\right)\right) \cdot y}{z - a} + \frac{y \cdot z}{z - a} \]

      14. associate-/l* N/A

          \[\leadsto \left(\mathsf{neg}\left(t\right)\right) \cdot \frac{y}{z - a} + \frac{\color{blue}{y \cdot z}}{z - a} \]

      15. *-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(t\right)\right) \cdot \frac{y}{z - a} + \frac{z \cdot y}{\color{blue}{z} - a} \]

      16. associate-/l* N/A

          \[\leadsto \left(\mathsf{neg}\left(t\right)\right) \cdot \frac{y}{z - a} + z \cdot \color{blue}{\frac{y}{z - a}} \]

      17. distribute-rgt-out N/A

          \[\leadsto \frac{y}{z - a} \cdot \color{blue}{\left(\left(\mathsf{neg}\left(t\right)\right) + z\right)} \]

      18. +-commutative N/A

          \[\leadsto \frac{y}{z - a} \cdot \left(z + \color{blue}{\left(\mathsf{neg}\left(t\right)\right)}\right) \]

      19. mul-1-neg N/A

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

      20. *-commutative N/A

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

   4. Applied rewrites 87.8%

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

   if -inf.0 < (+.f64 x (/.f64 (*.f64 y (-.f64 z t)) (-.f64 z a))) < 1.99999999999999997e307

   1. Initial program 99.6%

      \[x + \frac{y \cdot \left(z - t\right)}{z - a} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 86.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{z}{z - a}, y, x\right)\\ \mathbf{if}\;z \leq -2.05 \cdot 10^{+58}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.72 \cdot 10^{-23}:\\ \;\;\;\;x + \frac{y \cdot \left(-t\right)}{z - a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ z (- z a)) y x)))
   (if (<= z -2.05e+58)
     t_1
     (if (<= z 1.72e-23) (+ x (/ (* y (- t)) (- z a))) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((z / (z - a)), y, x);
	double tmp;
	if (z <= -2.05e+58) {
		tmp = t_1;
	} else if (z <= 1.72e-23) {
		tmp = x + ((y * -t) / (z - a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(z / Float64(z - a)), y, x)
	tmp = 0.0
	if (z <= -2.05e+58)
		tmp = t_1;
	elseif (z <= 1.72e-23)
		tmp = Float64(x + Float64(Float64(y * Float64(-t)) / Float64(z - a)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z / N[(z - a), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision]}, If[LessEqual[z, -2.05e+58], t$95$1, If[LessEqual[z, 1.72e-23], N[(x + N[(N[(y * (-t)), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.05e58 or 1.7200000000000001e-23 < z

   1. Initial program 73.7%

      \[x + \frac{y \cdot \left(z - t\right)}{z - a} \]

   2. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

         \[\leadsto y \cdot \frac{z}{z - a} + x \]

      3. *-commutative N/A

         \[\leadsto \frac{z}{z - a} \cdot y + x \]

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift--.f64 86.6

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

   4. Applied rewrites 86.6%

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

   if -2.05e58 < z < 1.7200000000000001e-23

   1. Initial program 95.4%

      \[x + \frac{y \cdot \left(z - t\right)}{z - a} \]

   2. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 86.7

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

   4. Applied rewrites 86.7%

      \[\leadsto x + \frac{y \cdot \color{blue}{\left(-t\right)}}{z - a} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 82.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{z - t}{z}, y, x\right)\\ \mathbf{if}\;z \leq -9.8 \cdot 10^{+26}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{-123}:\\ \;\;\;\;x + \left(-y\right) \cdot \frac{z - t}{a}\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{+89}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y}{z - a}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- z t) z) y x)))
   (if (<= z -9.8e+26)
     t_1
     (if (<= z 2.6e-123)
       (+ x (* (- y) (/ (- z t) a)))
       (if (<= z 3.5e+89) t_1 (fma z (/ y (- z a)) x))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(((z - t) / z), y, x);
	double tmp;
	if (z <= -9.8e+26) {
		tmp = t_1;
	} else if (z <= 2.6e-123) {
		tmp = x + (-y * ((z - t) / a));
	} else if (z <= 3.5e+89) {
		tmp = t_1;
	} else {
		tmp = fma(z, (y / (z - a)), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(z - t) / z), y, x)
	tmp = 0.0
	if (z <= -9.8e+26)
		tmp = t_1;
	elseif (z <= 2.6e-123)
		tmp = Float64(x + Float64(Float64(-y) * Float64(Float64(z - t) / a)));
	elseif (z <= 3.5e+89)
		tmp = t_1;
	else
		tmp = fma(z, Float64(y / Float64(z - a)), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(z - t), $MachinePrecision] / z), $MachinePrecision] * y + x), $MachinePrecision]}, If[LessEqual[z, -9.8e+26], t$95$1, If[LessEqual[z, 2.6e-123], N[(x + N[((-y) * N[(N[(z - t), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3.5e+89], t$95$1, N[(z * N[(y / N[(z - a), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -9.79999999999999947e26 or 2.59999999999999995e-123 < z < 3.5000000000000001e89

   1. Initial program 82.7%

      \[x + \frac{y \cdot \left(z - t\right)}{z - a} \]

   2. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

         \[\leadsto y \cdot \frac{z - t}{z} + x \]

      3. *-commutative N/A

         \[\leadsto \frac{z - t}{z} \cdot y + x \]

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift--.f64 80.2

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

   4. Applied rewrites 80.2%

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

   if -9.79999999999999947e26 < z < 2.59999999999999995e-123

   1. Initial program 95.3%

      \[x + \frac{y \cdot \left(z - t\right)}{z - a} \]

   2. Taylor expanded in a around inf

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift--.f64 82.4

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

   4. Applied rewrites 82.4%

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

   if 3.5000000000000001e89 < z

   1. Initial program 82.7%

      \[x + \frac{y \cdot \left(z - t\right)}{z - a} \]

   2. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

         \[\leadsto y \cdot \frac{z}{z - a} + x \]

      3. *-commutative N/A

         \[\leadsto \frac{z}{z - a} \cdot y + x \]

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift--.f64 80.2

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

   4. Applied rewrites 80.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z}{z - a}, y, x\right)} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-*l/ N/A

         \[\leadsto \frac{z \cdot y}{z - a} + x \]

      5. associate-/l* N/A

         \[\leadsto z \cdot \frac{y}{z - a} + x \]

      6. lower-fma.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift--.f64 78.4

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

   6. Applied rewrites 78.4%

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

Alternative 4: 81.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -9.8 \cdot 10^{+26}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{z}, y, x\right)\\ \mathbf{elif}\;z \leq 2.06 \cdot 10^{-137}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{z - a}, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -9.8e+26)
   (fma (/ (- z t) z) y x)
   (if (<= z 2.06e-137) (fma y (/ t a) x) (fma (/ z (- z a)) y x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -9.8e+26) {
		tmp = fma(((z - t) / z), y, x);
	} else if (z <= 2.06e-137) {
		tmp = fma(y, (t / a), x);
	} else {
		tmp = fma((z / (z - a)), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -9.8e+26)
		tmp = fma(Float64(Float64(z - t) / z), y, x);
	elseif (z <= 2.06e-137)
		tmp = fma(y, Float64(t / a), x);
	else
		tmp = fma(Float64(z / Float64(z - a)), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -9.8e+26], N[(N[(N[(z - t), $MachinePrecision] / z), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[z, 2.06e-137], N[(y * N[(t / a), $MachinePrecision] + x), $MachinePrecision], N[(N[(z / N[(z - a), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -9.79999999999999947e26

   1. Initial program 74.4%

      \[x + \frac{y \cdot \left(z - t\right)}{z - a} \]

   2. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

         \[\leadsto y \cdot \frac{z - t}{z} + x \]

      3. *-commutative N/A

         \[\leadsto \frac{z - t}{z} \cdot y + x \]

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift--.f64 87.6

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

   4. Applied rewrites 87.6%

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

   if -9.79999999999999947e26 < z < 2.06000000000000003e-137

   1. Initial program 95.3%

      \[x + \frac{y \cdot \left(z - t\right)}{z - a} \]

   2. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \frac{t \cdot y}{a}} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{t \cdot y}{a} + \color{blue}{x} \]

      2. associate-/l* N/A

         \[\leadsto t \cdot \frac{y}{a} + x \]

      3. *-commutative N/A

         \[\leadsto \frac{y}{a} \cdot t + x \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{y}{a}, \color{blue}{t}, x\right) \]

      5. lower-/.f64 80.0

         \[\leadsto \mathsf{fma}\left(\frac{y}{a}, t, x\right) \]

   4. Applied rewrites 80.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{a}, t, x\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{y}{a}, t, x\right) \]

      2. lift-fma.f64 N/A

         \[\leadsto \frac{y}{a} \cdot t + \color{blue}{x} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{y \cdot t}{a} + x \]

      4. associate-/l* N/A

         \[\leadsto y \cdot \frac{t}{a} + x \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{t}{a}}, x\right) \]

      6. lower-/.f64 79.7

         \[\leadsto \mathsf{fma}\left(y, \frac{t}{\color{blue}{a}}, x\right) \]

   6. Applied rewrites 79.7%

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{t}{a}}, x\right) \]

   if 2.06000000000000003e-137 < z

   1. Initial program 80.6%

      \[x + \frac{y \cdot \left(z - t\right)}{z - a} \]

   2. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

         \[\leadsto y \cdot \frac{z}{z - a} + x \]

      3. *-commutative N/A

         \[\leadsto \frac{z}{z - a} \cdot y + x \]

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift--.f64 79.7

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

   4. Applied rewrites 79.7%

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

Alternative 5: 81.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{z}{z - a}, y, x\right)\\ \mathbf{if}\;z \leq -5.2 \cdot 10^{+39}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.06 \cdot 10^{-137}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ z (- z a)) y x)))
   (if (<= z -5.2e+39) t_1 (if (<= z 2.06e-137) (fma y (/ t a) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((z / (z - a)), y, x);
	double tmp;
	if (z <= -5.2e+39) {
		tmp = t_1;
	} else if (z <= 2.06e-137) {
		tmp = fma(y, (t / a), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(z / Float64(z - a)), y, x)
	tmp = 0.0
	if (z <= -5.2e+39)
		tmp = t_1;
	elseif (z <= 2.06e-137)
		tmp = fma(y, Float64(t / a), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z / N[(z - a), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision]}, If[LessEqual[z, -5.2e+39], t$95$1, If[LessEqual[z, 2.06e-137], N[(y * N[(t / a), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -5.2e39 or 2.06000000000000003e-137 < z

   1. Initial program 77.8%

      \[x + \frac{y \cdot \left(z - t\right)}{z - a} \]

   2. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

         \[\leadsto y \cdot \frac{z}{z - a} + x \]

      3. *-commutative N/A

         \[\leadsto \frac{z}{z - a} \cdot y + x \]

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift--.f64 82.7

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

   4. Applied rewrites 82.7%

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

   if -5.2e39 < z < 2.06000000000000003e-137

   1. Initial program 95.4%

      \[x + \frac{y \cdot \left(z - t\right)}{z - a} \]

   2. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \frac{t \cdot y}{a}} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{t \cdot y}{a} + \color{blue}{x} \]

      2. associate-/l* N/A

         \[\leadsto t \cdot \frac{y}{a} + x \]

      3. *-commutative N/A

         \[\leadsto \frac{y}{a} \cdot t + x \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{y}{a}, \color{blue}{t}, x\right) \]

      5. lower-/.f64 79.4

         \[\leadsto \mathsf{fma}\left(\frac{y}{a}, t, x\right) \]

   4. Applied rewrites 79.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{a}, t, x\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{y}{a}, t, x\right) \]

      2. lift-fma.f64 N/A

         \[\leadsto \frac{y}{a} \cdot t + \color{blue}{x} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{y \cdot t}{a} + x \]

      4. associate-/l* N/A

         \[\leadsto y \cdot \frac{t}{a} + x \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{t}{a}}, x\right) \]

      6. lower-/.f64 79.1

         \[\leadsto \mathsf{fma}\left(y, \frac{t}{\color{blue}{a}}, x\right) \]

   6. Applied rewrites 79.1%

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{t}{a}}, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 77.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{+108}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 2.06 \cdot 10^{-137}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y}{z - a}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.5e+108)
   (+ x y)
   (if (<= z 2.06e-137) (fma y (/ t a) x) (fma z (/ y (- z a)) x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.5e+108) {
		tmp = x + y;
	} else if (z <= 2.06e-137) {
		tmp = fma(y, (t / a), x);
	} else {
		tmp = fma(z, (y / (z - a)), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.5e+108)
		tmp = Float64(x + y);
	elseif (z <= 2.06e-137)
		tmp = fma(y, Float64(t / a), x);
	else
		tmp = fma(z, Float64(y / Float64(z - a)), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.5e+108], N[(x + y), $MachinePrecision], If[LessEqual[z, 2.06e-137], N[(y * N[(t / a), $MachinePrecision] + x), $MachinePrecision], N[(z * N[(y / N[(z - a), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.49999999999999992e108

   1. Initial program 68.0%

      \[x + \frac{y \cdot \left(z - t\right)}{z - a} \]

   2. Taylor expanded in z around inf

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

   4. Applied rewrites 83.9%

      \[\leadsto x + \color{blue}{y} \]

   if -1.49999999999999992e108 < z < 2.06000000000000003e-137

   1. Initial program 94.3%

      \[x + \frac{y \cdot \left(z - t\right)}{z - a} \]

   2. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \frac{t \cdot y}{a}} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{t \cdot y}{a} + \color{blue}{x} \]

      2. associate-/l* N/A

         \[\leadsto t \cdot \frac{y}{a} + x \]

      3. *-commutative N/A

         \[\leadsto \frac{y}{a} \cdot t + x \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{y}{a}, \color{blue}{t}, x\right) \]

      5. lower-/.f64 76.0

         \[\leadsto \mathsf{fma}\left(\frac{y}{a}, t, x\right) \]

   4. Applied rewrites 76.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{a}, t, x\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{y}{a}, t, x\right) \]

      2. lift-fma.f64 N/A

         \[\leadsto \frac{y}{a} \cdot t + \color{blue}{x} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{y \cdot t}{a} + x \]

      4. associate-/l* N/A

         \[\leadsto y \cdot \frac{t}{a} + x \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{t}{a}}, x\right) \]

      6. lower-/.f64 75.7

         \[\leadsto \mathsf{fma}\left(y, \frac{t}{\color{blue}{a}}, x\right) \]

   6. Applied rewrites 75.7%

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{t}{a}}, x\right) \]

   if 2.06000000000000003e-137 < z

   1. Initial program 80.6%

      \[x + \frac{y \cdot \left(z - t\right)}{z - a} \]

   2. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

         \[\leadsto y \cdot \frac{z}{z - a} + x \]

      3. *-commutative N/A

         \[\leadsto \frac{z}{z - a} \cdot y + x \]

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift--.f64 79.7

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

   4. Applied rewrites 79.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z}{z - a}, y, x\right)} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-*l/ N/A

         \[\leadsto \frac{z \cdot y}{z - a} + x \]

      5. associate-/l* N/A

         \[\leadsto z \cdot \frac{y}{z - a} + x \]

      6. lower-fma.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift--.f64 78.1

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

   6. Applied rewrites 78.1%

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

Alternative 7: 76.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{+108}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 10^{-34}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.5e+108) (+ x y) (if (<= z 1e-34) (fma y (/ t a) x) (+ x y))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.5e+108) {
		tmp = x + y;
	} else if (z <= 1e-34) {
		tmp = fma(y, (t / a), x);
	} else {
		tmp = x + y;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.5e+108)
		tmp = Float64(x + y);
	elseif (z <= 1e-34)
		tmp = fma(y, Float64(t / a), x);
	else
		tmp = Float64(x + y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.5e+108], N[(x + y), $MachinePrecision], If[LessEqual[z, 1e-34], N[(y * N[(t / a), $MachinePrecision] + x), $MachinePrecision], N[(x + y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.49999999999999992e108 or 9.99999999999999928e-35 < z

   1. Initial program 72.9%

      \[x + \frac{y \cdot \left(z - t\right)}{z - a} \]

   2. Taylor expanded in z around inf

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

   4. Applied rewrites 77.9%

      \[\leadsto x + \color{blue}{y} \]

   if -1.49999999999999992e108 < z < 9.99999999999999928e-35

   1. Initial program 94.6%

      \[x + \frac{y \cdot \left(z - t\right)}{z - a} \]

   2. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \frac{t \cdot y}{a}} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{t \cdot y}{a} + \color{blue}{x} \]

      2. associate-/l* N/A

         \[\leadsto t \cdot \frac{y}{a} + x \]

      3. *-commutative N/A

         \[\leadsto \frac{y}{a} \cdot t + x \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{y}{a}, \color{blue}{t}, x\right) \]

      5. lower-/.f64 75.0

         \[\leadsto \mathsf{fma}\left(\frac{y}{a}, t, x\right) \]

   4. Applied rewrites 75.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{a}, t, x\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{y}{a}, t, x\right) \]

      2. lift-fma.f64 N/A

         \[\leadsto \frac{y}{a} \cdot t + \color{blue}{x} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{y \cdot t}{a} + x \]

      4. associate-/l* N/A

         \[\leadsto y \cdot \frac{t}{a} + x \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{t}{a}}, x\right) \]

      6. lower-/.f64 74.8

         \[\leadsto \mathsf{fma}\left(y, \frac{t}{\color{blue}{a}}, x\right) \]

   6. Applied rewrites 74.8%

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{t}{a}}, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 63.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.2 \cdot 10^{+148}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 3.6 \cdot 10^{+186}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -2.2e+148) x (if (<= a 3.6e+186) (+ x y) x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -2.2e+148) {
		tmp = x;
	} else if (a <= 3.6e+186) {
		tmp = x + y;
	} 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, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (a <= (-2.2d+148)) then
        tmp = x
    else if (a <= 3.6d+186) then
        tmp = x + y
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -2.2e+148) {
		tmp = x;
	} else if (a <= 3.6e+186) {
		tmp = x + y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if a <= -2.2e+148:
		tmp = x
	elif a <= 3.6e+186:
		tmp = x + y
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -2.2e+148)
		tmp = x;
	elseif (a <= 3.6e+186)
		tmp = Float64(x + y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (a <= -2.2e+148)
		tmp = x;
	elseif (a <= 3.6e+186)
		tmp = x + y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -2.2e+148], x, If[LessEqual[a, 3.6e+186], N[(x + y), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if a < -2.1999999999999999e148 or 3.6000000000000002e186 < a

   1. Initial program 80.5%

      \[x + \frac{y \cdot \left(z - t\right)}{z - a} \]

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 68.1%

      \[\leadsto \color{blue}{x} \]

   if -2.1999999999999999e148 < a < 3.6000000000000002e186

   1. Initial program 86.6%

      \[x + \frac{y \cdot \left(z - t\right)}{z - a} \]

   2. Taylor expanded in z around inf

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

   4. Applied rewrites 62.3%

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

Alternative 9: 55.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y \cdot \left(z - t\right)}{z - a}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+188}:\\ \;\;\;\;y\\ \mathbf{elif}\;t\_1 \leq 10^{+265}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (* y (- z t)) (- z a))))
   (if (<= t_1 -2e+188) y (if (<= t_1 1e+265) x y))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y * (z - t)) / (z - a);
	double tmp;
	if (t_1 <= -2e+188) {
		tmp = y;
	} else if (t_1 <= 1e+265) {
		tmp = x;
	} else {
		tmp = y;
	}
	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, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (y * (z - t)) / (z - a)
    if (t_1 <= (-2d+188)) then
        tmp = y
    else if (t_1 <= 1d+265) then
        tmp = x
    else
        tmp = y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (y * (z - t)) / (z - a);
	double tmp;
	if (t_1 <= -2e+188) {
		tmp = y;
	} else if (t_1 <= 1e+265) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (y * (z - t)) / (z - a)
	tmp = 0
	if t_1 <= -2e+188:
		tmp = y
	elif t_1 <= 1e+265:
		tmp = x
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y * Float64(z - t)) / Float64(z - a))
	tmp = 0.0
	if (t_1 <= -2e+188)
		tmp = y;
	elseif (t_1 <= 1e+265)
		tmp = x;
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (y * (z - t)) / (z - a);
	tmp = 0.0;
	if (t_1 <= -2e+188)
		tmp = y;
	elseif (t_1 <= 1e+265)
		tmp = x;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+188], y, If[LessEqual[t$95$1, 1e+265], x, y]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 y (-.f64 z t)) (-.f64 z a)) < -2e188 or 1.00000000000000007e265 < (/.f64 (*.f64 y (-.f64 z t)) (-.f64 z a))

   1. Initial program 47.6%

      \[x + \frac{y \cdot \left(z - t\right)}{z - a} \]

   2. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

         \[\leadsto y \cdot \frac{z}{z - a} + x \]

      3. *-commutative N/A

         \[\leadsto \frac{z}{z - a} \cdot y + x \]

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift--.f64 47.3

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

   4. Applied rewrites 47.3%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{y \cdot z}{\color{blue}{z - a}} \]
   6. Step-by-step derivation

      1. associate-/l* N/A

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

      2. *-commutative N/A

         \[\leadsto \frac{z}{z - a} \cdot y \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{z}{z - a} \cdot y \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{z}{z - a} \cdot y \]

      5. lift--.f64 35.6

         \[\leadsto \frac{z}{z - a} \cdot y \]

   7. Applied rewrites 35.6%

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

   8. Taylor expanded in z around inf

      \[\leadsto y \]
   9. Step-by-step derivation

   10. Applied rewrites 30.8%

       \[\leadsto y \]

   if -2e188 < (/.f64 (*.f64 y (-.f64 z t)) (-.f64 z a)) < 1.00000000000000007e265

   1. Initial program 99.6%

      \[x + \frac{y \cdot \left(z - t\right)}{z - a} \]

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 65.0%

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

Alternative 10: 50.7% accurate, 15.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_] := x

Derivation

1. Initial program 85.2%

   \[x + \frac{y \cdot \left(z - t\right)}{z - a} \]

2. Taylor expanded in x around inf

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

4. Applied rewrites 50.7%

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

Reproduce

herbie shell --seed 2025130 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTicks from plot-0.2.3.4, A"
  :precision binary64
  (+ x (/ (* y (- z t)) (- z a))))